id	sid	tid	token	lemma	pos
ejpam-4753	1	1	european	european	PROPN
ejpam-4753	1	2	journal	journal	PROPN
ejpam-4753	1	3	of	of	ADP
ejpam-4753	1	4	pure	pure	ADJ
ejpam-4753	1	5	and	and	CCONJ
ejpam-4753	1	6	applied	apply	VERB
ejpam-4753	1	7	mathematics	mathematic	NOUN
ejpam-4753	1	8	vol	vol	NOUN
ejpam-4753	1	9	.	.	PUNCT
ejpam-4753	2	1	16	16	NUM
ejpam-4753	2	2	,	,	PUNCT
ejpam-4753	2	3	no	no	INTJ
ejpam-4753	2	4	.	.	NOUN
ejpam-4753	2	5	3	3	NUM
ejpam-4753	2	6	,	,	PUNCT
ejpam-4753	2	7	2023	2023	NUM
ejpam-4753	2	8	,	,	PUNCT
ejpam-4753	2	9	1913	1913	NUM
ejpam-4753	2	10	-	-	SYM
ejpam-4753	2	11	1939	1939	NUM
ejpam-4753	2	12	issn	issn	PROPN
ejpam-4753	2	13	1307	1307	NUM
ejpam-4753	2	14	-	-	SYM
ejpam-4753	2	15	5543	5543	NUM
ejpam-4753	2	16	–	–	PUNCT
ejpam-4753	2	17	ejpam.com	ejpam.com	X
ejpam-4753	2	18	published	publish	VERB
ejpam-4753	2	19	by	by	ADP
ejpam-4753	2	20	new	new	PROPN
ejpam-4753	2	21	york	york	PROPN
ejpam-4753	2	22	business	business	PROPN
ejpam-4753	2	23	global	global	ADJ
ejpam-4753	2	24	localization	localization	NOUN
ejpam-4753	2	25	in	in	ADP
ejpam-4753	2	26	the	the	DET
ejpam-4753	2	27	category	category	NOUN
ejpam-4753	2	28	comp	comp	NOUN
ejpam-4753	2	29	(	(	PUNCT
ejpam-4753	2	30	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	2	31	)	)	PUNCT
ejpam-4753	2	32	)	)	PUNCT
ejpam-4753	2	33	of	of	ADP
ejpam-4753	2	34	complex	complex	ADJ
ejpam-4753	2	35	associated	associate	VERB
ejpam-4753	2	36	to	to	ADP
ejpam-4753	2	37	the	the	DET
ejpam-4753	2	38	category	category	NOUN
ejpam-4753	2	39	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	2	40	)	)	PUNCT
ejpam-4753	2	41	of	of	ADP
ejpam-4753	2	42	graded	grade	VERB
ejpam-4753	2	43	left	leave	VERB
ejpam-4753	2	44	a−modules	a−module	NOUN
ejpam-4753	2	45	over	over	ADP
ejpam-4753	2	46	a	a	DET
ejpam-4753	2	47	graded	grade	VERB
ejpam-4753	2	48	ring	ring	NOUN
ejpam-4753	2	49	ahmed	ahmed	PROPN
ejpam-4753	2	50	ould	ould	PROPN
ejpam-4753	2	51	chbih1,∗	chbih1,∗	PROPN
ejpam-4753	2	52	,	,	PUNCT
ejpam-4753	2	53	mohamed	mohamed	PROPN
ejpam-4753	2	54	ben	ben	PROPN
ejpam-4753	2	55	faraj	faraj	PROPN
ejpam-4753	2	56	ben	ben	PROPN
ejpam-4753	2	57	maaouia2	maaouia2	PROPN
ejpam-4753	2	58	,	,	PUNCT
ejpam-4753	2	59	mamadou	mamadou	X
ejpam-4753	2	60	sanghare3	sanghare3	NOUN
ejpam-4753	2	61	1	1	NUM
ejpam-4753	2	62	unité	unité	NOUN
ejpam-4753	2	63	de	de	X
ejpam-4753	2	64	recherche	recherche	X
ejpam-4753	2	65	géométrie	géométrie	PROPN
ejpam-4753	2	66	,	,	PUNCT
ejpam-4753	2	67	analyse	analyse	PROPN
ejpam-4753	2	68	,	,	PUNCT
ejpam-4753	2	69	algèbre	algèbre	PROPN
ejpam-4753	2	70	et	et	NOUN
ejpam-4753	2	71	applications	application	NOUN
ejpam-4753	2	72	(	(	PUNCT
ejpam-4753	2	73	g3a	g3a	NOUN
ejpam-4753	2	74	)	)	PUNCT
ejpam-4753	2	75	,	,	PUNCT
ejpam-4753	2	76	faculté	faculté	PROPN
ejpam-4753	2	77	des	des	PROPN
ejpam-4753	2	78	sciences	sciences	PROPN
ejpam-4753	2	79	et	et	PROPN
ejpam-4753	2	80	techniques	techniques	PROPN
ejpam-4753	2	81	/	/	SYM
ejpam-4753	2	82	universitéé	universitéé	PROPN
ejpam-4753	2	83	de	de	PROPN
ejpam-4753	2	84	nouakchott	nouakchott	PROPN
ejpam-4753	2	85	,	,	PUNCT
ejpam-4753	2	86	nouakchott	nouakchott	PROPN
ejpam-4753	2	87	,	,	PUNCT
ejpam-4753	2	88	mauritanie	mauritanie	X
ejpam-4753	2	89	2	2	NUM
ejpam-4753	2	90	applied	apply	VERB
ejpam-4753	2	91	mathematics	mathematic	NOUN
ejpam-4753	2	92	,	,	PUNCT
ejpam-4753	2	93	ufr	ufr	NOUN
ejpam-4753	2	94	-	-	PUNCT
ejpam-4753	2	95	sat	sit	VERB
ejpam-4753	2	96	/	/	SYM
ejpam-4753	2	97	gaston	gaston	PROPN
ejpam-4753	2	98	berger	berger	PROPN
ejpam-4753	2	99	,	,	PUNCT
ejpam-4753	2	100	university	university	NOUN
ejpam-4753	2	101	,	,	PUNCT
ejpam-4753	2	102	saint	saint	NOUN
ejpam-4753	2	103	-	-	PUNCT
ejpam-4753	2	104	louis	louis	NOUN
ejpam-4753	2	105	,	,	PUNCT
ejpam-4753	2	106	senegal	senegal	ADJ
ejpam-4753	2	107	3	3	NUM
ejpam-4753	2	108	université	université	NOUN
ejpam-4753	2	109	cheikh	cheikh	PROPN
ejpam-4753	2	110	anta	anta	PROPN
ejpam-4753	2	111	diop	diop	PROPN
ejpam-4753	2	112	,	,	PUNCT
ejpam-4753	2	113	dakar	dakar	NOUN
ejpam-4753	2	114	(	(	PUNCT
ejpam-4753	2	115	ucad	ucad	ADJ
ejpam-4753	2	116	)	)	PUNCT
ejpam-4753	2	117	,	,	PUNCT
ejpam-4753	2	118	sénégal	sénégal	ADJ
ejpam-4753	2	119	abstract	abstract	NOUN
ejpam-4753	2	120	.	.	PUNCT
ejpam-4753	3	1	the	the	DET
ejpam-4753	3	2	main	main	ADJ
ejpam-4753	3	3	results	result	NOUN
ejpam-4753	3	4	of	of	ADP
ejpam-4753	3	5	this	this	DET
ejpam-4753	3	6	paper	paper	NOUN
ejpam-4753	3	7	are	be	AUX
ejpam-4753	3	8	:	:	PUNCT
ejpam-4753	3	9	if	if	SCONJ
ejpam-4753	3	10	a	a	DET
ejpam-4753	3	11	=	=	PUNCT
ejpam-4753	3	12	⊕	⊕	PROPN
ejpam-4753	3	13	n∈z	n∈z	VERB
ejpam-4753	3	14	an	an	PRON
ejpam-4753	3	15	is	be	AUX
ejpam-4753	3	16	a	a	DET
ejpam-4753	3	17	graded	grade	VERB
ejpam-4753	3	18	duo	duo	NOUN
ejpam-4753	3	19	-	-	PUNCT
ejpam-4753	3	20	ring	ring	NOUN
ejpam-4753	3	21	,	,	PUNCT
ejpam-4753	3	22	sh	sh	PROPN
ejpam-4753	3	23	is	be	AUX
ejpam-4753	3	24	a	a	DET
ejpam-4753	3	25	part	part	NOUN
ejpam-4753	3	26	formed	form	VERB
ejpam-4753	3	27	of	of	ADP
ejpam-4753	3	28	regulars	regular	NOUN
ejpam-4753	3	29	homogeneous	homogeneous	ADJ
ejpam-4753	3	30	elements	element	NOUN
ejpam-4753	3	31	of	of	ADP
ejpam-4753	3	32	a	a	PRON
ejpam-4753	3	33	,	,	PUNCT
ejpam-4753	3	34	sh	sh	PROPN
ejpam-4753	3	35	is	be	AUX
ejpam-4753	3	36	the	the	DET
ejpam-4753	3	37	homogeneous	homogeneous	ADJ
ejpam-4753	3	38	multiplicatively	multiplicatively	ADV
ejpam-4753	3	39	closed	close	VERB
ejpam-4753	3	40	subset	subset	NOUN
ejpam-4753	3	41	of	of	ADP
ejpam-4753	3	42	a	a	DET
ejpam-4753	3	43	generated	generate	VERB
ejpam-4753	3	44	by	by	ADP
ejpam-4753	3	45	sh	sh	INTJ
ejpam-4753	3	46	,	,	PUNCT
ejpam-4753	3	47	then	then	ADV
ejpam-4753	3	48	:	:	PUNCT
ejpam-4753	3	49	(	(	PUNCT
ejpam-4753	3	50	i	i	NOUN
ejpam-4753	3	51	)	)	PUNCT
ejpam-4753	3	52	the	the	DET
ejpam-4753	3	53	relation	relation	NOUN
ejpam-4753	3	54	ch(−	ch(−	PUNCT
ejpam-4753	3	55	)	)	PUNCT
ejpam-4753	3	56	:	:	PUNCT
ejpam-4753	3	57	gr(s	gr(s	X
ejpam-4753	3	58	−1	−1	NOUN
ejpam-4753	3	59	h	h	NOUN
ejpam-4753	3	60	a	a	DET
ejpam-4753	3	61	−	−	PROPN
ejpam-4753	3	62	mod	mod	ADJ
ejpam-4753	3	63	)	)	PUNCT
ejpam-4753	3	64	−→	−→	NOUN
ejpam-4753	3	65	comp	comp	NOUN
ejpam-4753	3	66	(	(	PUNCT
ejpam-4753	3	67	gr(s	gr(s	NOUN
ejpam-4753	3	68	−1	−1	NOUN
ejpam-4753	3	69	h	h	NOUN
ejpam-4753	3	70	a	a	DET
ejpam-4753	3	71	−	−	PROPN
ejpam-4753	3	72	mod	mod	NOUN
ejpam-4753	3	73	)	)	PUNCT
ejpam-4753	3	74	)	)	PUNCT
ejpam-4753	3	75	which	which	PRON
ejpam-4753	3	76	that	that	SCONJ
ejpam-4753	3	77	for	for	ADP
ejpam-4753	3	78	all	all	PRON
ejpam-4753	3	79	graded	grade	VERB
ejpam-4753	3	80	left	leave	VERB
ejpam-4753	3	81	s	s	PRON
ejpam-4753	3	82	−1	−1	NOUN
ejpam-4753	3	83	h	h	NOUN
ejpam-4753	3	84	a−module	a−module	ADP
ejpam-4753	3	85	s	s	NUM
ejpam-4753	3	86	−1	−1	NOUN
ejpam-4753	3	87	h	h	NOUN
ejpam-4753	3	88	m	m	VERB
ejpam-4753	3	89	of	of	ADP
ejpam-4753	3	90	gr(s	gr(s	PUNCT
ejpam-4753	3	91	−1	−1	NOUN
ejpam-4753	3	92	h	h	PROPN
ejpam-4753	3	93	a−mod	a−mod	NOUN
ejpam-4753	3	94	)	)	PUNCT
ejpam-4753	4	1	we	we	PRON
ejpam-4753	4	2	correspond	correspond	VERB
ejpam-4753	4	3	the	the	DET
ejpam-4753	4	4	associate	associate	ADJ
ejpam-4753	4	5	complex	complex	ADJ
ejpam-4753	4	6	sequence	sequence	NOUN
ejpam-4753	4	7	(	(	PUNCT
ejpam-4753	4	8	s	s	NOUN
ejpam-4753	4	9	−1	−1	NOUN
ejpam-4753	4	10	h	h	NOUN
ejpam-4753	4	11	m)∗	m)∗	VERB
ejpam-4753	4	12	to	to	ADP
ejpam-4753	4	13	a	a	DET
ejpam-4753	4	14	graded	grade	VERB
ejpam-4753	4	15	s	s	PRON
ejpam-4753	4	16	−1	−1	NOUN
ejpam-4753	4	17	h	h	NOUN
ejpam-4753	4	18	a−module	a−module	ADP
ejpam-4753	4	19	s	s	X
ejpam-4753	4	20	−1	−1	NOUN
ejpam-4753	4	21	h	h	NOUN
ejpam-4753	4	22	m	m	PROPN
ejpam-4753	4	23	and	and	CCONJ
ejpam-4753	4	24	for	for	ADP
ejpam-4753	4	25	all	all	DET
ejpam-4753	4	26	graded	grade	VERB
ejpam-4753	4	27	morphism	morphism	NOUN
ejpam-4753	4	28	of	of	ADP
ejpam-4753	4	29	graded	grade	VERB
ejpam-4753	4	30	left	leave	VERB
ejpam-4753	4	31	s	s	PRON
ejpam-4753	4	32	−1	−1	NOUN
ejpam-4753	4	33	h	h	NOUN
ejpam-4753	5	1	a−modules	a−module	NOUN
ejpam-4753	5	2	s	s	PART
ejpam-4753	5	3	−1	−1	NOUN
ejpam-4753	5	4	h	h	NOUN
ejpam-4753	5	5	f	f	NOUN
ejpam-4753	5	6	:	:	PUNCT
ejpam-4753	5	7	s	s	VERB
ejpam-4753	5	8	−1	−1	NOUN
ejpam-4753	5	9	h	h	NOUN
ejpam-4753	5	10	m	m	VERB
ejpam-4753	5	11	−→	−→	NOUN
ejpam-4753	5	12	s	s	PART
ejpam-4753	5	13	−1	−1	NOUN
ejpam-4753	5	14	h	h	NOUN
ejpam-4753	5	15	n	n	PROPN
ejpam-4753	5	16	of	of	ADP
ejpam-4753	5	17	degree	degree	NOUN
ejpam-4753	6	1	k	k	NOUN
ejpam-4753	6	2	we	we	PRON
ejpam-4753	6	3	correspond	correspond	VERB
ejpam-4753	6	4	the	the	DET
ejpam-4753	6	5	associate	associate	ADJ
ejpam-4753	6	6	complex	complex	NOUN
ejpam-4753	6	7	chain	chain	NOUN
ejpam-4753	6	8	(	(	PUNCT
ejpam-4753	6	9	s	s	NOUN
ejpam-4753	6	10	−1	−1	NOUN
ejpam-4753	6	11	h	h	NOUN
ejpam-4753	6	12	f)k∗	f)k∗	ADJ
ejpam-4753	6	13	to	to	ADP
ejpam-4753	6	14	a	a	DET
ejpam-4753	6	15	morphism	morphism	NOUN
ejpam-4753	6	16	of	of	ADP
ejpam-4753	6	17	graded	grade	VERB
ejpam-4753	6	18	left	leave	VERB
ejpam-4753	6	19	s	s	PRON
ejpam-4753	6	20	−1	−1	NOUN
ejpam-4753	6	21	h	h	NOUN
ejpam-4753	6	22	a−module	a−module	ADP
ejpam-4753	6	23	s	s	NUM
ejpam-4753	6	24	−1	−1	NOUN
ejpam-4753	6	25	h	h	NOUN
ejpam-4753	6	26	f	f	NOUN
ejpam-4753	6	27	:	:	PUNCT
ejpam-4753	6	28	s	s	VERB
ejpam-4753	6	29	−1	−1	NOUN
ejpam-4753	6	30	h	h	NOUN
ejpam-4753	6	31	m	m	VERB
ejpam-4753	6	32	−→	−→	NOUN
ejpam-4753	6	33	s	s	PART
ejpam-4753	6	34	−1	−1	NOUN
ejpam-4753	6	35	h	h	NOUN
ejpam-4753	6	36	n	n	ADV
ejpam-4753	6	37	is	be	AUX
ejpam-4753	6	38	additively	additively	ADV
ejpam-4753	6	39	exact	exact	ADJ
ejpam-4753	6	40	covariant	covariant	PROPN
ejpam-4753	6	41	functor	functor	PROPN
ejpam-4753	6	42	.	.	PUNCT
ejpam-4753	7	1	(	(	PUNCT
ejpam-4753	7	2	ii	ii	PROPN
ejpam-4753	7	3	)	)	PUNCT
ejpam-4753	7	4	the	the	DET
ejpam-4753	7	5	relation	relation	NOUN
ejpam-4753	7	6	(	(	PUNCT
ejpam-4753	7	7	ch	ch	NOUN
ejpam-4753	7	8	◦	◦	NOUN
ejpam-4753	7	9	s−1	s−1	PROPN
ejpam-4753	7	10	h	h	NOUN
ejpam-4753	7	11	)	)	PUNCT
ejpam-4753	7	12	(	(	PUNCT
ejpam-4753	7	13	−	−	NOUN
ejpam-4753	7	14	)	)	PUNCT
ejpam-4753	7	15	:	:	PUNCT
ejpam-4753	7	16	gr(a	gr(a	X
ejpam-4753	7	17	−mod	−mod	ADJ
ejpam-4753	7	18	)	)	PUNCT
ejpam-4753	7	19	−→	−→	ADJ
ejpam-4753	7	20	comp	comp	NOUN
ejpam-4753	7	21	(	(	PUNCT
ejpam-4753	7	22	gr(s	gr(s	NOUN
ejpam-4753	7	23	−1	−1	NOUN
ejpam-4753	7	24	h	h	NOUN
ejpam-4753	7	25	a	a	DET
ejpam-4753	7	26	−mod	−mod	NOUN
ejpam-4753	7	27	)	)	PUNCT
ejpam-4753	7	28	)	)	PUNCT
ejpam-4753	7	29	which	which	PRON
ejpam-4753	7	30	that	that	SCONJ
ejpam-4753	7	31	for	for	ADP
ejpam-4753	7	32	all	all	PRON
ejpam-4753	7	33	graded	grade	VERB
ejpam-4753	7	34	left	leave	VERB
ejpam-4753	7	35	a−module	a−module	ADP
ejpam-4753	7	36	m	m	NOUN
ejpam-4753	7	37	of	of	ADP
ejpam-4753	7	38	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	7	39	)	)	PUNCT
ejpam-4753	7	40	we	we	PRON
ejpam-4753	7	41	correspond	correspond	VERB
ejpam-4753	7	42	the	the	DET
ejpam-4753	7	43	associate	associate	ADJ
ejpam-4753	7	44	complex	complex	ADJ
ejpam-4753	7	45	sequence	sequence	NOUN
ejpam-4753	7	46	(	(	PUNCT
ejpam-4753	7	47	ch	ch	NOUN
ejpam-4753	7	48	◦	◦	NOUN
ejpam-4753	7	49	s	s	PART
ejpam-4753	7	50	−1	−1	NOUN
ejpam-4753	7	51	h	h	NOUN
ejpam-4753	7	52	)	)	PUNCT
ejpam-4753	7	53	(	(	PUNCT
ejpam-4753	7	54	m	m	NOUN
ejpam-4753	7	55	)	)	PUNCT
ejpam-4753	7	56	=	=	SYM
ejpam-4753	8	1	(	(	PUNCT
ejpam-4753	8	2	s	s	NOUN
ejpam-4753	8	3	−1	−1	NOUN
ejpam-4753	8	4	h	h	NOUN
ejpam-4753	8	5	m)∗	m)∗	VERB
ejpam-4753	8	6	to	to	ADP
ejpam-4753	8	7	a	a	DET
ejpam-4753	8	8	graded	grade	VERB
ejpam-4753	8	9	a−module	a−module	ADP
ejpam-4753	8	10	m	m	NOUN
ejpam-4753	8	11	and	and	CCONJ
ejpam-4753	8	12	for	for	ADP
ejpam-4753	8	13	all	all	DET
ejpam-4753	8	14	graded	grade	VERB
ejpam-4753	8	15	morphism	morphism	NOUN
ejpam-4753	8	16	of	of	ADP
ejpam-4753	8	17	graded	grade	VERB
ejpam-4753	8	18	left	leave	VERB
ejpam-4753	8	19	a−modules	a−module	NOUN
ejpam-4753	8	20	f	f	X
ejpam-4753	8	21	:	:	PUNCT
ejpam-4753	8	22	m	m	VERB
ejpam-4753	8	23	−→	−→	ADJ
ejpam-4753	8	24	n	n	PRON
ejpam-4753	8	25	of	of	ADP
ejpam-4753	8	26	degree	degree	NOUN
ejpam-4753	9	1	k	k	NOUN
ejpam-4753	9	2	we	we	PRON
ejpam-4753	9	3	correspond	correspond	VERB
ejpam-4753	9	4	the	the	DET
ejpam-4753	9	5	associate	associate	ADJ
ejpam-4753	9	6	complex	complex	NOUN
ejpam-4753	9	7	chain	chain	NOUN
ejpam-4753	9	8	(	(	PUNCT
ejpam-4753	9	9	ch	ch	NOUN
ejpam-4753	9	10	◦	◦	NOUN
ejpam-4753	9	11	s−1	s−1	PROPN
ejpam-4753	9	12	h	h	NOUN
ejpam-4753	9	13	)	)	PUNCT
ejpam-4753	9	14	(	(	PUNCT
ejpam-4753	9	15	f	f	X
ejpam-4753	9	16	)	)	PUNCT
ejpam-4753	9	17	=	=	SYM
ejpam-4753	10	1	(	(	PUNCT
ejpam-4753	10	2	s	s	AUX
ejpam-4753	10	3	−1	−1	NOUN
ejpam-4753	10	4	h	h	NOUN
ejpam-4753	10	5	f)k∗	f)k∗	ADJ
ejpam-4753	10	6	to	to	ADP
ejpam-4753	10	7	a	a	DET
ejpam-4753	10	8	morphism	morphism	NOUN
ejpam-4753	10	9	of	of	ADP
ejpam-4753	10	10	graded	grade	VERB
ejpam-4753	10	11	left	leave	VERB
ejpam-4753	10	12	a−module	a−module	ADP
ejpam-4753	10	13	f	f	X
ejpam-4753	10	14	:	:	PUNCT
ejpam-4753	10	15	m	m	VERB
ejpam-4753	10	16	−→	−→	ADJ
ejpam-4753	11	1	n	n	NOUN
ejpam-4753	11	2	is	be	AUX
ejpam-4753	11	3	additively	additively	ADV
ejpam-4753	11	4	exact	exact	ADJ
ejpam-4753	11	5	covariant	covariant	PROPN
ejpam-4753	11	6	functor	functor	PROPN
ejpam-4753	11	7	.	.	PUNCT
ejpam-4753	12	1	(	(	PUNCT
ejpam-4753	12	2	iii	iii	NOUN
ejpam-4753	12	3	)	)	PUNCT
ejpam-4753	12	4	for	for	ADP
ejpam-4753	12	5	all	all	DET
ejpam-4753	12	6	n	n	PRON
ejpam-4753	12	7	∈	∈	PROPN
ejpam-4753	12	8	z	z	NOUN
ejpam-4753	12	9	fixed	fix	VERB
ejpam-4753	12	10	and	and	CCONJ
ejpam-4753	12	11	for	for	ADP
ejpam-4753	12	12	all	all	DET
ejpam-4753	12	13	m	m	NOUN
ejpam-4753	12	14	∈	∈	NOUN
ejpam-4753	12	15	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	12	16	)	)	PUNCT
ejpam-4753	12	17	we	we	PRON
ejpam-4753	12	18	have	have	VERB
ejpam-4753	12	19	:	:	PUNCT
ejpam-4753	12	20	s	s	VERB
ejpam-4753	13	1	−1	−1	NOUN
ejpam-4753	13	2	h	h	NOUN
ejpam-4753	13	3	(	(	PUNCT
ejpam-4753	13	4	(	(	PUNCT
ejpam-4753	13	5	hn	hn	PROPN
ejpam-4753	13	6	◦	◦	NOUN
ejpam-4753	13	7	c)(m	c)(m	VERB
ejpam-4753	13	8	)	)	PUNCT
ejpam-4753	13	9	)	)	PUNCT
ejpam-4753	14	1	∼=	∼=	PROPN
ejpam-4753	14	2	hn(ch	hn(ch	NOUN
ejpam-4753	14	3	◦	◦	VERB
ejpam-4753	14	4	s−1	s−1	PROPN
ejpam-4753	14	5	h	h	NOUN
ejpam-4753	14	6	)	)	PUNCT
ejpam-4753	14	7	(	(	PUNCT
ejpam-4753	14	8	m	m	NOUN
ejpam-4753	14	9	)	)	PUNCT
ejpam-4753	14	10	)	)	PUNCT
ejpam-4753	14	11	.	.	PUNCT
ejpam-4753	15	1	2020	2020	NUM
ejpam-4753	15	2	mathematics	mathematic	NOUN
ejpam-4753	15	3	subject	subject	NOUN
ejpam-4753	15	4	classifications	classification	NOUN
ejpam-4753	15	5	:	:	PUNCT
ejpam-4753	15	6	13a02	13a02	NUM
ejpam-4753	15	7	,	,	PUNCT
ejpam-4753	15	8	16w50	16w50	NUM
ejpam-4753	15	9	,	,	PUNCT
ejpam-4753	15	10	18c40	18c40	NUM
ejpam-4753	15	11	,	,	PUNCT
ejpam-4753	15	12	18g35	18g35	NUM
ejpam-4753	15	13	,	,	PUNCT
ejpam-4753	15	14	13d45	13d45	NOUN
ejpam-4753	15	15	key	key	ADJ
ejpam-4753	15	16	words	word	NOUN
ejpam-4753	15	17	and	and	CCONJ
ejpam-4753	15	18	phrases	phrase	NOUN
ejpam-4753	15	19	:	:	PUNCT
ejpam-4753	15	20	duo	duo	NOUN
ejpam-4753	15	21	-	-	PUNCT
ejpam-4753	15	22	ring	ring	NOUN
ejpam-4753	15	23	,	,	PUNCT
ejpam-4753	15	24	graded	grade	VERB
ejpam-4753	15	25	ring	ring	NOUN
ejpam-4753	15	26	,	,	PUNCT
ejpam-4753	15	27	graded	grade	VERB
ejpam-4753	15	28	module	module	NOUN
ejpam-4753	15	29	,	,	PUNCT
ejpam-4753	15	30	multiplicatively	multiplicatively	ADV
ejpam-4753	15	31	closed	close	VERB
ejpam-4753	15	32	subset	subset	NOUN
ejpam-4753	15	33	of	of	ADP
ejpam-4753	15	34	duo	duo	NOUN
ejpam-4753	15	35	-	-	PUNCT
ejpam-4753	15	36	ring	ring	NOUN
ejpam-4753	15	37	generated	generate	VERB
ejpam-4753	15	38	by	by	ADP
ejpam-4753	15	39	regular	regular	ADJ
ejpam-4753	15	40	homogeneous	homogeneous	ADJ
ejpam-4753	15	41	elements	element	NOUN
ejpam-4753	15	42	,	,	PUNCT
ejpam-4753	15	43	category	category	NOUN
ejpam-4753	15	44	,	,	PUNCT
ejpam-4753	15	45	sequence	sequence	NOUN
ejpam-4753	15	46	complex	complex	ADJ
ejpam-4753	15	47	,	,	PUNCT
ejpam-4753	15	48	complex	complex	ADJ
ejpam-4753	15	49	chain	chain	NOUN
ejpam-4753	15	50	and	and	CCONJ
ejpam-4753	15	51	homology	homology	NOUN
ejpam-4753	15	52	functor	functor	PROPN
ejpam-4753	15	53	∗corresponding	∗corresponde	VERB
ejpam-4753	15	54	author	author	NOUN
ejpam-4753	15	55	.	.	PUNCT
ejpam-4753	16	1	doi	doi	NOUN
ejpam-4753	16	2	:	:	PUNCT
ejpam-4753	16	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4753	https://doi.org/10.29020/nybg.ejpam.v16i3.4753	PRON
ejpam-4753	16	4	email	email	NOUN
ejpam-4753	16	5	addresses	address	VERB
ejpam-4753	16	6	:	:	PUNCT
ejpam-4753	16	7	achbih@gmail.com	achbih@gmail.com	X
ejpam-4753	16	8	(	(	PUNCT
ejpam-4753	16	9	a.	a.	PROPN
ejpam-4753	16	10	o.	o.	PROPN
ejpam-4753	16	11	chbih	chbih	PROPN
ejpam-4753	16	12	)	)	PUNCT
ejpam-4753	16	13	,	,	PUNCT
ejpam-4753	16	14	mohamed-ben.maaouia@ugb.edu.sn	mohamed-ben.maaouia@ugb.edu.sn	PROPN
ejpam-4753	16	15	(	(	PUNCT
ejpam-4753	16	16	m.	m.	PROPN
ejpam-4753	16	17	b.	b.	PROPN
ejpam-4753	16	18	maaouia	maaouia	PROPN
ejpam-4753	16	19	)	)	PUNCT
ejpam-4753	16	20	,	,	PUNCT
ejpam-4753	16	21	mamadou.sanghare@ucad.edu.sn	mamadou.sanghare@ucad.edu.sn	PROPN
ejpam-4753	16	22	(	(	PUNCT
ejpam-4753	16	23	m.	m.	NOUN
ejpam-4753	16	24	sanghare	sanghare	PROPN
ejpam-4753	16	25	)	)	PUNCT
ejpam-4753	16	26	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4753	16	27	1913	1913	NUM
ejpam-4753	17	1	©	©	PROPN
ejpam-4753	17	2	2023	2023	NUM
ejpam-4753	17	3	ejpam	ejpam	NOUN
ejpam-4753	17	4	all	all	DET
ejpam-4753	17	5	rights	right	NOUN
ejpam-4753	17	6	reserved	reserve	VERB
ejpam-4753	17	7	.	.	PUNCT
ejpam-4753	18	1	a.	a.	PROPN
ejpam-4753	18	2	o.	o.	PROPN
ejpam-4753	18	3	chbih	chbih	PROPN
ejpam-4753	18	4	,	,	PUNCT
ejpam-4753	18	5	m.	m.	PROPN
ejpam-4753	18	6	b.	b.	PROPN
ejpam-4753	18	7	maaouia	maaouia	PROPN
ejpam-4753	18	8	,	,	PUNCT
ejpam-4753	18	9	m.	m.	NOUN
ejpam-4753	18	10	sanghare	sanghare	PROPN
ejpam-4753	18	11	/	/	SYM
ejpam-4753	18	12	eur	eur	PROPN
ejpam-4753	18	13	.	.	PUNCT
ejpam-4753	19	1	j.	j.	PROPN
ejpam-4753	19	2	pure	pure	PROPN
ejpam-4753	19	3	appl	appl	PROPN
ejpam-4753	19	4	.	.	PROPN
ejpam-4753	19	5	math	math	PROPN
ejpam-4753	19	6	,	,	PUNCT
ejpam-4753	19	7	16	16	NUM
ejpam-4753	19	8	(	(	PUNCT
ejpam-4753	19	9	3	3	NUM
ejpam-4753	19	10	)	)	PUNCT
ejpam-4753	19	11	(	(	PUNCT
ejpam-4753	19	12	2023	2023	NUM
ejpam-4753	19	13	)	)	PUNCT
ejpam-4753	19	14	,	,	PUNCT
ejpam-4753	19	15	1913	1913	NUM
ejpam-4753	19	16	-	-	SYM
ejpam-4753	19	17	1939	1939	NUM
ejpam-4753	19	18	1914	1914	NUM
ejpam-4753	19	19	1	1	NUM
ejpam-4753	19	20	.	.	PUNCT
ejpam-4753	20	1	introduction	introduction	NOUN
ejpam-4753	20	2	in	in	ADP
ejpam-4753	20	3	this	this	DET
ejpam-4753	20	4	article	article	NOUN
ejpam-4753	20	5	a	a	PRON
ejpam-4753	20	6	is	be	AUX
ejpam-4753	20	7	supposed	suppose	VERB
ejpam-4753	20	8	unitary	unitary	ADJ
ejpam-4753	20	9	graded	grade	VERB
ejpam-4753	20	10	ring	ring	NOUN
ejpam-4753	20	11	and	and	CCONJ
ejpam-4753	20	12	all	all	PRON
ejpam-4753	20	13	left	leave	VERB
ejpam-4753	20	14	a−module	a−module	ADP
ejpam-4753	20	15	is	be	AUX
ejpam-4753	20	16	a	a	DET
ejpam-4753	20	17	unitary	unitary	ADJ
ejpam-4753	20	18	.	.	PUNCT
ejpam-4753	21	1	in	in	ADP
ejpam-4753	21	2	this	this	DET
ejpam-4753	21	3	article	article	NOUN
ejpam-4753	21	4	,	,	PUNCT
ejpam-4753	21	5	we	we	PRON
ejpam-4753	21	6	study	study	VERB
ejpam-4753	21	7	the	the	DET
ejpam-4753	21	8	localization	localization	NOUN
ejpam-4753	21	9	in	in	ADP
ejpam-4753	21	10	the	the	DET
ejpam-4753	21	11	category	category	NOUN
ejpam-4753	21	12	comp	comp	NOUN
ejpam-4753	21	13	(	(	PUNCT
ejpam-4753	21	14	gr(a	gr(a	NOUN
ejpam-4753	21	15	−mod	−mod	NOUN
ejpam-4753	21	16	)	)	PUNCT
ejpam-4753	21	17	)	)	PUNCT
ejpam-4753	21	18	of	of	ADP
ejpam-4753	21	19	complexes	complex	NOUN
ejpam-4753	21	20	of	of	ADP
ejpam-4753	21	21	graded	grade	VERB
ejpam-4753	21	22	left	leave	VERB
ejpam-4753	21	23	a−modules	a−module	NOUN
ejpam-4753	21	24	so	so	ADV
ejpam-4753	21	25	for	for	ADP
ejpam-4753	21	26	this	this	DET
ejpam-4753	21	27	gaol	gaol	NOUN
ejpam-4753	21	28	we	we	PRON
ejpam-4753	21	29	used	use	VERB
ejpam-4753	21	30	the	the	DET
ejpam-4753	21	31	localization	localization	NOUN
ejpam-4753	21	32	in	in	ADP
ejpam-4753	21	33	the	the	DET
ejpam-4753	21	34	category	category	NOUN
ejpam-4753	21	35	gr(a	gr(a	PUNCT
ejpam-4753	21	36	−	−	PROPN
ejpam-4753	21	37	mod	mod	PROPN
ejpam-4753	21	38	)	)	PUNCT
ejpam-4753	21	39	,	,	PUNCT
ejpam-4753	21	40	of	of	ADP
ejpam-4753	21	41	graded	grade	VERB
ejpam-4753	21	42	left	leave	VERB
ejpam-4753	21	43	a−modules	a−module	NOUN
ejpam-4753	21	44	,	,	PUNCT
ejpam-4753	21	45	the	the	DET
ejpam-4753	21	46	functor	functor	PROPN
ejpam-4753	21	47	s−1	s−1	PROPN
ejpam-4753	21	48	h	h	NOUN
ejpam-4753	21	49	:	:	PUNCT
ejpam-4753	21	50	gr(a	gr(a	NOUN
ejpam-4753	21	51	−	−	PROPN
ejpam-4753	21	52	mod	mod	ADJ
ejpam-4753	21	53	)	)	PUNCT
ejpam-4753	21	54	−→	−→	NOUN
ejpam-4753	21	55	gr(s	gr(s	PUNCT
ejpam-4753	21	56	−1	−1	NOUN
ejpam-4753	21	57	h	h	NOUN
ejpam-4753	21	58	a	a	DET
ejpam-4753	21	59	−	−	PROPN
ejpam-4753	21	60	mod	mod	NOUN
ejpam-4753	21	61	)	)	PUNCT
ejpam-4753	21	62	with	with	ADP
ejpam-4753	21	63	sh	sh	PROPN
ejpam-4753	21	64	is	be	AUX
ejpam-4753	21	65	a	a	DET
ejpam-4753	21	66	multiplicatively	multiplicatively	ADV
ejpam-4753	21	67	closed	close	VERB
ejpam-4753	21	68	subset	subset	NOUN
ejpam-4753	21	69	satisfying	satisfy	VERB
ejpam-4753	21	70	the	the	DET
ejpam-4753	21	71	left	left	ADJ
ejpam-4753	21	72	conditions	condition	NOUN
ejpam-4753	21	73	of	of	ADP
ejpam-4753	21	74	ore	ore	NOUN
ejpam-4753	21	75	formed	form	VERB
ejpam-4753	21	76	of	of	ADP
ejpam-4753	21	77	homogeneous	homogeneous	ADJ
ejpam-4753	21	78	elements	element	NOUN
ejpam-4753	21	79	of	of	ADP
ejpam-4753	21	80	a	a	DET
ejpam-4753	21	81	graded	grade	VERB
ejpam-4753	21	82	ring	ring	NOUN
ejpam-4753	21	83	a	a	PRON
ejpam-4753	21	84	and	and	CCONJ
ejpam-4753	21	85	the	the	DET
ejpam-4753	21	86	functor	functor	PROPN
ejpam-4753	22	1	hn	hn	PROPN
ejpam-4753	22	2	:	:	PUNCT
ejpam-4753	22	3	comp	comp	NOUN
ejpam-4753	22	4	(	(	PUNCT
ejpam-4753	22	5	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	22	6	)	)	PUNCT
ejpam-4753	22	7	)	)	PUNCT
ejpam-4753	22	8	−→	−→	NOUN
ejpam-4753	22	9	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	22	10	)	)	PUNCT
ejpam-4753	22	11	.	.	PUNCT
ejpam-4753	23	1	this	this	DET
ejpam-4753	23	2	work	work	NOUN
ejpam-4753	23	3	finds	find	VERB
ejpam-4753	23	4	its	its	PRON
ejpam-4753	23	5	roots	root	NOUN
ejpam-4753	23	6	in	in	ADP
ejpam-4753	23	7	particular	particular	ADJ
ejpam-4753	23	8	as	as	SCONJ
ejpam-4753	23	9	regards	regard	VERB
ejpam-4753	23	10	the	the	DET
ejpam-4753	23	11	functor	functor	NOUN
ejpam-4753	23	12	s−1	s−1	PROPN
ejpam-4753	23	13	:	:	PUNCT
ejpam-4753	23	14	a−mod	a−mod	NOUN
ejpam-4753	23	15	−→	−→	PROPN
ejpam-4753	23	16	s−1a−	s−1a−	PROPN
ejpam-4753	23	17	mod	mod	NOUN
ejpam-4753	23	18	in	in	ADP
ejpam-4753	23	19	[	[	X
ejpam-4753	23	20	8	8	NUM
ejpam-4753	23	21	]	]	PUNCT
ejpam-4753	23	22	,	,	PUNCT
ejpam-4753	23	23	and	and	CCONJ
ejpam-4753	23	24	as	as	SCONJ
ejpam-4753	23	25	regards	regard	VERB
ejpam-4753	23	26	the	the	DET
ejpam-4753	23	27	graduation	graduation	NOUN
ejpam-4753	23	28	of	of	ADP
ejpam-4753	23	29	graded	grade	VERB
ejpam-4753	23	30	module	module	NOUN
ejpam-4753	23	31	of	of	ADP
ejpam-4753	23	32	fractions	fraction	NOUN
ejpam-4753	23	33	in	in	ADP
ejpam-4753	23	34	[	[	X
ejpam-4753	23	35	2	2	NUM
ejpam-4753	23	36	]	]	PUNCT
ejpam-4753	23	37	and	and	CCONJ
ejpam-4753	23	38	[	[	X
ejpam-4753	23	39	1	1	NUM
ejpam-4753	23	40	]	]	PUNCT
ejpam-4753	23	41	.	.	PUNCT
ejpam-4753	24	1	this	this	DET
ejpam-4753	24	2	article	article	NOUN
ejpam-4753	24	3	is	be	AUX
ejpam-4753	24	4	presented	present	VERB
ejpam-4753	24	5	as	as	SCONJ
ejpam-4753	24	6	follows	follow	VERB
ejpam-4753	24	7	:	:	PUNCT
ejpam-4753	24	8	in	in	ADP
ejpam-4753	24	9	the	the	DET
ejpam-4753	24	10	second	second	ADJ
ejpam-4753	24	11	section	section	NOUN
ejpam-4753	24	12	we	we	PRON
ejpam-4753	24	13	present	present	VERB
ejpam-4753	24	14	a	a	DET
ejpam-4753	24	15	reminder	reminder	NOUN
ejpam-4753	24	16	containing	contain	VERB
ejpam-4753	24	17	the	the	DET
ejpam-4753	24	18	definitions	definition	NOUN
ejpam-4753	24	19	and	and	CCONJ
ejpam-4753	24	20	background	background	NOUN
ejpam-4753	24	21	results	result	NOUN
ejpam-4753	24	22	of	of	ADP
ejpam-4753	24	23	graded	grade	VERB
ejpam-4753	24	24	rings	ring	NOUN
ejpam-4753	24	25	and	and	CCONJ
ejpam-4753	24	26	modules	module	NOUN
ejpam-4753	24	27	and	and	CCONJ
ejpam-4753	24	28	homological	homological	ADJ
ejpam-4753	24	29	algebra	algebra	NOUN
ejpam-4753	24	30	extracted	extract	VERB
ejpam-4753	24	31	in	in	ADP
ejpam-4753	24	32	[	[	X
ejpam-4753	24	33	10	10	NUM
ejpam-4753	24	34	]	]	PUNCT
ejpam-4753	24	35	,	,	PUNCT
ejpam-4753	25	1	[	[	X
ejpam-4753	25	2	11],[4	11],[4	NUM
ejpam-4753	25	3	]	]	PUNCT
ejpam-4753	25	4	and	and	CCONJ
ejpam-4753	25	5	[	[	X
ejpam-4753	25	6	9	9	NUM
ejpam-4753	25	7	]	]	PUNCT
ejpam-4753	25	8	.	.	PUNCT
ejpam-4753	26	1	in	in	ADP
ejpam-4753	26	2	section	section	NOUN
ejpam-4753	26	3	3	3	NUM
ejpam-4753	26	4	,	,	PUNCT
ejpam-4753	26	5	the	the	DET
ejpam-4753	26	6	following	follow	VERB
ejpam-4753	26	7	results	result	NOUN
ejpam-4753	26	8	have	have	AUX
ejpam-4753	26	9	been	be	AUX
ejpam-4753	26	10	shown	show	VERB
ejpam-4753	26	11	,	,	PUNCT
ejpam-4753	26	12	among	among	ADP
ejpam-4753	26	13	others	other	NOUN
ejpam-4753	26	14	:	:	PUNCT
ejpam-4753	26	15	if	if	SCONJ
ejpam-4753	26	16	a	a	DET
ejpam-4753	26	17	=	=	PUNCT
ejpam-4753	26	18	⊕	⊕	PROPN
ejpam-4753	26	19	n∈z	n∈z	VERB
ejpam-4753	26	20	an	an	PRON
ejpam-4753	26	21	is	be	AUX
ejpam-4753	26	22	a	a	DET
ejpam-4753	26	23	graded	grade	VERB
ejpam-4753	26	24	duo	duo	NOUN
ejpam-4753	26	25	-	-	PUNCT
ejpam-4753	26	26	ring	ring	NOUN
ejpam-4753	26	27	,	,	PUNCT
ejpam-4753	26	28	sh	sh	PROPN
ejpam-4753	26	29	is	be	AUX
ejpam-4753	26	30	a	a	DET
ejpam-4753	26	31	part	part	NOUN
ejpam-4753	26	32	formed	form	VERB
ejpam-4753	26	33	of	of	ADP
ejpam-4753	26	34	regular	regular	ADJ
ejpam-4753	26	35	homogeneous	homogeneous	ADJ
ejpam-4753	26	36	elements	element	NOUN
ejpam-4753	26	37	of	of	ADP
ejpam-4753	26	38	a	a	PRON
ejpam-4753	26	39	,	,	PUNCT
ejpam-4753	26	40	sh	sh	PROPN
ejpam-4753	26	41	is	be	AUX
ejpam-4753	26	42	the	the	DET
ejpam-4753	26	43	homogeneous	homogeneous	ADJ
ejpam-4753	26	44	multiplicatively	multiplicatively	ADV
ejpam-4753	26	45	closed	close	VERB
ejpam-4753	26	46	subset	subset	NOUN
ejpam-4753	26	47	of	of	ADP
ejpam-4753	26	48	a	a	DET
ejpam-4753	26	49	generated	generate	VERB
ejpam-4753	26	50	by	by	ADP
ejpam-4753	26	51	sh	sh	PROPN
ejpam-4753	26	52	,	,	PUNCT
ejpam-4753	26	53	then	then	ADV
ejpam-4753	26	54	we	we	PRON
ejpam-4753	26	55	have	have	VERB
ejpam-4753	26	56	:	:	PUNCT
ejpam-4753	26	57	(	(	PUNCT
ejpam-4753	26	58	i	i	NOUN
ejpam-4753	26	59	)	)	PUNCT
ejpam-4753	26	60	s	s	VERB
ejpam-4753	26	61	−1	−1	NOUN
ejpam-4753	26	62	h	h	NOUN
ejpam-4753	26	63	(	(	PUNCT
ejpam-4753	26	64	f	f	X
ejpam-4753	26	65	)	)	PUNCT
ejpam-4753	26	66	:	:	PUNCT
ejpam-4753	26	67	s	s	VERB
ejpam-4753	26	68	−1	−1	NOUN
ejpam-4753	27	1	h	h	NOUN
ejpam-4753	27	2	m	m	VERB
ejpam-4753	27	3	−→	−→	NOUN
ejpam-4753	27	4	s	s	PART
ejpam-4753	27	5	−1	−1	NOUN
ejpam-4753	27	6	h	h	NOUN
ejpam-4753	28	1	n	n	NOUN
ejpam-4753	29	1	m	m	NOUN
ejpam-4753	29	2	s	s	PROPN
ejpam-4753	29	3	7−→	7−→	PROPN
ejpam-4753	29	4	s	s	PART
ejpam-4753	29	5	−1	−1	NOUN
ejpam-4753	29	6	h	h	NOUN
ejpam-4753	29	7	(	(	PUNCT
ejpam-4753	29	8	f	f	X
ejpam-4753	29	9	)	)	PUNCT
ejpam-4753	29	10	(	(	PUNCT
ejpam-4753	29	11	m	m	PROPN
ejpam-4753	29	12	s	s	PART
ejpam-4753	29	13	)	)	PUNCT
ejpam-4753	29	14	=	=	SYM
ejpam-4753	29	15	f(m	f(m	PROPN
ejpam-4753	29	16	)	)	PUNCT
ejpam-4753	29	17	s	s	VERB
ejpam-4753	29	18	is	be	AUX
ejpam-4753	29	19	a	a	DET
ejpam-4753	29	20	graded	grade	VERB
ejpam-4753	29	21	morphism	morphism	NOUN
ejpam-4753	29	22	of	of	ADP
ejpam-4753	29	23	degree	degree	NOUN
ejpam-4753	29	24	k	k	PROPN
ejpam-4753	29	25	∈	∈	PROPN
ejpam-4753	29	26	z	z	PROPN
ejpam-4753	29	27	of	of	ADP
ejpam-4753	29	28	graded	grade	VERB
ejpam-4753	29	29	left	leave	VERB
ejpam-4753	29	30	s	s	PRON
ejpam-4753	29	31	−1	−1	NOUN
ejpam-4753	29	32	h	h	NOUN
ejpam-4753	29	33	a−module	a−module	ADP
ejpam-4753	29	34	;	;	PUNCT
ejpam-4753	29	35	(	(	PUNCT
ejpam-4753	29	36	ii	ii	X
ejpam-4753	29	37	)	)	PUNCT
ejpam-4753	29	38	the	the	DET
ejpam-4753	29	39	relation	relation	NOUN
ejpam-4753	29	40	s	s	PART
ejpam-4753	29	41	−1	−1	NOUN
ejpam-4753	29	42	h	h	NOUN
ejpam-4753	29	43	(	(	PUNCT
ejpam-4753	29	44	−	−	PROPN
ejpam-4753	29	45	)	)	PUNCT
ejpam-4753	29	46	:	:	PUNCT
ejpam-4753	29	47	gr(a−mod	gr(a−mod	X
ejpam-4753	29	48	)	)	PUNCT
ejpam-4753	29	49	−→	−→	NOUN
ejpam-4753	29	50	gr(s	gr(s	PUNCT
ejpam-4753	29	51	−1	−1	NOUN
ejpam-4753	29	52	h	h	PROPN
ejpam-4753	29	53	a−mod	a−mod	NOUN
ejpam-4753	29	54	)	)	PUNCT
ejpam-4753	29	55	which	which	PRON
ejpam-4753	29	56	that	that	SCONJ
ejpam-4753	29	57	for	for	ADP
ejpam-4753	29	58	any	any	DET
ejpam-4753	29	59	graded	grade	VERB
ejpam-4753	29	60	left	leave	VERB
ejpam-4753	29	61	a−module	a−module	ADP
ejpam-4753	29	62	m	m	VERB
ejpam-4753	29	63	we	we	PRON
ejpam-4753	29	64	made	make	VERB
ejpam-4753	29	65	to	to	PART
ejpam-4753	29	66	correspond	correspond	VERB
ejpam-4753	29	67	s	s	PRON
ejpam-4753	29	68	−1	−1	NOUN
ejpam-4753	29	69	h	h	NOUN
ejpam-4753	29	70	(	(	PUNCT
ejpam-4753	29	71	m	m	NOUN
ejpam-4753	29	72	)	)	PUNCT
ejpam-4753	29	73	and	and	CCONJ
ejpam-4753	29	74	for	for	ADP
ejpam-4753	29	75	all	all	DET
ejpam-4753	29	76	graded	grade	VERB
ejpam-4753	29	77	morphism	morphism	NOUN
ejpam-4753	29	78	of	of	ADP
ejpam-4753	29	79	degree	degree	NOUN
ejpam-4753	29	80	k	k	PROPN
ejpam-4753	29	81	∈	∈	PROPN
ejpam-4753	29	82	z	z	PROPN
ejpam-4753	29	83	of	of	ADP
ejpam-4753	29	84	graded	grade	VERB
ejpam-4753	29	85	left	leave	VERB
ejpam-4753	29	86	a−modules	a−module	NOUN
ejpam-4753	29	87	f	f	X
ejpam-4753	29	88	:	:	PUNCT
ejpam-4753	29	89	m	m	VERB
ejpam-4753	29	90	−→	−→	ADJ
ejpam-4753	30	1	n	n	INTJ
ejpam-4753	30	2	we	we	PRON
ejpam-4753	30	3	correspond	correspond	VERB
ejpam-4753	30	4	s	s	PRON
ejpam-4753	30	5	−1	−1	NOUN
ejpam-4753	30	6	h	h	NOUN
ejpam-4753	30	7	(	(	PUNCT
ejpam-4753	30	8	f	f	X
ejpam-4753	30	9	)	)	PUNCT
ejpam-4753	30	10	of	of	ADP
ejpam-4753	30	11	degree	degree	NOUN
ejpam-4753	30	12	k	k	PROPN
ejpam-4753	30	13	∈	∈	PROPN
ejpam-4753	30	14	z	z	NOUN
ejpam-4753	30	15	is	be	AUX
ejpam-4753	30	16	a	a	DET
ejpam-4753	30	17	exact	exact	ADJ
ejpam-4753	30	18	additively	additively	ADV
ejpam-4753	30	19	covariant	covariant	ADJ
ejpam-4753	30	20	functor	functor	NOUN
ejpam-4753	30	21	;	;	PUNCT
ejpam-4753	30	22	(	(	PUNCT
ejpam-4753	30	23	iii	iii	NOUN
ejpam-4753	30	24	)	)	PUNCT
ejpam-4753	30	25	furthermore	furthermore	ADV
ejpam-4753	30	26	let	let	VERB
ejpam-4753	30	27	p	p	PRON
ejpam-4753	30	28	a	a	DET
ejpam-4753	30	29	prime	prime	ADJ
ejpam-4753	30	30	ideal	ideal	NOUN
ejpam-4753	30	31	of	of	ADP
ejpam-4753	30	32	a	a	PRON
ejpam-4753	30	33	and	and	CCONJ
ejpam-4753	30	34	sh	sh	PROPN
ejpam-4753	30	35	is	be	AUX
ejpam-4753	30	36	a	a	DET
ejpam-4753	30	37	part	part	NOUN
ejpam-4753	30	38	formed	form	VERB
ejpam-4753	30	39	of	of	ADP
ejpam-4753	30	40	regular	regular	ADJ
ejpam-4753	30	41	homogeneous	homogeneous	ADJ
ejpam-4753	30	42	elements	element	NOUN
ejpam-4753	30	43	of	of	ADP
ejpam-4753	30	44	a\p	a\p	PROPN
ejpam-4753	30	45	and	and	CCONJ
ejpam-4753	30	46	sph	sph	PROPN
ejpam-4753	30	47	is	be	AUX
ejpam-4753	30	48	the	the	DET
ejpam-4753	30	49	homogeneous	homogeneous	ADJ
ejpam-4753	30	50	multiplicatively	multiplicatively	ADV
ejpam-4753	30	51	closed	close	VERB
ejpam-4753	30	52	subset	subset	NOUN
ejpam-4753	30	53	of	of	ADP
ejpam-4753	30	54	a	a	DET
ejpam-4753	30	55	generated	generate	VERB
ejpam-4753	30	56	by	by	ADP
ejpam-4753	30	57	sh	sh	PROPN
ejpam-4753	30	58	,	,	PUNCT
ejpam-4753	30	59	then	then	ADV
ejpam-4753	30	60	the	the	DET
ejpam-4753	30	61	relation	relation	NOUN
ejpam-4753	30	62	s	s	PART
ejpam-4753	30	63	−1	−1	NOUN
ejpam-4753	30	64	ph	ph	NOUN
ejpam-4753	30	65	(	(	PUNCT
ejpam-4753	30	66	−	−	NOUN
ejpam-4753	30	67	)	)	PUNCT
ejpam-4753	30	68	:	:	PUNCT
ejpam-4753	30	69	gr(a−mod	gr(a−mod	X
ejpam-4753	30	70	)	)	PUNCT
ejpam-4753	30	71	−→	−→	NOUN
ejpam-4753	30	72	s	s	PART
ejpam-4753	30	73	−1	−1	NOUN
ejpam-4753	30	74	ph	ph	NOUN
ejpam-4753	30	75	a−mod	a−mod	NOUN
ejpam-4753	30	76	which	which	PRON
ejpam-4753	30	77	that	that	SCONJ
ejpam-4753	30	78	for	for	ADP
ejpam-4753	30	79	any	any	DET
ejpam-4753	30	80	graded	grade	VERB
ejpam-4753	30	81	left	leave	VERB
ejpam-4753	30	82	a−module	a−module	ADP
ejpam-4753	30	83	m	m	VERB
ejpam-4753	31	1	we	we	PRON
ejpam-4753	31	2	correspond	correspond	VERB
ejpam-4753	31	3	s	s	PRON
ejpam-4753	31	4	−1	−1	NOUN
ejpam-4753	31	5	ph	ph	NOUN
ejpam-4753	31	6	(	(	PUNCT
ejpam-4753	31	7	m	m	NOUN
ejpam-4753	31	8	)	)	PUNCT
ejpam-4753	31	9	and	and	CCONJ
ejpam-4753	31	10	for	for	ADP
ejpam-4753	31	11	all	all	DET
ejpam-4753	31	12	graded	grade	VERB
ejpam-4753	31	13	morphism	morphism	NOUN
ejpam-4753	31	14	of	of	ADP
ejpam-4753	31	15	degree	degree	NOUN
ejpam-4753	31	16	k	k	PROPN
ejpam-4753	31	17	∈	∈	PROPN
ejpam-4753	31	18	z	z	PROPN
ejpam-4753	31	19	of	of	ADP
ejpam-4753	31	20	graded	grade	VERB
ejpam-4753	31	21	left	leave	VERB
ejpam-4753	31	22	a−modules	a−module	NOUN
ejpam-4753	31	23	f	f	X
ejpam-4753	31	24	:	:	PUNCT
ejpam-4753	31	25	m	m	VERB
ejpam-4753	31	26	−→	−→	ADJ
ejpam-4753	32	1	n	n	INTJ
ejpam-4753	32	2	we	we	PRON
ejpam-4753	32	3	correspond	correspond	VERB
ejpam-4753	32	4	s	s	PRON
ejpam-4753	32	5	−1	−1	NOUN
ejpam-4753	32	6	ph	ph	NOUN
ejpam-4753	32	7	(	(	PUNCT
ejpam-4753	32	8	f	f	NOUN
ejpam-4753	32	9	)	)	PUNCT
ejpam-4753	32	10	of	of	ADP
ejpam-4753	32	11	degree	degree	NOUN
ejpam-4753	32	12	k	k	PROPN
ejpam-4753	32	13	∈	∈	PROPN
ejpam-4753	32	14	z	z	NOUN
ejpam-4753	32	15	is	be	AUX
ejpam-4753	32	16	a	a	DET
ejpam-4753	32	17	exact	exact	ADJ
ejpam-4753	32	18	additively	additively	ADV
ejpam-4753	32	19	covariant	covariant	ADJ
ejpam-4753	32	20	functor	functor	NOUN
ejpam-4753	32	21	.	.	PUNCT
ejpam-4753	33	1	in	in	ADP
ejpam-4753	33	2	the	the	DET
ejpam-4753	33	3	last	last	ADJ
ejpam-4753	33	4	section	section	NOUN
ejpam-4753	33	5	the	the	DET
ejpam-4753	33	6	following	follow	VERB
ejpam-4753	33	7	results	result	NOUN
ejpam-4753	33	8	among	among	ADP
ejpam-4753	33	9	others	other	NOUN
ejpam-4753	33	10	have	have	AUX
ejpam-4753	33	11	been	be	AUX
ejpam-4753	33	12	shown	show	VERB
ejpam-4753	33	13	:	:	PUNCT
ejpam-4753	33	14	if	if	SCONJ
ejpam-4753	33	15	a	a	DET
ejpam-4753	33	16	=	=	PUNCT
ejpam-4753	33	17	⊕	⊕	PROPN
ejpam-4753	33	18	n∈z	n∈z	VERB
ejpam-4753	33	19	an	an	PRON
ejpam-4753	33	20	is	be	AUX
ejpam-4753	33	21	a	a	DET
ejpam-4753	33	22	graded	grade	VERB
ejpam-4753	33	23	duo	duo	NOUN
ejpam-4753	33	24	-	-	PUNCT
ejpam-4753	33	25	ring	ring	NOUN
ejpam-4753	33	26	,	,	PUNCT
ejpam-4753	33	27	sh	sh	PROPN
ejpam-4753	33	28	is	be	AUX
ejpam-4753	33	29	a	a	DET
ejpam-4753	33	30	part	part	NOUN
ejpam-4753	33	31	formed	form	VERB
ejpam-4753	33	32	of	of	ADP
ejpam-4753	33	33	regular	regular	ADJ
ejpam-4753	33	34	homogeneous	homogeneous	ADJ
ejpam-4753	33	35	elements	element	NOUN
ejpam-4753	33	36	of	of	ADP
ejpam-4753	33	37	a	a	PRON
ejpam-4753	33	38	,	,	PUNCT
ejpam-4753	33	39	sh	sh	PROPN
ejpam-4753	33	40	is	be	AUX
ejpam-4753	33	41	the	the	DET
ejpam-4753	33	42	homogeneous	homogeneous	ADJ
ejpam-4753	33	43	multiplicatively	multiplicatively	ADV
ejpam-4753	33	44	closed	close	VERB
ejpam-4753	33	45	subset	subset	NOUN
ejpam-4753	33	46	of	of	ADP
ejpam-4753	33	47	a	a	DET
ejpam-4753	33	48	generated	generate	VERB
ejpam-4753	33	49	by	by	ADP
ejpam-4753	33	50	sh	sh	PROPN
ejpam-4753	33	51	,	,	PUNCT
ejpam-4753	33	52	then	then	ADV
ejpam-4753	33	53	we	we	PRON
ejpam-4753	33	54	have	have	VERB
ejpam-4753	33	55	the	the	DET
ejpam-4753	33	56	a.	a.	NOUN
ejpam-4753	33	57	o.	o.	PROPN
ejpam-4753	33	58	chbih	chbih	PROPN
ejpam-4753	33	59	,	,	PUNCT
ejpam-4753	33	60	m.	m.	PROPN
ejpam-4753	33	61	b.	b.	PROPN
ejpam-4753	33	62	maaouia	maaouia	PROPN
ejpam-4753	33	63	,	,	PUNCT
ejpam-4753	33	64	m.	m.	NOUN
ejpam-4753	33	65	sanghare	sanghare	PROPN
ejpam-4753	33	66	/	/	SYM
ejpam-4753	33	67	eur	eur	PROPN
ejpam-4753	33	68	.	.	PUNCT
ejpam-4753	34	1	j.	j.	PROPN
ejpam-4753	34	2	pure	pure	PROPN
ejpam-4753	34	3	appl	appl	PROPN
ejpam-4753	34	4	.	.	PROPN
ejpam-4753	34	5	math	math	PROPN
ejpam-4753	34	6	,	,	PUNCT
ejpam-4753	34	7	16	16	NUM
ejpam-4753	34	8	(	(	PUNCT
ejpam-4753	34	9	3	3	NUM
ejpam-4753	34	10	)	)	PUNCT
ejpam-4753	34	11	(	(	PUNCT
ejpam-4753	34	12	2023	2023	NUM
ejpam-4753	34	13	)	)	PUNCT
ejpam-4753	34	14	,	,	PUNCT
ejpam-4753	34	15	1913	1913	NUM
ejpam-4753	34	16	-	-	SYM
ejpam-4753	34	17	1939	1939	NUM
ejpam-4753	34	18	1915	1915	NUM
ejpam-4753	34	19	following	follow	VERB
ejpam-4753	34	20	results	result	NOUN
ejpam-4753	34	21	:	:	PUNCT
ejpam-4753	34	22	(	(	PUNCT
ejpam-4753	34	23	i	i	NOUN
ejpam-4753	34	24	)	)	PUNCT
ejpam-4753	34	25	the	the	DET
ejpam-4753	34	26	following	follow	VERB
ejpam-4753	34	27	complex	complex	ADJ
ejpam-4753	34	28	sequence	sequence	NOUN
ejpam-4753	34	29	:	:	PUNCT
ejpam-4753	34	30	s	s	VERB
ejpam-4753	34	31	−1	−1	NOUN
ejpam-4753	34	32	h	h	NOUN
ejpam-4753	34	33	(	(	PUNCT
ejpam-4753	34	34	m∗	m∗	PROPN
ejpam-4753	34	35	)	)	PUNCT
ejpam-4753	34	36	:	:	PUNCT
ejpam-4753	34	37	·	·	PUNCT
ejpam-4753	34	38	·	·	PUNCT
ejpam-4753	34	39	·	·	PUNCT
ejpam-4753	35	1	−→	−→	NOUN
ejpam-4753	35	2	s	s	PRON
ejpam-4753	35	3	−1	−1	NOUN
ejpam-4753	35	4	h	h	NOUN
ejpam-4753	35	5	(	(	PUNCT
ejpam-4753	35	6	m(n+1	m(n+1	NOUN
ejpam-4753	35	7	)	)	PUNCT
ejpam-4753	35	8	)	)	PUNCT
ejpam-4753	35	9	s	s	VERB
ejpam-4753	35	10	−1	−1	NOUN
ejpam-4753	35	11	h	h	NOUN
ejpam-4753	35	12	(	(	PUNCT
ejpam-4753	35	13	dn+1)−→	dn+1)−→	PROPN
ejpam-4753	35	14	s	s	PART
ejpam-4753	35	15	−1	−1	NOUN
ejpam-4753	35	16	h	h	NOUN
ejpam-4753	35	17	(	(	PUNCT
ejpam-4753	35	18	m(n	m(n	PROPN
ejpam-4753	35	19	)	)	PUNCT
ejpam-4753	35	20	)	)	PUNCT
ejpam-4753	36	1	s	s	VERB
ejpam-4753	36	2	−1	−1	NOUN
ejpam-4753	36	3	h	h	NOUN
ejpam-4753	36	4	(	(	PUNCT
ejpam-4753	36	5	dn)−→	dn)−→	PROPN
ejpam-4753	36	6	s	s	PART
ejpam-4753	36	7	−1	−1	NOUN
ejpam-4753	36	8	h	h	NOUN
ejpam-4753	36	9	(	(	PUNCT
ejpam-4753	36	10	m(n−1	m(n−1	PROPN
ejpam-4753	36	11	)	)	PUNCT
ejpam-4753	36	12	)	)	PUNCT
ejpam-4753	37	1	−→	−→	NOUN
ejpam-4753	37	2	·	·	PUNCT
ejpam-4753	37	3	·	·	PUNCT
ejpam-4753	37	4	·	·	PUNCT
ejpam-4753	37	5	with	with	ADP
ejpam-4753	37	6	dn	dn	NOUN
ejpam-4753	37	7	:	:	PUNCT
ejpam-4753	37	8	m(n	m(n	PROPN
ejpam-4753	37	9	)	)	PUNCT
ejpam-4753	37	10	−→	−→	NOUN
ejpam-4753	37	11	m(n−	m(n−	NOUN
ejpam-4753	37	12	1	1	NUM
ejpam-4753	37	13	)	)	PUNCT
ejpam-4753	37	14	x	x	X
ejpam-4753	38	1	=	=	PUNCT
ejpam-4753	38	2	y	y	PROPN
ejpam-4753	38	3	+	+	NOUN
ejpam-4753	38	4	z	z	PROPN
ejpam-4753	38	5	7−→	7−→	NUM
ejpam-4753	38	6	y	y	NOUN
ejpam-4753	38	7	with	with	ADP
ejpam-4753	38	8	(	(	PUNCT
ejpam-4753	38	9	y	y	PROPN
ejpam-4753	38	10	,	,	PUNCT
ejpam-4753	38	11	z	z	NOUN
ejpam-4753	38	12	)	)	PUNCT
ejpam-4753	38	13	∈	∈	PROPN
ejpam-4753	38	14	mn	mn	PROPN
ejpam-4753	38	15	×m(n+	×m(n+	PROPN
ejpam-4753	38	16	1	1	NUM
ejpam-4753	38	17	)	)	PUNCT
ejpam-4753	38	18	;	;	PUNCT
ejpam-4753	38	19	(	(	PUNCT
ejpam-4753	38	20	ii	ii	X
ejpam-4753	38	21	)	)	PUNCT
ejpam-4753	38	22	the	the	DET
ejpam-4753	38	23	following	follow	VERB
ejpam-4753	38	24	complex	complex	ADJ
ejpam-4753	38	25	chain	chain	NOUN
ejpam-4753	38	26	:	:	PUNCT
ejpam-4753	38	27	s	s	VERB
ejpam-4753	38	28	−1	−1	NOUN
ejpam-4753	38	29	h	h	NOUN
ejpam-4753	38	30	(	(	PUNCT
ejpam-4753	38	31	m∗	m∗	PROPN
ejpam-4753	38	32	)	)	PUNCT
ejpam-4753	38	33	:	:	PUNCT
ejpam-4753	38	34	·	·	PUNCT
ejpam-4753	38	35	·	·	PUNCT
ejpam-4753	38	36	·	·	PUNCT
ejpam-4753	39	1	//	//	PUNCT
ejpam-4753	39	2	s	s	PART
ejpam-4753	39	3	−1	−1	NOUN
ejpam-4753	39	4	h	h	NOUN
ejpam-4753	39	5	(	(	PUNCT
ejpam-4753	39	6	fk	fk	INTJ
ejpam-4753	39	7	∗	∗	NOUN
ejpam-4753	39	8	)	)	PUNCT
ejpam-4753	39	9	�	�	PROPN
ejpam-4753	39	10	�	�	PROPN
ejpam-4753	39	11	//	//	PROPN
ejpam-4753	39	12	s	s	PART
ejpam-4753	39	13	−1	−1	NOUN
ejpam-4753	39	14	h	h	NOUN
ejpam-4753	39	15	(	(	PUNCT
ejpam-4753	39	16	m(n+	m(n+	NOUN
ejpam-4753	39	17	1	1	NUM
ejpam-4753	39	18	)	)	PUNCT
ejpam-4753	39	19	)	)	PUNCT
ejpam-4753	39	20	s	s	VERB
ejpam-4753	39	21	−1	−1	NOUN
ejpam-4753	39	22	h	h	NOUN
ejpam-4753	39	23	(	(	PUNCT
ejpam-4753	39	24	dn+1)//	dn+1)//	PROPN
ejpam-4753	39	25	s	s	PART
ejpam-4753	39	26	−1	−1	NOUN
ejpam-4753	39	27	h	h	NOUN
ejpam-4753	39	28	(	(	PUNCT
ejpam-4753	39	29	fk(n+1	fk(n+1	PROPN
ejpam-4753	39	30	)	)	PUNCT
ejpam-4753	39	31	)	)	PUNCT
ejpam-4753	39	32	�	�	PROPN
ejpam-4753	39	33	�	�	PROPN
ejpam-4753	39	34	s	s	PART
ejpam-4753	39	35	−1	−1	NOUN
ejpam-4753	39	36	h	h	NOUN
ejpam-4753	39	37	(	(	PUNCT
ejpam-4753	39	38	m(n	m(n	PROPN
ejpam-4753	39	39	)	)	PUNCT
ejpam-4753	39	40	)	)	PUNCT
ejpam-4753	40	1	s	s	VERB
ejpam-4753	40	2	−1	−1	NOUN
ejpam-4753	40	3	h	h	NOUN
ejpam-4753	40	4	(	(	PUNCT
ejpam-4753	40	5	dn)//	dn)//	PROPN
ejpam-4753	40	6	s	s	PART
ejpam-4753	40	7	−1	−1	NOUN
ejpam-4753	40	8	h	h	NOUN
ejpam-4753	40	9	(	(	PUNCT
ejpam-4753	40	10	fk(n	fk(n	NOUN
ejpam-4753	40	11	)	)	PUNCT
ejpam-4753	40	12	)	)	PUNCT
ejpam-4753	40	13	�	�	PROPN
ejpam-4753	40	14	�	�	PROPN
ejpam-4753	40	15	s	s	PART
ejpam-4753	40	16	−1	−1	NOUN
ejpam-4753	40	17	h	h	NOUN
ejpam-4753	40	18	(	(	PUNCT
ejpam-4753	40	19	m(n−	m(n−	NOUN
ejpam-4753	40	20	1	1	NUM
ejpam-4753	40	21	)	)	PUNCT
ejpam-4753	40	22	)	)	PUNCT
ejpam-4753	41	1	//	//	PUNCT
ejpam-4753	41	2	s	s	PART
ejpam-4753	41	3	−1	−1	NOUN
ejpam-4753	41	4	h	h	NOUN
ejpam-4753	41	5	(	(	PUNCT
ejpam-4753	41	6	fk(n−1	fk(n−1	PROPN
ejpam-4753	41	7	)	)	PUNCT
ejpam-4753	41	8	)	)	PUNCT
ejpam-4753	41	9	�	�	PROPN
ejpam-4753	41	10	�	�	PROPN
ejpam-4753	41	11	·	·	PUNCT
ejpam-4753	41	12	·	·	PUNCT
ejpam-4753	41	13	·	·	PUNCT
ejpam-4753	41	14	s	s	X
ejpam-4753	41	15	−1	−1	NOUN
ejpam-4753	41	16	h	h	NOUN
ejpam-4753	41	17	(	(	PUNCT
ejpam-4753	41	18	n∗	n∗	PROPN
ejpam-4753	41	19	)	)	PUNCT
ejpam-4753	41	20	:	:	PUNCT
ejpam-4753	41	21	·	·	PUNCT
ejpam-4753	41	22	·	·	PUNCT
ejpam-4753	41	23	·	·	PUNCT
ejpam-4753	42	1	//	//	PUNCT
ejpam-4753	42	2	s	s	PART
ejpam-4753	42	3	−1	−1	NOUN
ejpam-4753	42	4	h	h	NOUN
ejpam-4753	42	5	(	(	PUNCT
ejpam-4753	42	6	n(n+	n(n+	NOUN
ejpam-4753	42	7	1	1	NUM
ejpam-4753	42	8	)	)	PUNCT
ejpam-4753	42	9	)	)	PUNCT
ejpam-4753	42	10	s	s	VERB
ejpam-4753	42	11	−1	−1	NOUN
ejpam-4753	42	12	h	h	NOUN
ejpam-4753	42	13	(	(	PUNCT
ejpam-4753	42	14	d′n+1+k)//	d′n+1+k)//	PROPN
ejpam-4753	42	15	s	s	PART
ejpam-4753	42	16	−1	−1	NOUN
ejpam-4753	42	17	h	h	NOUN
ejpam-4753	42	18	(	(	PUNCT
ejpam-4753	42	19	n(n	n(n	NOUN
ejpam-4753	42	20	)	)	PUNCT
ejpam-4753	42	21	)	)	PUNCT
ejpam-4753	43	1	s	s	VERB
ejpam-4753	43	2	−1	−1	NOUN
ejpam-4753	43	3	h	h	NOUN
ejpam-4753	43	4	(	(	PUNCT
ejpam-4753	43	5	d′n+k)//	d′n+k)//	NOUN
ejpam-4753	43	6	s	s	PART
ejpam-4753	43	7	−1	−1	NOUN
ejpam-4753	43	8	h	h	NOUN
ejpam-4753	43	9	(	(	PUNCT
ejpam-4753	43	10	n(n−	n(n−	PROPN
ejpam-4753	43	11	1	1	NUM
ejpam-4753	43	12	)	)	PUNCT
ejpam-4753	43	13	)	)	PUNCT
ejpam-4753	43	14	//	//	X
ejpam-4753	43	15	·	·	PUNCT
ejpam-4753	43	16	·	·	PUNCT
ejpam-4753	43	17	·	·	PUNCT
ejpam-4753	43	18	(	(	PUNCT
ejpam-4753	43	19	iii	iii	X
ejpam-4753	43	20	)	)	PUNCT
ejpam-4753	43	21	the	the	DET
ejpam-4753	43	22	relation	relation	NOUN
ejpam-4753	43	23	ch(−	ch(−	PUNCT
ejpam-4753	43	24	)	)	PUNCT
ejpam-4753	43	25	:	:	PUNCT
ejpam-4753	43	26	gr(s	gr(s	X
ejpam-4753	43	27	−1	−1	NOUN
ejpam-4753	43	28	h	h	PROPN
ejpam-4753	43	29	a−mod	a−mod	NOUN
ejpam-4753	43	30	)	)	PUNCT
ejpam-4753	43	31	−→	−→	NOUN
ejpam-4753	43	32	comp	comp	NOUN
ejpam-4753	43	33	(	(	PUNCT
ejpam-4753	43	34	gr(s	gr(s	NOUN
ejpam-4753	43	35	−1	−1	NOUN
ejpam-4753	43	36	h	h	NOUN
ejpam-4753	43	37	a−mod	a−mod	NOUN
ejpam-4753	43	38	)	)	PUNCT
ejpam-4753	43	39	)	)	PUNCT
ejpam-4753	43	40	which	which	PRON
ejpam-4753	43	41	that	that	SCONJ
ejpam-4753	43	42	for	for	ADP
ejpam-4753	43	43	all	all	PRON
ejpam-4753	43	44	graded	grade	VERB
ejpam-4753	43	45	left	leave	VERB
ejpam-4753	43	46	s	s	PRON
ejpam-4753	43	47	−1	−1	NOUN
ejpam-4753	43	48	h	h	NOUN
ejpam-4753	43	49	a−module	a−module	ADP
ejpam-4753	43	50	s	s	NUM
ejpam-4753	43	51	−1	−1	NOUN
ejpam-4753	43	52	h	h	NOUN
ejpam-4753	43	53	m	m	VERB
ejpam-4753	43	54	of	of	ADP
ejpam-4753	43	55	gr(s	gr(s	PUNCT
ejpam-4753	43	56	−1	−1	NOUN
ejpam-4753	43	57	h	h	NOUN
ejpam-4753	43	58	a	a	DET
ejpam-4753	43	59	−	−	PROPN
ejpam-4753	43	60	mod	mod	NOUN
ejpam-4753	43	61	)	)	PUNCT
ejpam-4753	43	62	we	we	PRON
ejpam-4753	43	63	correspond	correspond	VERB
ejpam-4753	43	64	the	the	DET
ejpam-4753	43	65	associate	associate	ADJ
ejpam-4753	43	66	complex	complex	ADJ
ejpam-4753	43	67	sequence	sequence	NOUN
ejpam-4753	43	68	(	(	PUNCT
ejpam-4753	43	69	s	s	NOUN
ejpam-4753	43	70	−1	−1	NOUN
ejpam-4753	43	71	h	h	NOUN
ejpam-4753	43	72	m)∗	m)∗	VERB
ejpam-4753	43	73	to	to	ADP
ejpam-4753	43	74	a	a	DET
ejpam-4753	43	75	graded	grade	VERB
ejpam-4753	43	76	s	s	PRON
ejpam-4753	43	77	−1	−1	NOUN
ejpam-4753	43	78	h	h	NOUN
ejpam-4753	43	79	a−module	a−module	ADP
ejpam-4753	43	80	s	s	X
ejpam-4753	43	81	−1	−1	NOUN
ejpam-4753	43	82	h	h	NOUN
ejpam-4753	43	83	m	m	PROPN
ejpam-4753	43	84	and	and	CCONJ
ejpam-4753	43	85	for	for	ADP
ejpam-4753	43	86	all	all	DET
ejpam-4753	43	87	graded	grade	VERB
ejpam-4753	43	88	morphism	morphism	NOUN
ejpam-4753	43	89	of	of	ADP
ejpam-4753	43	90	graded	grade	VERB
ejpam-4753	43	91	left	leave	VERB
ejpam-4753	43	92	s	s	PRON
ejpam-4753	43	93	−1	−1	NOUN
ejpam-4753	43	94	h	h	NOUN
ejpam-4753	44	1	a−modules	a−module	NOUN
ejpam-4753	44	2	s	s	PART
ejpam-4753	44	3	−1	−1	NOUN
ejpam-4753	44	4	h	h	NOUN
ejpam-4753	44	5	f	f	NOUN
ejpam-4753	44	6	:	:	PUNCT
ejpam-4753	44	7	s	s	VERB
ejpam-4753	44	8	−1	−1	NOUN
ejpam-4753	44	9	h	h	NOUN
ejpam-4753	44	10	m	m	VERB
ejpam-4753	44	11	−→	−→	NOUN
ejpam-4753	44	12	s	s	PART
ejpam-4753	44	13	−1	−1	NOUN
ejpam-4753	44	14	h	h	NOUN
ejpam-4753	44	15	n	n	PROPN
ejpam-4753	44	16	of	of	ADP
ejpam-4753	44	17	degree	degree	NOUN
ejpam-4753	45	1	k	k	NOUN
ejpam-4753	45	2	we	we	PRON
ejpam-4753	45	3	correspond	correspond	VERB
ejpam-4753	45	4	the	the	DET
ejpam-4753	45	5	associate	associate	ADJ
ejpam-4753	45	6	complex	complex	NOUN
ejpam-4753	45	7	chain	chain	NOUN
ejpam-4753	45	8	(	(	PUNCT
ejpam-4753	45	9	s	s	NOUN
ejpam-4753	45	10	−1	−1	NOUN
ejpam-4753	45	11	h	h	NOUN
ejpam-4753	45	12	f)k∗	f)k∗	ADJ
ejpam-4753	45	13	to	to	ADP
ejpam-4753	45	14	a	a	DET
ejpam-4753	45	15	morphism	morphism	NOUN
ejpam-4753	45	16	of	of	ADP
ejpam-4753	45	17	graded	grade	VERB
ejpam-4753	45	18	left	leave	VERB
ejpam-4753	45	19	s	s	PRON
ejpam-4753	45	20	−1	−1	NOUN
ejpam-4753	45	21	h	h	NOUN
ejpam-4753	45	22	a−module	a−module	ADP
ejpam-4753	45	23	s	s	NUM
ejpam-4753	45	24	−1	−1	NOUN
ejpam-4753	45	25	h	h	NOUN
ejpam-4753	45	26	f	f	NOUN
ejpam-4753	45	27	:	:	PUNCT
ejpam-4753	45	28	s	s	VERB
ejpam-4753	45	29	−1	−1	NOUN
ejpam-4753	45	30	h	h	NOUN
ejpam-4753	45	31	m	m	VERB
ejpam-4753	45	32	−→	−→	NOUN
ejpam-4753	45	33	s	s	PART
ejpam-4753	45	34	−1	−1	NOUN
ejpam-4753	45	35	h	h	NOUN
ejpam-4753	45	36	n	n	ADV
ejpam-4753	45	37	is	be	AUX
ejpam-4753	45	38	additively	additively	ADV
ejpam-4753	45	39	exact	exact	ADJ
ejpam-4753	45	40	covariant	covariant	PROPN
ejpam-4753	45	41	functor	functor	PROPN
ejpam-4753	45	42	.	.	PUNCT
ejpam-4753	46	1	(	(	PUNCT
ejpam-4753	46	2	iv	iv	X
ejpam-4753	46	3	)	)	PUNCT
ejpam-4753	46	4	the	the	DET
ejpam-4753	46	5	relation	relation	NOUN
ejpam-4753	46	6	(	(	PUNCT
ejpam-4753	46	7	ch	ch	NOUN
ejpam-4753	46	8	◦	◦	NOUN
ejpam-4753	46	9	s−1	s−1	PROPN
ejpam-4753	46	10	h	h	NOUN
ejpam-4753	46	11	)	)	PUNCT
ejpam-4753	46	12	(	(	PUNCT
ejpam-4753	46	13	−	−	NOUN
ejpam-4753	46	14	)	)	PUNCT
ejpam-4753	46	15	:	:	PUNCT
ejpam-4753	46	16	gr(a	gr(a	X
ejpam-4753	46	17	−mod	−mod	ADJ
ejpam-4753	46	18	)	)	PUNCT
ejpam-4753	46	19	−→	−→	ADJ
ejpam-4753	46	20	comp	comp	NOUN
ejpam-4753	46	21	(	(	PUNCT
ejpam-4753	46	22	gr(s	gr(s	NOUN
ejpam-4753	46	23	−1	−1	NOUN
ejpam-4753	46	24	h	h	NOUN
ejpam-4753	46	25	a	a	DET
ejpam-4753	46	26	−mod	−mod	NOUN
ejpam-4753	46	27	)	)	PUNCT
ejpam-4753	46	28	)	)	PUNCT
ejpam-4753	46	29	which	which	PRON
ejpam-4753	46	30	that	that	SCONJ
ejpam-4753	46	31	for	for	ADP
ejpam-4753	46	32	all	all	PRON
ejpam-4753	46	33	graded	grade	VERB
ejpam-4753	46	34	left	leave	VERB
ejpam-4753	46	35	a−module	a−module	ADP
ejpam-4753	46	36	m	m	NOUN
ejpam-4753	46	37	of	of	ADP
ejpam-4753	46	38	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	46	39	)	)	PUNCT
ejpam-4753	46	40	we	we	PRON
ejpam-4753	46	41	correspond	correspond	VERB
ejpam-4753	46	42	the	the	DET
ejpam-4753	46	43	associate	associate	ADJ
ejpam-4753	46	44	complex	complex	ADJ
ejpam-4753	46	45	sequence	sequence	NOUN
ejpam-4753	46	46	(	(	PUNCT
ejpam-4753	46	47	ch	ch	NOUN
ejpam-4753	46	48	◦	◦	NOUN
ejpam-4753	46	49	s−1	s−1	PROPN
ejpam-4753	46	50	h	h	NOUN
ejpam-4753	46	51	)	)	PUNCT
ejpam-4753	46	52	(	(	PUNCT
ejpam-4753	46	53	m	m	NOUN
ejpam-4753	46	54	)	)	PUNCT
ejpam-4753	46	55	=	=	SYM
ejpam-4753	47	1	(	(	PUNCT
ejpam-4753	47	2	s	s	NOUN
ejpam-4753	47	3	−1	−1	NOUN
ejpam-4753	47	4	h	h	NOUN
ejpam-4753	47	5	m)∗	m)∗	VERB
ejpam-4753	47	6	to	to	ADP
ejpam-4753	47	7	a	a	DET
ejpam-4753	47	8	graded	grade	VERB
ejpam-4753	47	9	a−module	a−module	ADP
ejpam-4753	47	10	m	m	NOUN
ejpam-4753	47	11	and	and	CCONJ
ejpam-4753	47	12	for	for	ADP
ejpam-4753	47	13	all	all	DET
ejpam-4753	47	14	graded	grade	VERB
ejpam-4753	47	15	morphism	morphism	NOUN
ejpam-4753	47	16	of	of	ADP
ejpam-4753	47	17	graded	grade	VERB
ejpam-4753	47	18	left	leave	VERB
ejpam-4753	47	19	a−modules	a−module	NOUN
ejpam-4753	47	20	f	f	X
ejpam-4753	47	21	:	:	PUNCT
ejpam-4753	47	22	m	m	VERB
ejpam-4753	47	23	−→	−→	ADJ
ejpam-4753	47	24	n	n	PRON
ejpam-4753	47	25	of	of	ADP
ejpam-4753	47	26	degree	degree	NOUN
ejpam-4753	48	1	k	k	NOUN
ejpam-4753	48	2	we	we	PRON
ejpam-4753	48	3	correspond	correspond	VERB
ejpam-4753	48	4	the	the	DET
ejpam-4753	48	5	associate	associate	ADJ
ejpam-4753	48	6	complex	complex	NOUN
ejpam-4753	48	7	chain	chain	NOUN
ejpam-4753	48	8	(	(	PUNCT
ejpam-4753	48	9	ch	ch	NOUN
ejpam-4753	48	10	◦	◦	NOUN
ejpam-4753	48	11	s−1	s−1	PROPN
ejpam-4753	48	12	h	h	NOUN
ejpam-4753	48	13	)	)	PUNCT
ejpam-4753	48	14	(	(	PUNCT
ejpam-4753	48	15	f	f	X
ejpam-4753	48	16	)	)	PUNCT
ejpam-4753	48	17	=	=	SYM
ejpam-4753	49	1	(	(	PUNCT
ejpam-4753	49	2	s	s	AUX
ejpam-4753	49	3	−1	−1	NOUN
ejpam-4753	49	4	h	h	NOUN
ejpam-4753	49	5	f)k∗	f)k∗	ADJ
ejpam-4753	49	6	to	to	ADP
ejpam-4753	49	7	a	a	DET
ejpam-4753	49	8	morphism	morphism	NOUN
ejpam-4753	49	9	of	of	ADP
ejpam-4753	49	10	graded	grade	VERB
ejpam-4753	49	11	left	leave	VERB
ejpam-4753	49	12	a−module	a−module	ADP
ejpam-4753	49	13	f	f	X
ejpam-4753	49	14	:	:	PUNCT
ejpam-4753	49	15	m	m	VERB
ejpam-4753	49	16	−→	−→	ADJ
ejpam-4753	50	1	n	n	NOUN
ejpam-4753	50	2	is	be	AUX
ejpam-4753	50	3	additively	additively	ADV
ejpam-4753	50	4	exact	exact	ADJ
ejpam-4753	50	5	covariant	covariant	PROPN
ejpam-4753	50	6	functor	functor	PROPN
ejpam-4753	50	7	.	.	PUNCT
ejpam-4753	51	1	(	(	PUNCT
ejpam-4753	51	2	v	v	X
ejpam-4753	51	3	)	)	PUNCT
ejpam-4753	51	4	we	we	PRON
ejpam-4753	51	5	have	have	VERB
ejpam-4753	51	6	the	the	DET
ejpam-4753	51	7	composed	compose	VERB
ejpam-4753	51	8	functor	functor	NOUN
ejpam-4753	52	1	hn	hn	PROPN
ejpam-4753	52	2	=	=	PUNCT
ejpam-4753	52	3	hn	hn	PROPN
ejpam-4753	52	4	◦	◦	NOUN
ejpam-4753	52	5	c	c	X
ejpam-4753	52	6	,	,	PUNCT
ejpam-4753	52	7	hn	hn	PROPN
ejpam-4753	52	8	:	:	PUNCT
ejpam-4753	52	9	gr(a	gr(a	NOUN
ejpam-4753	52	10	−	−	PROPN
ejpam-4753	52	11	mod	mod	PROPN
ejpam-4753	52	12	)	)	PUNCT
ejpam-4753	52	13	−→	−→	NOUN
ejpam-4753	52	14	gr(a	gr(a	PUNCT
ejpam-4753	52	15	−	−	PROPN
ejpam-4753	52	16	mod	mod	PROPN
ejpam-4753	52	17	)	)	PUNCT
ejpam-4753	52	18	.	.	PUNCT
ejpam-4753	53	1	with	with	ADP
ejpam-4753	53	2	c	c	PROPN
ejpam-4753	53	3	(	(	PUNCT
ejpam-4753	53	4	)	)	PUNCT
ejpam-4753	53	5	:	:	PUNCT
ejpam-4753	53	6	gr(a	gr(a	PUNCT
ejpam-4753	53	7	−	−	PROPN
ejpam-4753	53	8	mod	mod	ADJ
ejpam-4753	53	9	)	)	PUNCT
ejpam-4753	53	10	−→	−→	NOUN
ejpam-4753	53	11	comp	comp	NOUN
ejpam-4753	53	12	(	(	PUNCT
ejpam-4753	53	13	gr(a	gr(a	NOUN
ejpam-4753	53	14	−	−	PROPN
ejpam-4753	53	15	mod	mod	PROPN
ejpam-4753	53	16	)	)	PUNCT
ejpam-4753	53	17	)	)	PUNCT
ejpam-4753	53	18	and	and	CCONJ
ejpam-4753	53	19	hn	hn	PRON
ejpam-4753	53	20	:	:	PUNCT
ejpam-4753	53	21	comp	comp	NOUN
ejpam-4753	53	22	(	(	PUNCT
ejpam-4753	53	23	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	53	24	)	)	PUNCT
ejpam-4753	53	25	)	)	PUNCT
ejpam-4753	53	26	−→	−→	NOUN
ejpam-4753	53	27	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	53	28	)	)	PUNCT
ejpam-4753	53	29	.	.	PUNCT
ejpam-4753	54	1	(	(	PUNCT
ejpam-4753	54	2	vi	vi	X
ejpam-4753	54	3	)	)	PUNCT
ejpam-4753	54	4	for	for	ADP
ejpam-4753	54	5	all	all	DET
ejpam-4753	54	6	n	n	PRON
ejpam-4753	54	7	∈	∈	PROPN
ejpam-4753	54	8	z	z	NOUN
ejpam-4753	54	9	fixed	fix	VERB
ejpam-4753	54	10	and	and	CCONJ
ejpam-4753	54	11	for	for	ADP
ejpam-4753	54	12	all	all	DET
ejpam-4753	54	13	m	m	NOUN
ejpam-4753	54	14	∈	∈	NOUN
ejpam-4753	54	15	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	54	16	)	)	PUNCT
ejpam-4753	54	17	we	we	PRON
ejpam-4753	54	18	have	have	VERB
ejpam-4753	54	19	:	:	PUNCT
ejpam-4753	54	20	s	s	VERB
ejpam-4753	55	1	−1	−1	NOUN
ejpam-4753	55	2	h	h	NOUN
ejpam-4753	55	3	(	(	PUNCT
ejpam-4753	55	4	(	(	PUNCT
ejpam-4753	55	5	hn	hn	PROPN
ejpam-4753	55	6	◦	◦	NOUN
ejpam-4753	55	7	c)(m	c)(m	VERB
ejpam-4753	55	8	)	)	PUNCT
ejpam-4753	55	9	)	)	PUNCT
ejpam-4753	56	1	∼=	∼=	PROPN
ejpam-4753	56	2	hn(ch	hn(ch	NOUN
ejpam-4753	56	3	◦	◦	VERB
ejpam-4753	56	4	s−1	s−1	PROPN
ejpam-4753	56	5	h	h	NOUN
ejpam-4753	56	6	)	)	PUNCT
ejpam-4753	56	7	(	(	PUNCT
ejpam-4753	56	8	m	m	NOUN
ejpam-4753	56	9	)	)	PUNCT
ejpam-4753	56	10	)	)	PUNCT
ejpam-4753	56	11	.	.	PUNCT
ejpam-4753	57	1	a.	a.	PROPN
ejpam-4753	57	2	o.	o.	PROPN
ejpam-4753	57	3	chbih	chbih	PROPN
ejpam-4753	57	4	,	,	PUNCT
ejpam-4753	57	5	m.	m.	PROPN
ejpam-4753	57	6	b.	b.	PROPN
ejpam-4753	57	7	maaouia	maaouia	PROPN
ejpam-4753	57	8	,	,	PUNCT
ejpam-4753	57	9	m.	m.	NOUN
ejpam-4753	57	10	sanghare	sanghare	PROPN
ejpam-4753	57	11	/	/	SYM
ejpam-4753	57	12	eur	eur	PROPN
ejpam-4753	57	13	.	.	PUNCT
ejpam-4753	58	1	j.	j.	PROPN
ejpam-4753	58	2	pure	pure	PROPN
ejpam-4753	58	3	appl	appl	PROPN
ejpam-4753	58	4	.	.	PROPN
ejpam-4753	58	5	math	math	PROPN
ejpam-4753	58	6	,	,	PUNCT
ejpam-4753	58	7	16	16	NUM
ejpam-4753	58	8	(	(	PUNCT
ejpam-4753	58	9	3	3	NUM
ejpam-4753	58	10	)	)	PUNCT
ejpam-4753	58	11	(	(	PUNCT
ejpam-4753	58	12	2023	2023	NUM
ejpam-4753	58	13	)	)	PUNCT
ejpam-4753	58	14	,	,	PUNCT
ejpam-4753	58	15	1913	1913	NUM
ejpam-4753	58	16	-	-	SYM
ejpam-4753	58	17	1939	1939	NUM
ejpam-4753	58	18	1916	1916	NUM
ejpam-4753	58	19	2	2	NUM
ejpam-4753	58	20	.	.	X
ejpam-4753	58	21	reminder	reminder	NOUN
ejpam-4753	58	22	and	and	CCONJ
ejpam-4753	58	23	preliminary	preliminary	ADJ
ejpam-4753	58	24	results	result	NOUN
ejpam-4753	58	25	definition	definition	NOUN
ejpam-4753	58	26	1	1	NUM
ejpam-4753	58	27	.	.	PUNCT
ejpam-4753	59	1	let	let	VERB
ejpam-4753	59	2	a	a	PRON
ejpam-4753	59	3	be	be	AUX
ejpam-4753	59	4	a	a	DET
ejpam-4753	59	5	ring	ring	NOUN
ejpam-4753	59	6	,	,	PUNCT
ejpam-4753	59	7	then	then	ADV
ejpam-4753	59	8	we	we	PRON
ejpam-4753	59	9	say	say	VERB
ejpam-4753	59	10	that	that	SCONJ
ejpam-4753	59	11	a	a	PRON
ejpam-4753	59	12	is	be	AUX
ejpam-4753	59	13	a	a	DET
ejpam-4753	59	14	graded	grade	VERB
ejpam-4753	59	15	ring	ring	NOUN
ejpam-4753	59	16	if	if	SCONJ
ejpam-4753	59	17	there	there	PRON
ejpam-4753	59	18	exists	exist	VERB
ejpam-4753	59	19	a	a	DET
ejpam-4753	59	20	suite	suite	NOUN
ejpam-4753	59	21	(	(	PUNCT
ejpam-4753	59	22	an)n∈z	an)n∈z	NUM
ejpam-4753	59	23	of	of	ADP
ejpam-4753	59	24	additive	additive	ADJ
ejpam-4753	59	25	subgroups	subgroup	NOUN
ejpam-4753	59	26	of	of	ADP
ejpam-4753	59	27	a	a	DET
ejpam-4753	59	28	such	such	ADJ
ejpam-4753	59	29	that	that	PRON
ejpam-4753	59	30	(	(	PUNCT
ejpam-4753	59	31	i	i	NOUN
ejpam-4753	59	32	)	)	PUNCT
ejpam-4753	59	33	a	a	DET
ejpam-4753	59	34	=	=	PROPN
ejpam-4753	59	35	⊕	⊕	PROPN
ejpam-4753	59	36	n∈z	n∈z	VERB
ejpam-4753	59	37	an	an	PRON
ejpam-4753	59	38	;	;	PUNCT
ejpam-4753	59	39	(	(	PUNCT
ejpam-4753	59	40	ii	ii	NOUN
ejpam-4753	59	41	)	)	PUNCT
ejpam-4753	59	42	an	an	DET
ejpam-4753	59	43	·	·	PUNCT
ejpam-4753	59	44	am	am	NOUN
ejpam-4753	59	45	⊂	⊂	PROPN
ejpam-4753	59	46	an+m	an+m	PROPN
ejpam-4753	59	47	,	,	PUNCT
ejpam-4753	59	48	∀	∀	X
ejpam-4753	59	49	n	n	CCONJ
ejpam-4753	59	50	,	,	PUNCT
ejpam-4753	59	51	m	m	PROPN
ejpam-4753	59	52	∈	∈	PROPN
ejpam-4753	59	53	z.	z.	PROPN
ejpam-4753	59	54	definition	definition	NOUN
ejpam-4753	59	55	2	2	NUM
ejpam-4753	59	56	.	.	PUNCT
ejpam-4753	60	1	let	let	VERB
ejpam-4753	60	2	a	a	DET
ejpam-4753	60	3	be	be	AUX
ejpam-4753	60	4	a	a	DET
ejpam-4753	60	5	graded	grade	VERB
ejpam-4753	60	6	ring	ring	NOUN
ejpam-4753	60	7	,	,	PUNCT
ejpam-4753	60	8	and	and	CCONJ
ejpam-4753	60	9	x	x	ADJ
ejpam-4753	60	10	be	be	AUX
ejpam-4753	60	11	a	a	DET
ejpam-4753	60	12	non	non	ADJ
ejpam-4753	60	13	-	-	ADJ
ejpam-4753	60	14	zero	zero	NUM
ejpam-4753	60	15	element	element	NOUN
ejpam-4753	60	16	of	of	ADP
ejpam-4753	60	17	a.	a.	NOUN
ejpam-4753	60	18	then	then	ADV
ejpam-4753	60	19	we	we	PRON
ejpam-4753	60	20	say	say	VERB
ejpam-4753	60	21	that	that	SCONJ
ejpam-4753	60	22	x	x	PRON
ejpam-4753	60	23	is	be	AUX
ejpam-4753	60	24	homogeneous	homogeneous	ADJ
ejpam-4753	60	25	of	of	ADP
ejpam-4753	60	26	degree	degree	NOUN
ejpam-4753	60	27	n	n	CCONJ
ejpam-4753	60	28	,	,	PUNCT
ejpam-4753	60	29	if	if	SCONJ
ejpam-4753	60	30	there	there	PRON
ejpam-4753	60	31	exist	exist	VERB
ejpam-4753	60	32	n	n	PRON
ejpam-4753	60	33	such	such	ADJ
ejpam-4753	60	34	that	that	SCONJ
ejpam-4753	60	35	x	x	SYM
ejpam-4753	60	36	∈	∈	PROPN
ejpam-4753	60	37	an	an	PRON
ejpam-4753	61	1	and	and	CCONJ
ejpam-4753	61	2	we	we	PRON
ejpam-4753	61	3	note	note	VERB
ejpam-4753	61	4	deg(x	deg(x	ADV
ejpam-4753	61	5	)	)	PUNCT
ejpam-4753	61	6	=	=	VERB
ejpam-4753	62	1	n.	n.	NOUN
ejpam-4753	62	2	in	in	ADP
ejpam-4753	62	3	all	all	PRON
ejpam-4753	62	4	that	that	PRON
ejpam-4753	62	5	follows	follow	VERB
ejpam-4753	62	6	,	,	PUNCT
ejpam-4753	62	7	a	a	PRON
ejpam-4753	62	8	and	and	CCONJ
ejpam-4753	62	9	m	m	VERB
ejpam-4753	62	10	are	be	AUX
ejpam-4753	62	11	supposed	suppose	VERB
ejpam-4753	62	12	unitary	unitary	ADJ
ejpam-4753	62	13	.	.	PUNCT
ejpam-4753	63	1	definition	definition	NOUN
ejpam-4753	63	2	3	3	NUM
ejpam-4753	63	3	.	.	PUNCT
ejpam-4753	64	1	let	let	VERB
ejpam-4753	64	2	a	a	DET
ejpam-4753	64	3	=	=	SYM
ejpam-4753	64	4	⊕	⊕	PROPN
ejpam-4753	64	5	n∈z	n∈z	VERB
ejpam-4753	64	6	an	an	DET
ejpam-4753	64	7	be	be	AUX
ejpam-4753	64	8	a	a	DET
ejpam-4753	64	9	graded	grade	VERB
ejpam-4753	64	10	ring	ring	NOUN
ejpam-4753	64	11	and	and	CCONJ
ejpam-4753	64	12	m	m	AUX
ejpam-4753	64	13	be	be	AUX
ejpam-4753	64	14	a	a	DET
ejpam-4753	64	15	left	left	ADJ
ejpam-4753	64	16	a−module	a−module	ADP
ejpam-4753	64	17	,	,	PUNCT
ejpam-4753	64	18	we	we	PRON
ejpam-4753	64	19	say	say	VERB
ejpam-4753	64	20	that	that	SCONJ
ejpam-4753	64	21	m	m	PROPN
ejpam-4753	64	22	is	be	AUX
ejpam-4753	64	23	a	a	DET
ejpam-4753	64	24	graded	grade	VERB
ejpam-4753	64	25	left	leave	VERB
ejpam-4753	64	26	a−module	a−module	ADP
ejpam-4753	64	27	if	if	SCONJ
ejpam-4753	64	28	there	there	PRON
ejpam-4753	64	29	exists	exist	VERB
ejpam-4753	64	30	a	a	DET
ejpam-4753	64	31	suite	suite	NOUN
ejpam-4753	64	32	(	(	PUNCT
ejpam-4753	64	33	mn)n∈z	mn)n∈z	NUM
ejpam-4753	64	34	of	of	ADP
ejpam-4753	64	35	sub	sub	NOUN
ejpam-4753	64	36	-	-	NOUN
ejpam-4753	64	37	groups	group	NOUN
ejpam-4753	64	38	of	of	ADP
ejpam-4753	64	39	m	m	PRON
ejpam-4753	64	40	such	such	ADJ
ejpam-4753	64	41	that	that	SCONJ
ejpam-4753	64	42	:	:	PUNCT
ejpam-4753	64	43	(	(	PUNCT
ejpam-4753	64	44	i	i	NOUN
ejpam-4753	64	45	)	)	PUNCT
ejpam-4753	64	46	m	m	VERB
ejpam-4753	64	47	=	=	PROPN
ejpam-4753	64	48	⊕	⊕	PROPN
ejpam-4753	64	49	n∈z	n∈z	PROPN
ejpam-4753	64	50	mn	mn	PROPN
ejpam-4753	64	51	;	;	PUNCT
ejpam-4753	64	52	(	(	PUNCT
ejpam-4753	64	53	ii	ii	NOUN
ejpam-4753	64	54	)	)	PUNCT
ejpam-4753	64	55	an	an	DET
ejpam-4753	64	56	·	·	SYM
ejpam-4753	64	57	md	md	PROPN
ejpam-4753	64	58	⊂	⊂	PROPN
ejpam-4753	64	59	mn+d	mn+d	PROPN
ejpam-4753	64	60	,	,	PUNCT
ejpam-4753	64	61	∀	∀	X
ejpam-4753	64	62	n	n	CCONJ
ejpam-4753	64	63	,	,	PUNCT
ejpam-4753	64	64	d	d	PROPN
ejpam-4753	64	65	∈	∈	PROPN
ejpam-4753	64	66	z.	z.	PROPN
ejpam-4753	64	67	definition	definition	NOUN
ejpam-4753	64	68	4	4	X
ejpam-4753	64	69	.	.	PUNCT
ejpam-4753	65	1	let	let	VERB
ejpam-4753	65	2	a	a	DET
ejpam-4753	65	3	=	=	SYM
ejpam-4753	65	4	⊕	⊕	PROPN
ejpam-4753	65	5	n∈z	n∈z	VERB
ejpam-4753	65	6	an	an	DET
ejpam-4753	65	7	be	be	AUX
ejpam-4753	65	8	a	a	DET
ejpam-4753	65	9	graded	grade	VERB
ejpam-4753	65	10	ring	ring	NOUN
ejpam-4753	65	11	,	,	PUNCT
ejpam-4753	65	12	m	m	VERB
ejpam-4753	65	13	=	=	ADJ
ejpam-4753	65	14	⊕	⊕	PROPN
ejpam-4753	66	1	n∈z	n∈z	ADJ
ejpam-4753	66	2	mn	mn	PROPN
ejpam-4753	66	3	be	be	AUX
ejpam-4753	66	4	a	a	DET
ejpam-4753	66	5	graded	grade	VERB
ejpam-4753	66	6	left	leave	VERB
ejpam-4753	66	7	a−module	a−module	NOUN
ejpam-4753	66	8	and	and	CCONJ
ejpam-4753	66	9	n	n	PRON
ejpam-4753	66	10	is	be	AUX
ejpam-4753	66	11	a	a	DET
ejpam-4753	66	12	sub	sub	NOUN
ejpam-4753	66	13	-	-	NOUN
ejpam-4753	66	14	module	module	NOUN
ejpam-4753	66	15	of	of	ADP
ejpam-4753	66	16	m	m	PROPN
ejpam-4753	66	17	,	,	PUNCT
ejpam-4753	66	18	then	then	ADV
ejpam-4753	66	19	we	we	PRON
ejpam-4753	66	20	say	say	VERB
ejpam-4753	66	21	that	that	SCONJ
ejpam-4753	66	22	n	n	PRON
ejpam-4753	66	23	is	be	AUX
ejpam-4753	66	24	a	a	DET
ejpam-4753	66	25	graded	grade	VERB
ejpam-4753	66	26	sub	sub	NOUN
ejpam-4753	66	27	-	-	NOUN
ejpam-4753	66	28	module	module	NOUN
ejpam-4753	66	29	of	of	ADP
ejpam-4753	66	30	m	m	PRON
ejpam-4753	66	31	,	,	PUNCT
ejpam-4753	66	32	if	if	SCONJ
ejpam-4753	66	33	∀x	∀x	NUM
ejpam-4753	66	34	∈	∈	PROPN
ejpam-4753	66	35	n	n	PRON
ejpam-4753	66	36	such	such	ADJ
ejpam-4753	66	37	that	that	SCONJ
ejpam-4753	66	38	x	x	PUNCT
ejpam-4753	66	39	=	=	PUNCT
ejpam-4753	66	40	∑	∑	PROPN
ejpam-4753	66	41	n∈z	n∈z	PROPN
ejpam-4753	66	42	xn	xn	PROPN
ejpam-4753	66	43	,	,	PUNCT
ejpam-4753	66	44	then	then	ADV
ejpam-4753	66	45	xn	xn	PROPN
ejpam-4753	66	46	∈	∈	PROPN
ejpam-4753	66	47	n	n	PRON
ejpam-4753	66	48	,	,	PUNCT
ejpam-4753	66	49	∀n	∀n	CCONJ
ejpam-4753	66	50	∈	∈	PROPN
ejpam-4753	66	51	z.	z.	PROPN
ejpam-4753	66	52	proposition	proposition	NOUN
ejpam-4753	66	53	1	1	X
ejpam-4753	66	54	.	.	PUNCT
ejpam-4753	67	1	let	let	VERB
ejpam-4753	67	2	a	a	DET
ejpam-4753	67	3	=	=	SYM
ejpam-4753	67	4	⊕	⊕	PROPN
ejpam-4753	67	5	n∈z	n∈z	VERB
ejpam-4753	67	6	an	an	DET
ejpam-4753	67	7	be	be	AUX
ejpam-4753	67	8	a	a	DET
ejpam-4753	67	9	graded	grade	VERB
ejpam-4753	67	10	ring	ring	NOUN
ejpam-4753	67	11	and	and	CCONJ
ejpam-4753	67	12	m	m	NOUN
ejpam-4753	67	13	=	=	PROPN
ejpam-4753	67	14	⊕	⊕	PROPN
ejpam-4753	67	15	n∈z	n∈z	ADJ
ejpam-4753	67	16	mn	mn	PROPN
ejpam-4753	67	17	is	be	AUX
ejpam-4753	67	18	graded	grade	VERB
ejpam-4753	67	19	left	leave	VERB
ejpam-4753	67	20	a−module	a−module	ADP
ejpam-4753	67	21	,	,	PUNCT
ejpam-4753	67	22	then	then	ADV
ejpam-4753	67	23	for	for	SCONJ
ejpam-4753	67	24	all	all	DET
ejpam-4753	67	25	n	n	PRON
ejpam-4753	67	26	∈	∈	PROPN
ejpam-4753	67	27	z	z	NOUN
ejpam-4753	67	28	fixed	fix	VERB
ejpam-4753	67	29	,	,	PUNCT
ejpam-4753	67	30	we	we	PRON
ejpam-4753	67	31	have	have	VERB
ejpam-4753	67	32	m(n	m(n	NOUN
ejpam-4753	67	33	)	)	PUNCT
ejpam-4753	68	1	=	=	SYM
ejpam-4753	68	2	⊕	⊕	PROPN
ejpam-4753	68	3	k≥n	k≥n	PROPN
ejpam-4753	68	4	mk	mk	PROPN
ejpam-4753	68	5	is	be	AUX
ejpam-4753	68	6	a	a	DET
ejpam-4753	68	7	graded	grade	VERB
ejpam-4753	68	8	sub	sub	NOUN
ejpam-4753	68	9	-	-	NOUN
ejpam-4753	68	10	module	module	NOUN
ejpam-4753	68	11	of	of	ADP
ejpam-4753	68	12	m	m	PRON
ejpam-4753	68	13	and	and	CCONJ
ejpam-4753	68	14	we	we	PRON
ejpam-4753	68	15	have	have	VERB
ejpam-4753	68	16	the	the	DET
ejpam-4753	68	17	descendant	descendant	ADJ
ejpam-4753	68	18	sequence	sequence	NOUN
ejpam-4753	68	19	:	:	PUNCT
ejpam-4753	68	20	·	·	PUNCT
ejpam-4753	68	21	·	·	PUNCT
ejpam-4753	68	22	·	·	PUNCT
ejpam-4753	68	23	m(n+	m(n+	X
ejpam-4753	68	24	2	2	NUM
ejpam-4753	68	25	)	)	PUNCT
ejpam-4753	68	26	⊂	⊂	PRON
ejpam-4753	68	27	m(n+	m(n+	PROPN
ejpam-4753	69	1	1	1	NUM
ejpam-4753	69	2	)	)	PUNCT
ejpam-4753	69	3	⊂	⊂	PROPN
ejpam-4753	69	4	m(n	m(n	PROPN
ejpam-4753	69	5	)	)	PUNCT
ejpam-4753	69	6	⊂	⊂	X
ejpam-4753	69	7	·	·	PUNCT
ejpam-4753	69	8	·	·	PUNCT
ejpam-4753	69	9	·	·	PUNCT
ejpam-4753	69	10	.	.	PUNCT
ejpam-4753	70	1	proof	proof	NOUN
ejpam-4753	70	2	.	.	PUNCT
ejpam-4753	71	1	for	for	SCONJ
ejpam-4753	71	2	all	all	DET
ejpam-4753	71	3	n	n	PRON
ejpam-4753	71	4	∈	∈	PROPN
ejpam-4753	71	5	z	z	NOUN
ejpam-4753	71	6	fixed	fix	VERB
ejpam-4753	71	7	,	,	PUNCT
ejpam-4753	71	8	m(n	m(n	PROPN
ejpam-4753	71	9	)	)	PUNCT
ejpam-4753	71	10	=	=	SYM
ejpam-4753	71	11	⊕	⊕	PROPN
ejpam-4753	71	12	k≥n	k≥n	PROPN
ejpam-4753	71	13	mk	mk	PROPN
ejpam-4753	71	14	is	be	AUX
ejpam-4753	71	15	a	a	DET
ejpam-4753	71	16	sub	sub	NOUN
ejpam-4753	71	17	-	-	NOUN
ejpam-4753	71	18	group	group	NOUN
ejpam-4753	71	19	of	of	ADP
ejpam-4753	71	20	m	m	PROPN
ejpam-4753	71	21	and	and	CCONJ
ejpam-4753	71	22	as	as	ADP
ejpam-4753	71	23	·	·	PUNCT
ejpam-4753	71	24	m(n)k	m(n)k	NOUN
ejpam-4753	71	25	=	=	PUNCT
ejpam-4753	71	26	as	as	ADP
ejpam-4753	71	27	·	·	PUNCT
ejpam-4753	71	28	mn+k	mn+k	PROPN
ejpam-4753	71	29	⊂	⊂	X
ejpam-4753	71	30	mn+k+s	mn+k+s	X
ejpam-4753	71	31	=	=	SYM
ejpam-4753	71	32	mn+(k+s	mn+(k+s	NOUN
ejpam-4753	71	33	)	)	PUNCT
ejpam-4753	71	34	=	=	SYM
ejpam-4753	71	35	m(n)k+s	m(n)k+s	PROPN
ejpam-4753	71	36	.	.	PUNCT
ejpam-4753	72	1	in	in	ADP
ejpam-4753	72	2	the	the	DET
ejpam-4753	72	3	other	other	ADJ
ejpam-4753	72	4	hand	hand	NOUN
ejpam-4753	72	5	,	,	PUNCT
ejpam-4753	72	6	it	it	PRON
ejpam-4753	72	7	suffices	suffice	VERB
ejpam-4753	72	8	to	to	PART
ejpam-4753	72	9	remark	remark	VERB
ejpam-4753	72	10	that	that	SCONJ
ejpam-4753	72	11	m(n	m(n	NOUN
ejpam-4753	72	12	)	)	PUNCT
ejpam-4753	72	13	=	=	SYM
ejpam-4753	72	14	⊕	⊕	PROPN
ejpam-4753	72	15	k≥n	k≥n	PROPN
ejpam-4753	72	16	mk	mk	PROPN
ejpam-4753	72	17	=	=	PROPN
ejpam-4753	72	18	mn	mn	PROPN
ejpam-4753	72	19	⊕	⊕	PROPN
ejpam-4753	72	20	m(n+	m(n+	PROPN
ejpam-4753	72	21	1	1	NUM
ejpam-4753	72	22	)	)	PUNCT
ejpam-4753	72	23	.	.	PUNCT
ejpam-4753	73	1	hence	hence	ADV
ejpam-4753	73	2	m(n+	m(n+	NOUN
ejpam-4753	73	3	1	1	NUM
ejpam-4753	73	4	)	)	PUNCT
ejpam-4753	73	5	⊂	⊂	PROPN
ejpam-4753	73	6	m(n	m(n	NOUN
ejpam-4753	73	7	)	)	PUNCT
ejpam-4753	73	8	.	.	PUNCT
ejpam-4753	74	1	thus	thus	ADV
ejpam-4753	74	2	·	·	PUNCT
ejpam-4753	74	3	·	·	PUNCT
ejpam-4753	74	4	·	·	PUNCT
ejpam-4753	74	5	m(n+	m(n+	X
ejpam-4753	74	6	2	2	NUM
ejpam-4753	74	7	)	)	PUNCT
ejpam-4753	74	8	⊂	⊂	PRON
ejpam-4753	74	9	m(n+	m(n+	PROPN
ejpam-4753	74	10	1	1	NUM
ejpam-4753	74	11	)	)	PUNCT
ejpam-4753	74	12	⊂	⊂	PROPN
ejpam-4753	74	13	m(n	m(n	PROPN
ejpam-4753	74	14	)	)	PUNCT
ejpam-4753	75	1	⊂	⊂	X
ejpam-4753	75	2	·	·	PUNCT
ejpam-4753	75	3	·	·	PUNCT
ejpam-4753	75	4	·	·	PUNCT
ejpam-4753	75	5	.	.	PUNCT
ejpam-4753	75	6	a.	a.	PROPN
ejpam-4753	75	7	o.	o.	PROPN
ejpam-4753	75	8	chbih	chbih	PROPN
ejpam-4753	75	9	,	,	PUNCT
ejpam-4753	75	10	m.	m.	PROPN
ejpam-4753	75	11	b.	b.	PROPN
ejpam-4753	75	12	maaouia	maaouia	PROPN
ejpam-4753	75	13	,	,	PUNCT
ejpam-4753	75	14	m.	m.	NOUN
ejpam-4753	75	15	sanghare	sanghare	PROPN
ejpam-4753	75	16	/	/	SYM
ejpam-4753	75	17	eur	eur	PROPN
ejpam-4753	75	18	.	.	PUNCT
ejpam-4753	76	1	j.	j.	PROPN
ejpam-4753	76	2	pure	pure	PROPN
ejpam-4753	76	3	appl	appl	PROPN
ejpam-4753	76	4	.	.	PROPN
ejpam-4753	76	5	math	math	PROPN
ejpam-4753	76	6	,	,	PUNCT
ejpam-4753	76	7	16	16	NUM
ejpam-4753	76	8	(	(	PUNCT
ejpam-4753	76	9	3	3	NUM
ejpam-4753	76	10	)	)	PUNCT
ejpam-4753	76	11	(	(	PUNCT
ejpam-4753	76	12	2023	2023	NUM
ejpam-4753	76	13	)	)	PUNCT
ejpam-4753	76	14	,	,	PUNCT
ejpam-4753	76	15	1913	1913	NUM
ejpam-4753	76	16	-	-	SYM
ejpam-4753	76	17	1939	1939	NUM
ejpam-4753	76	18	1917	1917	NUM
ejpam-4753	76	19	definition	definition	NOUN
ejpam-4753	76	20	5	5	NUM
ejpam-4753	76	21	.	.	PUNCT
ejpam-4753	77	1	let	let	VERB
ejpam-4753	77	2	a	a	DET
ejpam-4753	77	3	=	=	PROPN
ejpam-4753	77	4	⊕	⊕	PROPN
ejpam-4753	77	5	n∈zan	n∈zan	PROPN
ejpam-4753	77	6	be	be	AUX
ejpam-4753	77	7	a	a	DET
ejpam-4753	77	8	graded	grade	VERB
ejpam-4753	77	9	ring	ring	NOUN
ejpam-4753	77	10	,	,	PUNCT
ejpam-4753	77	11	m	m	VERB
ejpam-4753	77	12	=	=	ADJ
ejpam-4753	77	13	⊕	⊕	PROPN
ejpam-4753	77	14	n∈z	n∈z	VERB
ejpam-4753	77	15	mn	mn	PROPN
ejpam-4753	77	16	and	and	CCONJ
ejpam-4753	77	17	n	n	PROPN
ejpam-4753	77	18	=	=	PROPN
ejpam-4753	77	19	⊕	⊕	PROPN
ejpam-4753	77	20	n∈z	n∈z	VERB
ejpam-4753	77	21	nn	nn	PROPN
ejpam-4753	77	22	two	two	NUM
ejpam-4753	77	23	graded	grade	VERB
ejpam-4753	77	24	left	leave	VERB
ejpam-4753	77	25	a−modules	a−module	NOUN
ejpam-4753	77	26	and	and	CCONJ
ejpam-4753	77	27	f	f	X
ejpam-4753	77	28	:	:	PUNCT
ejpam-4753	77	29	m	m	VERB
ejpam-4753	77	30	−→	−→	ADJ
ejpam-4753	78	1	n	n	NOUN
ejpam-4753	78	2	is	be	AUX
ejpam-4753	78	3	a	a	DET
ejpam-4753	78	4	morphism	morphism	NOUN
ejpam-4753	78	5	of	of	ADP
ejpam-4753	78	6	left	left	ADJ
ejpam-4753	78	7	a−modules	a−module	NOUN
ejpam-4753	78	8	,	,	PUNCT
ejpam-4753	78	9	then	then	ADV
ejpam-4753	78	10	we	we	PRON
ejpam-4753	78	11	say	say	VERB
ejpam-4753	78	12	that	that	SCONJ
ejpam-4753	78	13	f	f	PROPN
ejpam-4753	78	14	is	be	AUX
ejpam-4753	78	15	a	a	DET
ejpam-4753	78	16	graded	grade	VERB
ejpam-4753	78	17	morphism	morphism	NOUN
ejpam-4753	78	18	of	of	ADP
ejpam-4753	78	19	degree	degree	NOUN
ejpam-4753	78	20	k	k	PROPN
ejpam-4753	78	21	∈	∈	PROPN
ejpam-4753	78	22	z	z	NOUN
ejpam-4753	79	1	if	if	SCONJ
ejpam-4753	79	2	for	for	ADP
ejpam-4753	79	3	any	any	DET
ejpam-4753	79	4	m	m	NOUN
ejpam-4753	79	5	∈	∈	NOUN
ejpam-4753	79	6	ms	ms	NOUN
ejpam-4753	79	7	then	then	ADV
ejpam-4753	79	8	f(m	f(m	PROPN
ejpam-4753	79	9	)	)	PUNCT
ejpam-4753	79	10	∈	∈	PROPN
ejpam-4753	79	11	ns+k	ns+k	PROPN
ejpam-4753	79	12	.	.	PUNCT
ejpam-4753	79	13	theorem	theorem	NOUN
ejpam-4753	79	14	1	1	NUM
ejpam-4753	79	15	.	.	PUNCT
ejpam-4753	80	1	let	let	VERB
ejpam-4753	80	2	a	a	DET
ejpam-4753	80	3	be	be	AUX
ejpam-4753	80	4	a	a	DET
ejpam-4753	80	5	graded	grade	VERB
ejpam-4753	80	6	ring	ring	NOUN
ejpam-4753	80	7	,	,	PUNCT
ejpam-4753	80	8	then	then	ADV
ejpam-4753	80	9	the	the	DET
ejpam-4753	80	10	following	follow	VERB
ejpam-4753	80	11	information	information	NOUN
ejpam-4753	80	12	:	:	PUNCT
ejpam-4753	80	13	(	(	PUNCT
ejpam-4753	80	14	i	i	NOUN
ejpam-4753	80	15	)	)	PUNCT
ejpam-4753	80	16	the	the	DET
ejpam-4753	80	17	class	class	NOUN
ejpam-4753	80	18	of	of	ADP
ejpam-4753	80	19	objects	object	NOUN
ejpam-4753	80	20	are	be	AUX
ejpam-4753	80	21	the	the	DET
ejpam-4753	80	22	graded	grade	VERB
ejpam-4753	80	23	left	leave	VERB
ejpam-4753	80	24	a−modules	a−module	NOUN
ejpam-4753	80	25	;	;	PUNCT
ejpam-4753	80	26	(	(	PUNCT
ejpam-4753	80	27	ii	ii	NOUN
ejpam-4753	80	28	)	)	PUNCT
ejpam-4753	80	29	the	the	DET
ejpam-4753	80	30	class	class	NOUN
ejpam-4753	80	31	of	of	ADP
ejpam-4753	80	32	morphisms	morphism	NOUN
ejpam-4753	80	33	are	be	AUX
ejpam-4753	80	34	the	the	DET
ejpam-4753	80	35	graded	grade	VERB
ejpam-4753	80	36	morphisms	morphism	NOUN
ejpam-4753	80	37	of	of	ADP
ejpam-4753	80	38	degree	degree	NOUN
ejpam-4753	80	39	k	k	PROPN
ejpam-4753	80	40	∈	∈	PROPN
ejpam-4753	80	41	z.	z.	PROPN
ejpam-4753	80	42	constitute	constitute	VERB
ejpam-4753	80	43	a	a	DET
ejpam-4753	80	44	category	category	NOUN
ejpam-4753	80	45	called	call	VERB
ejpam-4753	80	46	the	the	DET
ejpam-4753	80	47	category	category	NOUN
ejpam-4753	80	48	of	of	ADP
ejpam-4753	80	49	graded	grade	VERB
ejpam-4753	80	50	left	leave	VERB
ejpam-4753	80	51	a−module	a−module	NOUN
ejpam-4753	80	52	and	and	CCONJ
ejpam-4753	80	53	it	it	PRON
ejpam-4753	80	54	is	be	AUX
ejpam-4753	80	55	denoted	denote	VERB
ejpam-4753	80	56	by	by	ADP
ejpam-4753	80	57	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	80	58	)	)	PUNCT
ejpam-4753	80	59	.	.	PUNCT
ejpam-4753	81	1	proof	proof	NOUN
ejpam-4753	81	2	.	.	PUNCT
ejpam-4753	82	1	see	see	VERB
ejpam-4753	82	2	[	[	X
ejpam-4753	82	3	3	3	NUM
ejpam-4753	82	4	]	]	PUNCT
ejpam-4753	82	5	definition	definition	NOUN
ejpam-4753	82	6	6	6	NUM
ejpam-4753	82	7	.	.	PUNCT
ejpam-4753	83	1	a	a	DET
ejpam-4753	83	2	complex	complex	ADJ
ejpam-4753	83	3	sequence	sequence	NOUN
ejpam-4753	83	4	(	(	PUNCT
ejpam-4753	83	5	c	c	X
ejpam-4753	83	6	,	,	PUNCT
ejpam-4753	83	7	d	d	NOUN
ejpam-4753	83	8	)	)	PUNCT
ejpam-4753	83	9	:	:	PUNCT
ejpam-4753	83	10	.	.	PUNCT
ejpam-4753	83	11	.	.	PUNCT
ejpam-4753	83	12	.	.	PUNCT
ejpam-4753	84	1	→	→	PUNCT
ejpam-4753	84	2	cn+1	cn+1	NUM
ejpam-4753	84	3	dn+1→	dn+1→	NOUN
ejpam-4753	84	4	cn	cn	PROPN
ejpam-4753	84	5	dn→	dn→	PROPN
ejpam-4753	84	6	cn−1	cn−1	PROPN
ejpam-4753	84	7	dn−1→	dn−1→	PROPN
ejpam-4753	84	8	.	.	PUNCT
ejpam-4753	84	9	.	.	PUNCT
ejpam-4753	84	10	.	.	PUNCT
ejpam-4753	84	11	is	be	AUX
ejpam-4753	84	12	a	a	DET
ejpam-4753	84	13	sequence	sequence	NOUN
ejpam-4753	84	14	of	of	ADP
ejpam-4753	84	15	morphisms	morphism	NOUN
ejpam-4753	84	16	of	of	ADP
ejpam-4753	84	17	a−	a−	NOUN
ejpam-4753	84	18	modules	module	NOUN
ejpam-4753	84	19	satisfying	satisfy	VERB
ejpam-4753	84	20	dn	dn	ADP
ejpam-4753	84	21	◦	◦	NOUN
ejpam-4753	84	22	dn+1	dn+1	NOUN
ejpam-4753	84	23	=	=	SYM
ejpam-4753	84	24	0	0	NUM
ejpam-4753	84	25	,	,	PUNCT
ejpam-4753	84	26	for	for	ADP
ejpam-4753	84	27	all	all	DET
ejpam-4753	84	28	n	n	PRON
ejpam-4753	84	29	∈	∈	PROPN
ejpam-4753	84	30	z.	z.	NOUN
ejpam-4753	84	31	definition	definition	NOUN
ejpam-4753	84	32	7	7	NUM
ejpam-4753	84	33	.	.	PUNCT
ejpam-4753	85	1	a	a	DET
ejpam-4753	85	2	complex	complex	ADJ
ejpam-4753	85	3	chain	chain	NOUN
ejpam-4753	85	4	f	f	NOUN
ejpam-4753	85	5	:	:	PUNCT
ejpam-4753	85	6	(	(	PUNCT
ejpam-4753	85	7	c	c	X
ejpam-4753	85	8	,	,	PUNCT
ejpam-4753	85	9	d	d	NOUN
ejpam-4753	85	10	)	)	PUNCT
ejpam-4753	85	11	→	→	SYM
ejpam-4753	85	12	(	(	PUNCT
ejpam-4753	85	13	c	c	NOUN
ejpam-4753	85	14	′	′	NUM
ejpam-4753	85	15	,	,	PUNCT
ejpam-4753	85	16	d′	d′	NUM
ejpam-4753	85	17	)	)	PUNCT
ejpam-4753	85	18	is	be	AUX
ejpam-4753	85	19	a	a	DET
ejpam-4753	85	20	sequence	sequence	NOUN
ejpam-4753	85	21	of	of	ADP
ejpam-4753	85	22	homomorphisms	homomorphism	NOUN
ejpam-4753	85	23	(	(	PUNCT
ejpam-4753	85	24	fn	fn	NOUN
ejpam-4753	85	25	:	:	PUNCT
ejpam-4753	85	26	cn	cn	INTJ
ejpam-4753	86	1	−→	−→	NOUN
ejpam-4753	87	1	c	c	PROPN
ejpam-4753	87	2	′	′	NUM
ejpam-4753	88	1	n)n∈z	n)n∈z	NOUN
ejpam-4753	88	2	of	of	ADP
ejpam-4753	88	3	a−	a−	PROPN
ejpam-4753	88	4	modules	module	NOUN
ejpam-4753	88	5	making	make	VERB
ejpam-4753	88	6	the	the	DET
ejpam-4753	88	7	following	follow	VERB
ejpam-4753	88	8	diagram	diagram	NOUN
ejpam-4753	88	9	commute	commute	NOUN
ejpam-4753	88	10	:	:	PUNCT
ejpam-4753	88	11	(	(	PUNCT
ejpam-4753	88	12	c	c	X
ejpam-4753	88	13	,	,	PUNCT
ejpam-4753	88	14	d	d	NOUN
ejpam-4753	88	15	)	)	PUNCT
ejpam-4753	88	16	:	:	PUNCT
ejpam-4753	88	17	·	·	PUNCT
ejpam-4753	88	18	·	·	PUNCT
ejpam-4753	88	19	·	·	PUNCT
ejpam-4753	89	1	//	//	PUNCT
ejpam-4753	89	2	f	f	PROPN
ejpam-4753	89	3	�	�	PROPN
ejpam-4753	89	4	�	�	PROPN
ejpam-4753	89	5	cn+1	cn+1	NUM
ejpam-4753	89	6	dn+1	dn+1	PROPN
ejpam-4753	89	7	//	//	X
ejpam-4753	89	8	fn+1	fn+1	X
ejpam-4753	89	9	�	�	PROPN
ejpam-4753	89	10	�	�	PROPN
ejpam-4753	89	11	cn	cn	PROPN
ejpam-4753	90	1	dn	dn	PROPN
ejpam-4753	90	2	//	//	PROPN
ejpam-4753	90	3	fn	fn	PROPN
ejpam-4753	90	4	�	�	PROPN
ejpam-4753	90	5	�	�	PROPN
ejpam-4753	90	6	cn−1	cn−1	PROPN
ejpam-4753	90	7	//	//	X
ejpam-4753	90	8	fn−1	fn−1	PROPN
ejpam-4753	90	9	�	�	PROPN
ejpam-4753	90	10	�	�	PROPN
ejpam-4753	90	11	·	·	PUNCT
ejpam-4753	90	12	·	·	PUNCT
ejpam-4753	90	13	·	·	PUNCT
ejpam-4753	91	1	(	(	PUNCT
ejpam-4753	91	2	c	c	NOUN
ejpam-4753	91	3	′	′	NUM
ejpam-4753	91	4	,	,	PUNCT
ejpam-4753	91	5	d′	d′	NUM
ejpam-4753	91	6	)	)	PUNCT
ejpam-4753	91	7	:	:	PUNCT
ejpam-4753	91	8	·	·	PUNCT
ejpam-4753	91	9	·	·	PUNCT
ejpam-4753	91	10	·	·	PUNCT
ejpam-4753	92	1	//	//	PUNCT
ejpam-4753	93	1	c	c	NOUN
ejpam-4753	93	2	′	′	NOUN
ejpam-4753	94	1	n+1	n+1	PROPN
ejpam-4753	94	2	d′n+1	d′n+1	PROPN
ejpam-4753	94	3	//	//	PROPN
ejpam-4753	95	1	c	c	NOUN
ejpam-4753	95	2	′	′	NUM
ejpam-4753	96	1	n	n	PROPN
ejpam-4753	96	2	d′n	d′n	PROPN
ejpam-4753	97	1	//	//	NOUN
ejpam-4753	98	1	c	c	NOUN
ejpam-4753	98	2	′	′	NUM
ejpam-4753	98	3	n−1	n−1	PROPN
ejpam-4753	98	4	//	//	X
ejpam-4753	98	5	·	·	PUNCT
ejpam-4753	98	6	·	·	PUNCT
ejpam-4753	98	7	·	·	PUNCT
ejpam-4753	99	1	i.e	i.e	X
ejpam-4753	99	2	d′n+1	d′n+1	VERB
ejpam-4753	99	3	◦	◦	NOUN
ejpam-4753	99	4	fn+1	fn+1	NOUN
ejpam-4753	99	5	=	=	SYM
ejpam-4753	99	6	fn	fn	NOUN
ejpam-4753	99	7	◦	◦	NOUN
ejpam-4753	99	8	dn+1	dn+1	PROPN
ejpam-4753	99	9	,	,	PUNCT
ejpam-4753	99	10	for	for	ADP
ejpam-4753	99	11	all	all	DET
ejpam-4753	99	12	n	n	PRON
ejpam-4753	99	13	∈	∈	PROPN
ejpam-4753	99	14	z.	z.	NOUN
ejpam-4753	99	15	proposition	proposition	NOUN
ejpam-4753	99	16	2	2	X
ejpam-4753	99	17	.	.	PUNCT
ejpam-4753	100	1	we	we	PRON
ejpam-4753	100	2	called	call	VERB
ejpam-4753	100	3	the	the	DET
ejpam-4753	100	4	category	category	NOUN
ejpam-4753	100	5	of	of	ADP
ejpam-4753	100	6	complexes	complex	NOUN
ejpam-4753	100	7	of	of	ADP
ejpam-4753	100	8	a−modules	a−module	NOUN
ejpam-4753	100	9	and	and	CCONJ
ejpam-4753	100	10	we	we	PRON
ejpam-4753	100	11	denote	denote	VERB
ejpam-4753	100	12	comp	comp	NOUN
ejpam-4753	100	13	,	,	PUNCT
ejpam-4753	100	14	the	the	DET
ejpam-4753	100	15	category	category	NOUN
ejpam-4753	100	16	whose	whose	DET
ejpam-4753	100	17	:	:	PUNCT
ejpam-4753	100	18	(	(	PUNCT
ejpam-4753	100	19	i	i	NOUN
ejpam-4753	100	20	)	)	PUNCT
ejpam-4753	100	21	the	the	DET
ejpam-4753	100	22	objects	object	NOUN
ejpam-4753	100	23	are	be	AUX
ejpam-4753	100	24	the	the	DET
ejpam-4753	100	25	sequences	sequence	NOUN
ejpam-4753	100	26	complex	complex	NOUN
ejpam-4753	100	27	;	;	PUNCT
ejpam-4753	100	28	(	(	PUNCT
ejpam-4753	100	29	ii	ii	NOUN
ejpam-4753	100	30	)	)	PUNCT
ejpam-4753	100	31	the	the	DET
ejpam-4753	100	32	morphisms	morphism	NOUN
ejpam-4753	100	33	are	be	AUX
ejpam-4753	100	34	the	the	DET
ejpam-4753	100	35	complex	complex	ADJ
ejpam-4753	100	36	chains	chain	NOUN
ejpam-4753	100	37	.	.	PUNCT
ejpam-4753	101	1	proof	proof	NOUN
ejpam-4753	101	2	.	.	PUNCT
ejpam-4753	102	1	see	see	VERB
ejpam-4753	102	2	[	[	X
ejpam-4753	102	3	3	3	X
ejpam-4753	102	4	]	]	PUNCT
ejpam-4753	102	5	proposition	proposition	NOUN
ejpam-4753	102	6	3	3	X
ejpam-4753	102	7	.	.	PUNCT
ejpam-4753	103	1	we	we	PRON
ejpam-4753	103	2	called	call	VERB
ejpam-4753	103	3	functor	functor	PROPN
ejpam-4753	103	4	homology	homology	PROPN
ejpam-4753	103	5	hn	hn	PROPN
ejpam-4753	103	6	the	the	DET
ejpam-4753	103	7	functor	functor	PROPN
ejpam-4753	103	8	hn	hn	PROPN
ejpam-4753	103	9	:	:	PUNCT
ejpam-4753	103	10	comp	comp	PROPN
ejpam-4753	103	11	−→	−→	PROPN
ejpam-4753	103	12	ab	ab	PROPN
ejpam-4753	103	13	defined	define	VERB
ejpam-4753	103	14	by	by	ADP
ejpam-4753	103	15	:	:	PUNCT
ejpam-4753	103	16	(	(	PUNCT
ejpam-4753	103	17	i	i	NOUN
ejpam-4753	103	18	)	)	PUNCT
ejpam-4753	103	19	for	for	ADP
ejpam-4753	103	20	all	all	DET
ejpam-4753	103	21	objet	objet	NOUN
ejpam-4753	103	22	(	(	PUNCT
ejpam-4753	103	23	c	c	X
ejpam-4753	103	24	,	,	PUNCT
ejpam-4753	103	25	d	d	NOUN
ejpam-4753	103	26	)	)	PUNCT
ejpam-4753	103	27	of	of	ADP
ejpam-4753	103	28	comp	comp	NOUN
ejpam-4753	103	29	,	,	PUNCT
ejpam-4753	103	30	hn((c	hn((c	PROPN
ejpam-4753	103	31	,	,	PUNCT
ejpam-4753	103	32	d	d	NOUN
ejpam-4753	103	33	)	)	PUNCT
ejpam-4753	103	34	)	)	PUNCT
ejpam-4753	104	1	=	=	PRON
ejpam-4753	104	2	ker	ker	NOUN
ejpam-4753	104	3	dn	dn	PROPN
ejpam-4753	104	4	/	/	SYM
ejpam-4753	104	5	imdn+1	imdn+1	PROPN
ejpam-4753	104	6	(	(	PUNCT
ejpam-4753	104	7	ii	ii	NOUN
ejpam-4753	104	8	)	)	PUNCT
ejpam-4753	104	9	for	for	ADP
ejpam-4753	104	10	all	all	DET
ejpam-4753	104	11	chain	chain	NOUN
ejpam-4753	104	12	f	f	NOUN
ejpam-4753	104	13	:	:	PUNCT
ejpam-4753	104	14	(	(	PUNCT
ejpam-4753	104	15	c	c	X
ejpam-4753	104	16	,	,	PUNCT
ejpam-4753	104	17	d	d	NOUN
ejpam-4753	104	18	)	)	PUNCT
ejpam-4753	104	19	→	→	SYM
ejpam-4753	104	20	(	(	PUNCT
ejpam-4753	104	21	c	c	NOUN
ejpam-4753	104	22	′	′	NUM
ejpam-4753	104	23	,	,	PUNCT
ejpam-4753	104	24	d′	d′	NUM
ejpam-4753	104	25	)	)	PUNCT
ejpam-4753	104	26	of	of	ADP
ejpam-4753	104	27	comp	comp	NOUN
ejpam-4753	104	28	hn(f	hn(f	PUNCT
ejpam-4753	104	29	)	)	PUNCT
ejpam-4753	104	30	:	:	PUNCT
ejpam-4753	105	1	hn((c	hn((c	X
ejpam-4753	105	2	,	,	PUNCT
ejpam-4753	105	3	d	d	NOUN
ejpam-4753	105	4	)	)	PUNCT
ejpam-4753	105	5	)	)	PUNCT
ejpam-4753	106	1	−→	−→	ADJ
ejpam-4753	106	2	hn((c	hn((c	PROPN
ejpam-4753	106	3	′	′	NUM
ejpam-4753	106	4	,	,	PUNCT
ejpam-4753	106	5	d′	d′	NUM
ejpam-4753	106	6	)	)	PUNCT
ejpam-4753	106	7	)	)	PUNCT
ejpam-4753	107	1	zn	zn	PROPN
ejpam-4753	107	2	7−→	7−→	PROPN
ejpam-4753	107	3	fn(zn	fn(zn	PROPN
ejpam-4753	107	4	)	)	PUNCT
ejpam-4753	107	5	a.	a.	NOUN
ejpam-4753	107	6	o.	o.	PROPN
ejpam-4753	107	7	chbih	chbih	PROPN
ejpam-4753	107	8	,	,	PUNCT
ejpam-4753	107	9	m.	m.	PROPN
ejpam-4753	107	10	b.	b.	PROPN
ejpam-4753	107	11	maaouia	maaouia	PROPN
ejpam-4753	107	12	,	,	PUNCT
ejpam-4753	107	13	m.	m.	NOUN
ejpam-4753	107	14	sanghare	sanghare	PROPN
ejpam-4753	107	15	/	/	SYM
ejpam-4753	107	16	eur	eur	PROPN
ejpam-4753	107	17	.	.	PUNCT
ejpam-4753	108	1	j.	j.	PROPN
ejpam-4753	108	2	pure	pure	PROPN
ejpam-4753	108	3	appl	appl	PROPN
ejpam-4753	108	4	.	.	PROPN
ejpam-4753	108	5	math	math	PROPN
ejpam-4753	108	6	,	,	PUNCT
ejpam-4753	108	7	16	16	NUM
ejpam-4753	108	8	(	(	PUNCT
ejpam-4753	108	9	3	3	NUM
ejpam-4753	108	10	)	)	PUNCT
ejpam-4753	108	11	(	(	PUNCT
ejpam-4753	108	12	2023	2023	NUM
ejpam-4753	108	13	)	)	PUNCT
ejpam-4753	108	14	,	,	PUNCT
ejpam-4753	108	15	1913	1913	NUM
ejpam-4753	108	16	-	-	SYM
ejpam-4753	108	17	1939	1939	NUM
ejpam-4753	108	18	1918	1918	NUM
ejpam-4753	108	19	proof	proof	NOUN
ejpam-4753	108	20	.	.	PUNCT
ejpam-4753	109	1	see	see	VERB
ejpam-4753	109	2	[	[	X
ejpam-4753	109	3	3	3	NUM
ejpam-4753	109	4	]	]	X
ejpam-4753	109	5	theorem	theorem	NOUN
ejpam-4753	109	6	2	2	NUM
ejpam-4753	109	7	.	.	PUNCT
ejpam-4753	110	1	let	let	VERB
ejpam-4753	110	2	(	(	PUNCT
ejpam-4753	110	3	0	0	NUM
ejpam-4753	110	4	)	)	PUNCT
ejpam-4753	110	5	−→	−→	NOUN
ejpam-4753	110	6	(	(	PUNCT
ejpam-4753	110	7	(	(	PUNCT
ejpam-4753	110	8	m	m	PROPN
ejpam-4753	110	9	,	,	PUNCT
ejpam-4753	110	10	d	d	NOUN
ejpam-4753	110	11	)	)	PUNCT
ejpam-4753	110	12	)	)	PUNCT
ejpam-4753	111	1	f−→	f−→	NOUN
ejpam-4753	111	2	(	(	PUNCT
ejpam-4753	111	3	(	(	PUNCT
ejpam-4753	111	4	n	n	CCONJ
ejpam-4753	111	5	,	,	PUNCT
ejpam-4753	111	6	d	d	NOUN
ejpam-4753	111	7	′	′	NUM
ejpam-4753	111	8	)	)	PUNCT
ejpam-4753	111	9	)	)	PUNCT
ejpam-4753	111	10	g−→	g−→	NOUN
ejpam-4753	111	11	(	(	PUNCT
ejpam-4753	111	12	(	(	PUNCT
ejpam-4753	111	13	l	l	NOUN
ejpam-4753	111	14	,	,	PUNCT
ejpam-4753	111	15	d	d	PROPN
ejpam-4753	111	16	′′	′′	PROPN
ejpam-4753	111	17	)	)	PUNCT
ejpam-4753	111	18	)	)	PUNCT
ejpam-4753	111	19	−→	−→	NOUN
ejpam-4753	111	20	(	(	PUNCT
ejpam-4753	111	21	0	0	NUM
ejpam-4753	111	22	)	)	PUNCT
ejpam-4753	111	23	be	be	AUX
ejpam-4753	111	24	a	a	DET
ejpam-4753	111	25	short	short	ADJ
ejpam-4753	111	26	exact	exact	ADJ
ejpam-4753	111	27	complex	complex	ADJ
ejpam-4753	111	28	sequence	sequence	NOUN
ejpam-4753	111	29	,	,	PUNCT
ejpam-4753	111	30	then	then	ADV
ejpam-4753	111	31	for	for	ADP
ejpam-4753	111	32	all	all	DET
ejpam-4753	111	33	n	n	PRON
ejpam-4753	111	34	∈	∈	NOUN
ejpam-4753	111	35	z	z	NOUN
ejpam-4753	111	36	there	there	PRON
ejpam-4753	111	37	exist	exist	VERB
ejpam-4753	111	38	a	a	DET
ejpam-4753	111	39	morphism	morphism	NOUN
ejpam-4753	111	40	of	of	ADP
ejpam-4753	111	41	left	left	ADJ
ejpam-4753	111	42	a−module	a−module	ADP
ejpam-4753	111	43	δn	δn	NOUN
ejpam-4753	111	44	:	:	PUNCT
ejpam-4753	111	45	hn((l	hn((l	NOUN
ejpam-4753	111	46	,	,	PUNCT
ejpam-4753	111	47	d	d	NOUN
ejpam-4753	111	48	′	′	NUM
ejpam-4753	111	49	)	)	PUNCT
ejpam-4753	111	50	−→	−→	NOUN
ejpam-4753	111	51	hn−1((m	hn−1((m	NOUN
ejpam-4753	111	52	,	,	PUNCT
ejpam-4753	111	53	d	d	NOUN
ejpam-4753	111	54	)	)	PUNCT
ejpam-4753	111	55	)	)	PUNCT
ejpam-4753	111	56	called	call	VERB
ejpam-4753	111	57	connecting	connect	VERB
ejpam-4753	111	58	morphism	morphism	NOUN
ejpam-4753	111	59	such	such	ADJ
ejpam-4753	111	60	that	that	SCONJ
ejpam-4753	111	61	the	the	DET
ejpam-4753	111	62	following	follow	VERB
ejpam-4753	111	63	long	long	ADJ
ejpam-4753	111	64	exact	exact	ADJ
ejpam-4753	111	65	sequence	sequence	NOUN
ejpam-4753	111	66	is	be	AUX
ejpam-4753	111	67	exact	exact	ADJ
ejpam-4753	111	68	·	·	PUNCT
ejpam-4753	111	69	·	·	PUNCT
ejpam-4753	111	70	·	·	PUNCT
ejpam-4753	112	1	−→	−→	ADJ
ejpam-4753	112	2	hn((m	hn((m	NOUN
ejpam-4753	112	3	,	,	PUNCT
ejpam-4753	112	4	d	d	NOUN
ejpam-4753	112	5	)	)	PUNCT
ejpam-4753	112	6	)	)	PUNCT
ejpam-4753	113	1	hn(f)−→	hn(f)−→	PROPN
ejpam-4753	113	2	hn((n	hn((n	NOUN
ejpam-4753	113	3	,	,	PUNCT
ejpam-4753	113	4	d	d	NOUN
ejpam-4753	113	5	′	′	NUM
ejpam-4753	113	6	)	)	PUNCT
ejpam-4753	113	7	)	)	PUNCT
ejpam-4753	113	8	hn(g)−→	hn(g)−→	NOUN
ejpam-4753	113	9	hn((l	hn((l	VERB
ejpam-4753	113	10	,	,	PUNCT
ejpam-4753	113	11	d	d	PROPN
ejpam-4753	113	12	′′	′′	PROPN
ejpam-4753	113	13	)	)	PUNCT
ejpam-4753	113	14	)	)	PUNCT
ejpam-4753	113	15	δn−→	δn−→	NUM
ejpam-4753	113	16	hn−1(m	hn−1(m	ADJ
ejpam-4753	113	17	,	,	PUNCT
ejpam-4753	113	18	d	d	NOUN
ejpam-4753	113	19	)	)	PUNCT
ejpam-4753	113	20	hn−1(f)−→	hn−1(f)−→	PROPN
ejpam-4753	113	21	hn−1(n	hn−1(n	PROPN
ejpam-4753	113	22	,	,	PUNCT
ejpam-4753	113	23	d	d	NOUN
ejpam-4753	113	24	′	′	NUM
ejpam-4753	113	25	)	)	PUNCT
ejpam-4753	114	1	−→	−→	NOUN
ejpam-4753	114	2	·	·	PUNCT
ejpam-4753	114	3	·	·	PUNCT
ejpam-4753	115	1	·	·	PUNCT
ejpam-4753	115	2	proof	proof	NOUN
ejpam-4753	115	3	.	.	PUNCT
ejpam-4753	116	1	see	see	VERB
ejpam-4753	116	2	[	[	X
ejpam-4753	116	3	3	3	NUM
ejpam-4753	116	4	]	]	PUNCT
ejpam-4753	116	5	definition	definition	NOUN
ejpam-4753	116	6	8	8	NUM
ejpam-4753	116	7	.	.	PUNCT
ejpam-4753	117	1	let	let	VERB
ejpam-4753	117	2	a	a	PRON
ejpam-4753	117	3	be	be	AUX
ejpam-4753	117	4	a	a	DET
ejpam-4753	117	5	ring	ring	NOUN
ejpam-4753	117	6	,	,	PUNCT
ejpam-4753	117	7	we	we	PRON
ejpam-4753	117	8	say	say	VERB
ejpam-4753	117	9	that	that	SCONJ
ejpam-4753	117	10	a	a	PRON
ejpam-4753	117	11	is	be	AUX
ejpam-4753	117	12	duo	duo	NOUN
ejpam-4753	117	13	ring	ring	NOUN
ejpam-4753	117	14	if	if	SCONJ
ejpam-4753	117	15	every	every	PRON
ejpam-4753	117	16	left	leave	VERB
ejpam-4753	117	17	ideal	ideal	NOUN
ejpam-4753	117	18	of	of	ADP
ejpam-4753	117	19	a	a	PRON
ejpam-4753	117	20	is	be	AUX
ejpam-4753	117	21	two	two	NUM
ejpam-4753	117	22	-	-	PUNCT
ejpam-4753	117	23	sided	sided	ADJ
ejpam-4753	117	24	,	,	PUNCT
ejpam-4753	117	25	and	and	CCONJ
ejpam-4753	117	26	any	any	DET
ejpam-4753	117	27	right	right	ADJ
ejpam-4753	117	28	ideal	ideal	NOUN
ejpam-4753	117	29	of	of	ADP
ejpam-4753	117	30	a	a	PRON
ejpam-4753	117	31	is	be	AUX
ejpam-4753	117	32	two	two	NUM
ejpam-4753	117	33	-	-	PUNCT
ejpam-4753	117	34	sided	sided	ADJ
ejpam-4753	117	35	.	.	PUNCT
ejpam-4753	118	1	proposition	proposition	NOUN
ejpam-4753	118	2	4	4	NUM
ejpam-4753	118	3	.	.	PUNCT
ejpam-4753	118	4	let	let	VERB
ejpam-4753	118	5	a	a	PRON
ejpam-4753	118	6	be	be	AUX
ejpam-4753	118	7	a	a	DET
ejpam-4753	118	8	ring	ring	NOUN
ejpam-4753	118	9	,	,	PUNCT
ejpam-4753	118	10	then	then	ADV
ejpam-4753	118	11	a	a	PRON
ejpam-4753	118	12	is	be	AUX
ejpam-4753	118	13	a	a	DET
ejpam-4753	118	14	duo	duo	NOUN
ejpam-4753	118	15	-	-	PUNCT
ejpam-4753	118	16	ring	ring	NOUN
ejpam-4753	118	17	if	if	SCONJ
ejpam-4753	118	18	,	,	PUNCT
ejpam-4753	118	19	and	and	CCONJ
ejpam-4753	118	20	only	only	ADV
ejpam-4753	118	21	if	if	SCONJ
ejpam-4753	118	22	,	,	PUNCT
ejpam-4753	118	23	∀a	∀a	VERB
ejpam-4753	118	24	∈	∈	PROPN
ejpam-4753	118	25	a	a	X
ejpam-4753	118	26	,	,	PUNCT
ejpam-4753	118	27	aa	aa	NOUN
ejpam-4753	118	28	=	=	ADJ
ejpam-4753	118	29	aa	aa	NOUN
ejpam-4753	118	30	.	.	PUNCT
ejpam-4753	119	1	proof	proof	NOUN
ejpam-4753	119	2	.	.	PUNCT
ejpam-4753	120	1	see	see	VERB
ejpam-4753	120	2	[	[	X
ejpam-4753	120	3	6	6	NUM
ejpam-4753	120	4	]	]	PUNCT
ejpam-4753	120	5	.	.	PUNCT
ejpam-4753	121	1	proposition	proposition	NOUN
ejpam-4753	121	2	5	5	NUM
ejpam-4753	121	3	.	.	PUNCT
ejpam-4753	121	4	let	let	VERB
ejpam-4753	121	5	a	a	PRON
ejpam-4753	121	6	be	be	AUX
ejpam-4753	121	7	a	a	DET
ejpam-4753	121	8	duo	duo	NOUN
ejpam-4753	121	9	-	-	PUNCT
ejpam-4753	121	10	ring	ring	NOUN
ejpam-4753	121	11	then	then	ADV
ejpam-4753	121	12	,	,	PUNCT
ejpam-4753	121	13	the	the	DET
ejpam-4753	121	14	set	set	NOUN
ejpam-4753	121	15	of	of	ADP
ejpam-4753	121	16	all	all	DET
ejpam-4753	121	17	regular	regular	ADJ
ejpam-4753	121	18	elements	element	NOUN
ejpam-4753	121	19	of	of	ADP
ejpam-4753	121	20	a	a	PRON
ejpam-4753	121	21	is	be	AUX
ejpam-4753	121	22	a	a	DET
ejpam-4753	121	23	multiplicatively	multiplicatively	ADV
ejpam-4753	121	24	closed	close	VERB
ejpam-4753	121	25	subset	subset	NOUN
ejpam-4753	121	26	of	of	ADP
ejpam-4753	121	27	a	a	DET
ejpam-4753	121	28	verifies	verifie	NOUN
ejpam-4753	121	29	the	the	DET
ejpam-4753	121	30	conditions	condition	NOUN
ejpam-4753	121	31	ore	ore	NOUN
ejpam-4753	121	32	.	.	PUNCT
ejpam-4753	122	1	proof	proof	NOUN
ejpam-4753	122	2	.	.	PUNCT
ejpam-4753	123	1	see	see	VERB
ejpam-4753	123	2	[	[	X
ejpam-4753	123	3	6	6	NUM
ejpam-4753	123	4	]	]	PUNCT
ejpam-4753	123	5	.	.	PUNCT
ejpam-4753	124	1	proposition	proposition	NOUN
ejpam-4753	124	2	6	6	NUM
ejpam-4753	124	3	.	.	PUNCT
ejpam-4753	125	1	let	let	VERB
ejpam-4753	125	2	a	a	PRON
ejpam-4753	125	3	be	be	AUX
ejpam-4753	125	4	a	a	DET
ejpam-4753	125	5	duo	duo	NOUN
ejpam-4753	125	6	-	-	PUNCT
ejpam-4753	125	7	ring	ring	NOUN
ejpam-4753	125	8	and	and	CCONJ
ejpam-4753	125	9	s	s	AUX
ejpam-4753	125	10	be	be	AUX
ejpam-4753	125	11	a	a	DET
ejpam-4753	125	12	nonempty	nonempty	NOUN
ejpam-4753	125	13	subset	subset	NOUN
ejpam-4753	125	14	formed	form	VERB
ejpam-4753	125	15	of	of	ADP
ejpam-4753	125	16	regular	regular	ADJ
ejpam-4753	125	17	elements	element	NOUN
ejpam-4753	125	18	of	of	ADP
ejpam-4753	125	19	a	a	PRON
ejpam-4753	125	20	,	,	PUNCT
ejpam-4753	125	21	then	then	ADV
ejpam-4753	125	22	there	there	PRON
ejpam-4753	125	23	exists	exist	VERB
ejpam-4753	125	24	a	a	DET
ejpam-4753	125	25	multiplicatively	multiplicatively	ADV
ejpam-4753	125	26	closed	close	VERB
ejpam-4753	125	27	subset	subset	NOUN
ejpam-4753	125	28	of	of	ADP
ejpam-4753	125	29	a	a	DET
ejpam-4753	125	30	satisfying	satisfying	NOUN
ejpam-4753	125	31	the	the	DET
ejpam-4753	125	32	left	left	ADJ
ejpam-4753	125	33	conditions	condition	NOUN
ejpam-4753	125	34	of	of	ADP
ejpam-4753	125	35	ore	ore	NOUN
ejpam-4753	125	36	containing	contain	VERB
ejpam-4753	125	37	s.	s.	PROPN
ejpam-4753	125	38	proof	proof	NOUN
ejpam-4753	125	39	.	.	PUNCT
ejpam-4753	126	1	it	it	PRON
ejpam-4753	126	2	suffices	suffice	VERB
ejpam-4753	126	3	to	to	PART
ejpam-4753	126	4	note	note	VERB
ejpam-4753	126	5	that	that	SCONJ
ejpam-4753	126	6	the	the	DET
ejpam-4753	126	7	set	set	NOUN
ejpam-4753	126	8	of	of	ADP
ejpam-4753	126	9	all	all	DET
ejpam-4753	126	10	regular	regular	ADJ
ejpam-4753	126	11	elements	element	NOUN
ejpam-4753	126	12	of	of	ADP
ejpam-4753	126	13	a	a	PRON
ejpam-4753	126	14	is	be	AUX
ejpam-4753	126	15	a	a	DET
ejpam-4753	126	16	multiplicatively	multiplicatively	ADV
ejpam-4753	126	17	closed	close	VERB
ejpam-4753	126	18	subset	subset	NOUN
ejpam-4753	126	19	satisfying	satisfy	VERB
ejpam-4753	126	20	the	the	DET
ejpam-4753	126	21	conditions	condition	NOUN
ejpam-4753	126	22	ore	ore	NOUN
ejpam-4753	126	23	and	and	CCONJ
ejpam-4753	126	24	containing	contain	VERB
ejpam-4753	126	25	s.	s.	PROPN
ejpam-4753	126	26	definition	definition	NOUN
ejpam-4753	126	27	9	9	NUM
ejpam-4753	126	28	.	.	PUNCT
ejpam-4753	127	1	let	let	VERB
ejpam-4753	127	2	a	a	PRON
ejpam-4753	127	3	be	be	AUX
ejpam-4753	127	4	a	a	DET
ejpam-4753	127	5	duo	duo	NOUN
ejpam-4753	127	6	-	-	PUNCT
ejpam-4753	127	7	ring	ring	NOUN
ejpam-4753	127	8	and	and	CCONJ
ejpam-4753	127	9	s	s	AUX
ejpam-4753	127	10	be	be	AUX
ejpam-4753	127	11	a	a	DET
ejpam-4753	127	12	nonempty	nonempty	NOUN
ejpam-4753	127	13	subset	subset	NOUN
ejpam-4753	127	14	formed	form	VERB
ejpam-4753	127	15	of	of	ADP
ejpam-4753	127	16	regular	regular	ADJ
ejpam-4753	127	17	elements	element	NOUN
ejpam-4753	127	18	of	of	ADP
ejpam-4753	127	19	a	a	PRON
ejpam-4753	127	20	,	,	PUNCT
ejpam-4753	127	21	then	then	ADV
ejpam-4753	127	22	the	the	DET
ejpam-4753	127	23	smaller	small	ADJ
ejpam-4753	127	24	multiplicatively	multiplicatively	ADV
ejpam-4753	127	25	closed	close	VERB
ejpam-4753	127	26	subset	subset	NOUN
ejpam-4753	127	27	of	of	ADP
ejpam-4753	127	28	a	a	DET
ejpam-4753	127	29	satisfying	satisfying	NOUN
ejpam-4753	127	30	the	the	DET
ejpam-4753	127	31	conditions	condition	NOUN
ejpam-4753	127	32	of	of	ADP
ejpam-4753	127	33	ore	ore	NOUN
ejpam-4753	127	34	containing	contain	VERB
ejpam-4753	127	35	s	s	NOUN
ejpam-4753	127	36	is	be	AUX
ejpam-4753	127	37	called	call	VERB
ejpam-4753	127	38	the	the	DET
ejpam-4753	127	39	multiplicatively	multiplicatively	ADV
ejpam-4753	127	40	closed	close	VERB
ejpam-4753	127	41	subset	subset	NOUN
ejpam-4753	127	42	of	of	ADP
ejpam-4753	127	43	a	a	DET
ejpam-4753	127	44	satisfying	satisfying	NOUN
ejpam-4753	127	45	the	the	DET
ejpam-4753	127	46	left	left	ADJ
ejpam-4753	127	47	conditions	condition	NOUN
ejpam-4753	127	48	of	of	ADP
ejpam-4753	127	49	ore	ore	NOUN
ejpam-4753	127	50	generated	generate	VERB
ejpam-4753	127	51	by	by	ADP
ejpam-4753	127	52	s	s	PRON
ejpam-4753	127	53	and	and	CCONJ
ejpam-4753	127	54	denoted	denote	VERB
ejpam-4753	127	55	by	by	ADP
ejpam-4753	127	56	s.	s.	PROPN
ejpam-4753	127	57	proposition	proposition	PROPN
ejpam-4753	127	58	7	7	NUM
ejpam-4753	127	59	.	.	PUNCT
ejpam-4753	128	1	let	let	VERB
ejpam-4753	128	2	a	a	DET
ejpam-4753	128	3	=	=	SYM
ejpam-4753	128	4	⊕	⊕	PROPN
ejpam-4753	128	5	n∈z	n∈z	VERB
ejpam-4753	128	6	an	an	DET
ejpam-4753	128	7	be	be	AUX
ejpam-4753	128	8	a	a	DET
ejpam-4753	128	9	graded	grade	VERB
ejpam-4753	128	10	duo	duo	NOUN
ejpam-4753	128	11	-	-	PUNCT
ejpam-4753	128	12	ring	ring	NOUN
ejpam-4753	128	13	and	and	CCONJ
ejpam-4753	128	14	sh	sh	INTJ
ejpam-4753	128	15	be	be	AUX
ejpam-4753	128	16	a	a	DET
ejpam-4753	128	17	nonempty	nonempty	NOUN
ejpam-4753	128	18	subset	subset	NOUN
ejpam-4753	128	19	formed	form	VERB
ejpam-4753	128	20	of	of	ADP
ejpam-4753	128	21	regular	regular	ADJ
ejpam-4753	128	22	homogeneous	homogeneous	ADJ
ejpam-4753	128	23	elements	element	NOUN
ejpam-4753	128	24	of	of	ADP
ejpam-4753	128	25	a	a	PRON
ejpam-4753	128	26	,	,	PUNCT
ejpam-4753	128	27	then	then	ADV
ejpam-4753	128	28	there	there	PRON
ejpam-4753	128	29	exists	exist	VERB
ejpam-4753	128	30	a	a	DET
ejpam-4753	128	31	homogeneous	homogeneous	ADJ
ejpam-4753	128	32	multiplicatively	multiplicatively	ADV
ejpam-4753	128	33	closed	close	VERB
ejpam-4753	128	34	subset	subset	NOUN
ejpam-4753	128	35	of	of	ADP
ejpam-4753	128	36	a	a	DET
ejpam-4753	128	37	satisfying	satisfying	NOUN
ejpam-4753	128	38	the	the	DET
ejpam-4753	128	39	left	left	ADJ
ejpam-4753	128	40	conditions	condition	NOUN
ejpam-4753	128	41	of	of	ADP
ejpam-4753	128	42	ore	ore	NOUN
ejpam-4753	128	43	containing	contain	VERB
ejpam-4753	128	44	s	s	NOUN
ejpam-4753	128	45	,	,	PUNCT
ejpam-4753	128	46	and	and	CCONJ
ejpam-4753	128	47	denoted	denote	VERB
ejpam-4753	128	48	by	by	ADP
ejpam-4753	128	49	sh	sh	PROPN
ejpam-4753	128	50	.	.	PUNCT
ejpam-4753	129	1	a.	a.	PROPN
ejpam-4753	129	2	o.	o.	PROPN
ejpam-4753	129	3	chbih	chbih	PROPN
ejpam-4753	129	4	,	,	PUNCT
ejpam-4753	129	5	m.	m.	PROPN
ejpam-4753	129	6	b.	b.	PROPN
ejpam-4753	129	7	maaouia	maaouia	PROPN
ejpam-4753	129	8	,	,	PUNCT
ejpam-4753	129	9	m.	m.	NOUN
ejpam-4753	129	10	sanghare	sanghare	PROPN
ejpam-4753	129	11	/	/	SYM
ejpam-4753	129	12	eur	eur	PROPN
ejpam-4753	129	13	.	.	PUNCT
ejpam-4753	130	1	j.	j.	PROPN
ejpam-4753	130	2	pure	pure	PROPN
ejpam-4753	130	3	appl	appl	PROPN
ejpam-4753	130	4	.	.	PROPN
ejpam-4753	130	5	math	math	PROPN
ejpam-4753	130	6	,	,	PUNCT
ejpam-4753	130	7	16	16	NUM
ejpam-4753	130	8	(	(	PUNCT
ejpam-4753	130	9	3	3	NUM
ejpam-4753	130	10	)	)	PUNCT
ejpam-4753	130	11	(	(	PUNCT
ejpam-4753	130	12	2023	2023	NUM
ejpam-4753	130	13	)	)	PUNCT
ejpam-4753	130	14	,	,	PUNCT
ejpam-4753	130	15	1913	1913	NUM
ejpam-4753	130	16	-	-	SYM
ejpam-4753	130	17	1939	1939	NUM
ejpam-4753	130	18	1919	1919	NUM
ejpam-4753	130	19	proof	proof	NOUN
ejpam-4753	130	20	.	.	PUNCT
ejpam-4753	131	1	put	put	VERB
ejpam-4753	131	2	sh	sh	PROPN
ejpam-4753	131	3	the	the	DET
ejpam-4753	131	4	the	the	DET
ejpam-4753	131	5	smaller	smaller	ADV
ejpam-4753	131	6	multiplicatively	multiplicatively	ADV
ejpam-4753	131	7	closed	close	VERB
ejpam-4753	131	8	subset	subset	NOUN
ejpam-4753	131	9	of	of	ADP
ejpam-4753	131	10	a	a	DET
ejpam-4753	131	11	satisfying	satisfying	NOUN
ejpam-4753	131	12	the	the	DET
ejpam-4753	131	13	conditions	condition	NOUN
ejpam-4753	131	14	ore	ore	NOUN
ejpam-4753	131	15	containing	contain	VERB
ejpam-4753	131	16	s	s	PRON
ejpam-4753	131	17	,	,	PUNCT
ejpam-4753	131	18	sh	sh	PROPN
ejpam-4753	131	19	exist	exist	VERB
ejpam-4753	131	20	because	because	SCONJ
ejpam-4753	131	21	the	the	DET
ejpam-4753	131	22	set	set	NOUN
ejpam-4753	131	23	of	of	ADP
ejpam-4753	131	24	regular	regular	ADJ
ejpam-4753	131	25	elements	element	NOUN
ejpam-4753	131	26	of	of	ADP
ejpam-4753	131	27	a	a	PRON
ejpam-4753	131	28	is	be	AUX
ejpam-4753	131	29	a	a	DET
ejpam-4753	131	30	multiplicatively	multiplicatively	ADV
ejpam-4753	131	31	closed	close	VERB
ejpam-4753	131	32	subset	subset	NOUN
ejpam-4753	131	33	of	of	ADP
ejpam-4753	131	34	a	a	DET
ejpam-4753	131	35	satisfying	satisfying	NOUN
ejpam-4753	131	36	the	the	DET
ejpam-4753	131	37	conditions	condition	NOUN
ejpam-4753	131	38	ore	ore	NOUN
ejpam-4753	131	39	containing	contain	VERB
ejpam-4753	131	40	s.	s.	PROPN
ejpam-4753	131	41	then	then	ADV
ejpam-4753	131	42	it	it	PRON
ejpam-4753	131	43	is	be	AUX
ejpam-4753	131	44	enough	enough	ADJ
ejpam-4753	131	45	to	to	PART
ejpam-4753	131	46	proof	proof	VERB
ejpam-4753	131	47	that	that	SCONJ
ejpam-4753	131	48	sh	sh	PROPN
ejpam-4753	131	49	is	be	AUX
ejpam-4753	131	50	homogeneous	homogeneous	ADJ
ejpam-4753	131	51	.	.	PUNCT
ejpam-4753	132	1	we	we	PRON
ejpam-4753	132	2	have	have	VERB
ejpam-4753	132	3	the	the	DET
ejpam-4753	132	4	elements	element	NOUN
ejpam-4753	132	5	of	of	ADP
ejpam-4753	132	6	sh	sh	PROPN
ejpam-4753	132	7	are	be	AUX
ejpam-4753	132	8	of	of	ADP
ejpam-4753	132	9	the	the	DET
ejpam-4753	132	10	form	form	NOUN
ejpam-4753	132	11	∏	∏	NUM
ejpam-4753	132	12	i	i	PRON
ejpam-4753	132	13	si	si	VERB
ejpam-4753	132	14	,	,	PUNCT
ejpam-4753	132	15	si	si	PROPN
ejpam-4753	132	16	∈	∈	PROPN
ejpam-4753	132	17	s	s	X
ejpam-4753	132	18	which	which	PRON
ejpam-4753	132	19	∏	∏	PROPN
ejpam-4753	132	20	i	i	PRON
ejpam-4753	132	21	si	si	VERB
ejpam-4753	132	22	is	be	AUX
ejpam-4753	132	23	homogeneous	homogeneous	ADJ
ejpam-4753	132	24	.	.	PUNCT
ejpam-4753	133	1	corollary	corollary	ADJ
ejpam-4753	133	2	1	1	NUM
ejpam-4753	133	3	.	.	PUNCT
ejpam-4753	134	1	let	let	VERB
ejpam-4753	134	2	a	a	DET
ejpam-4753	134	3	=	=	SYM
ejpam-4753	134	4	⊕	⊕	PROPN
ejpam-4753	134	5	n∈z	n∈z	VERB
ejpam-4753	134	6	an	an	DET
ejpam-4753	134	7	be	be	AUX
ejpam-4753	134	8	a	a	DET
ejpam-4753	134	9	graded	grade	VERB
ejpam-4753	134	10	duo	duo	NOUN
ejpam-4753	134	11	-	-	PUNCT
ejpam-4753	134	12	ring	ring	NOUN
ejpam-4753	134	13	then	then	ADV
ejpam-4753	134	14	the	the	DET
ejpam-4753	134	15	set	set	NOUN
ejpam-4753	134	16	of	of	ADP
ejpam-4753	134	17	all	all	DET
ejpam-4753	134	18	regular	regular	ADJ
ejpam-4753	134	19	homogeneous	homogeneous	ADJ
ejpam-4753	134	20	of	of	ADP
ejpam-4753	134	21	a	a	PRON
ejpam-4753	134	22	is	be	AUX
ejpam-4753	134	23	multiplicatively	multiplicatively	ADV
ejpam-4753	134	24	closed	close	VERB
ejpam-4753	134	25	subset	subset	NOUN
ejpam-4753	134	26	satisfying	satisfy	VERB
ejpam-4753	134	27	the	the	DET
ejpam-4753	134	28	left	left	ADJ
ejpam-4753	134	29	conditions	condition	NOUN
ejpam-4753	134	30	of	of	ADP
ejpam-4753	134	31	ore	ore	NOUN
ejpam-4753	134	32	.	.	PUNCT
ejpam-4753	135	1	proof	proof	NOUN
ejpam-4753	135	2	.	.	PUNCT
ejpam-4753	136	1	put	put	VERB
ejpam-4753	136	2	s	s	PRON
ejpam-4753	136	3	the	the	DET
ejpam-4753	136	4	set	set	NOUN
ejpam-4753	136	5	of	of	ADP
ejpam-4753	136	6	all	all	DET
ejpam-4753	136	7	regular	regular	ADJ
ejpam-4753	136	8	homogeneous	homogeneous	ADJ
ejpam-4753	136	9	of	of	ADP
ejpam-4753	136	10	a	a	DET
ejpam-4753	136	11	then	then	ADV
ejpam-4753	136	12	sh	sh	PROPN
ejpam-4753	136	13	=	=	SYM
ejpam-4753	136	14	s.	s.	PROPN
ejpam-4753	136	15	proposition	proposition	NOUN
ejpam-4753	136	16	8	8	NUM
ejpam-4753	136	17	.	.	PUNCT
ejpam-4753	137	1	let	let	VERB
ejpam-4753	137	2	a	a	DET
ejpam-4753	137	3	=	=	SYM
ejpam-4753	137	4	⊕	⊕	PROPN
ejpam-4753	137	5	n∈z	n∈z	VERB
ejpam-4753	137	6	an	an	DET
ejpam-4753	137	7	be	be	AUX
ejpam-4753	137	8	a	a	DET
ejpam-4753	137	9	graded	grade	VERB
ejpam-4753	137	10	duo	duo	NOUN
ejpam-4753	137	11	-	-	PUNCT
ejpam-4753	137	12	ring	ring	NOUN
ejpam-4753	137	13	,	,	PUNCT
ejpam-4753	137	14	p	p	PRON
ejpam-4753	137	15	is	be	AUX
ejpam-4753	137	16	a	a	DET
ejpam-4753	137	17	prime	prime	ADJ
ejpam-4753	137	18	ideal	ideal	NOUN
ejpam-4753	137	19	of	of	ADP
ejpam-4753	137	20	a	a	PRON
ejpam-4753	137	21	and	and	CCONJ
ejpam-4753	137	22	sph	sph	PROPN
ejpam-4753	137	23	is	be	AUX
ejpam-4753	137	24	the	the	DET
ejpam-4753	137	25	set	set	NOUN
ejpam-4753	137	26	formed	form	VERB
ejpam-4753	137	27	of	of	ADP
ejpam-4753	137	28	homogeneous	homogeneous	ADJ
ejpam-4753	137	29	regular	regular	ADJ
ejpam-4753	137	30	elements	element	NOUN
ejpam-4753	137	31	of	of	ADP
ejpam-4753	137	32	a\p	a\p	PROPN
ejpam-4753	137	33	,	,	PUNCT
ejpam-4753	137	34	then	then	ADV
ejpam-4753	137	35	sph	sph	PROPN
ejpam-4753	137	36	⊂	⊂	PROPN
ejpam-4753	137	37	(	(	PUNCT
ejpam-4753	137	38	a\p	a\p	PROPN
ejpam-4753	137	39	)	)	PUNCT
ejpam-4753	137	40	.	.	PUNCT
ejpam-4753	138	1	proof	proof	NOUN
ejpam-4753	138	2	.	.	PUNCT
ejpam-4753	139	1	the	the	DET
ejpam-4753	139	2	set	set	NOUN
ejpam-4753	139	3	of	of	ADP
ejpam-4753	139	4	regular	regular	ADJ
ejpam-4753	139	5	elements	element	NOUN
ejpam-4753	139	6	of	of	ADP
ejpam-4753	139	7	a\p	a\p	NOUN
ejpam-4753	139	8	is	be	AUX
ejpam-4753	139	9	a	a	DET
ejpam-4753	139	10	multiplicatively	multiplicatively	ADV
ejpam-4753	139	11	closed	close	VERB
ejpam-4753	139	12	subset	subset	NOUN
ejpam-4753	139	13	satisfying	satisfy	VERB
ejpam-4753	139	14	the	the	DET
ejpam-4753	139	15	conditions	condition	NOUN
ejpam-4753	139	16	of	of	ADP
ejpam-4753	139	17	ore	ore	NOUN
ejpam-4753	139	18	,	,	PUNCT
ejpam-4753	139	19	(	(	PUNCT
ejpam-4753	139	20	see	see	VERB
ejpam-4753	139	21	[	[	X
ejpam-4753	139	22	5	5	NUM
ejpam-4753	139	23	]	]	PUNCT
ejpam-4753	139	24	and	and	CCONJ
ejpam-4753	139	25	[	[	X
ejpam-4753	139	26	7	7	NUM
ejpam-4753	139	27	]	]	PUNCT
ejpam-4753	139	28	)	)	PUNCT
ejpam-4753	139	29	and	and	CCONJ
ejpam-4753	139	30	containing	contain	VERB
ejpam-4753	139	31	sph	sph	NOUN
ejpam-4753	139	32	,	,	PUNCT
ejpam-4753	139	33	then	then	ADV
ejpam-4753	139	34	sph	sph	PROPN
ejpam-4753	139	35	⊂	⊂	PROPN
ejpam-4753	139	36	(	(	PUNCT
ejpam-4753	139	37	a\p	a\p	PROPN
ejpam-4753	139	38	)	)	PUNCT
ejpam-4753	139	39	.	.	PUNCT
ejpam-4753	140	1	corollary	corollary	ADJ
ejpam-4753	140	2	2	2	NUM
ejpam-4753	140	3	.	.	PUNCT
ejpam-4753	141	1	let	let	VERB
ejpam-4753	141	2	a	a	DET
ejpam-4753	141	3	=	=	SYM
ejpam-4753	141	4	⊕	⊕	PROPN
ejpam-4753	141	5	n∈z	n∈z	VERB
ejpam-4753	141	6	an	an	DET
ejpam-4753	141	7	be	be	AUX
ejpam-4753	141	8	a	a	DET
ejpam-4753	141	9	graded	grade	VERB
ejpam-4753	141	10	duo	duo	NOUN
ejpam-4753	141	11	-	-	PUNCT
ejpam-4753	141	12	ring	ring	NOUN
ejpam-4753	141	13	and	and	CCONJ
ejpam-4753	141	14	p	p	NOUN
ejpam-4753	141	15	is	be	AUX
ejpam-4753	141	16	a	a	DET
ejpam-4753	141	17	prime	prime	ADJ
ejpam-4753	141	18	ideal	ideal	NOUN
ejpam-4753	141	19	of	of	ADP
ejpam-4753	141	20	a	a	PRON
ejpam-4753	141	21	,	,	PUNCT
ejpam-4753	141	22	then	then	ADV
ejpam-4753	141	23	the	the	DET
ejpam-4753	141	24	set	set	NOUN
ejpam-4753	141	25	of	of	ADP
ejpam-4753	141	26	regular	regular	ADJ
ejpam-4753	141	27	homogeneous	homogeneous	ADJ
ejpam-4753	141	28	of	of	ADP
ejpam-4753	141	29	a\p	a\p	PROPN
ejpam-4753	141	30	is	be	AUX
ejpam-4753	141	31	a	a	DET
ejpam-4753	141	32	multiplicatively	multiplicatively	ADV
ejpam-4753	141	33	closed	close	VERB
ejpam-4753	141	34	subset	subset	NOUN
ejpam-4753	141	35	satisfying	satisfy	VERB
ejpam-4753	141	36	the	the	DET
ejpam-4753	141	37	conditions	condition	NOUN
ejpam-4753	141	38	of	of	ADP
ejpam-4753	141	39	ore	ore	NOUN
ejpam-4753	141	40	.	.	PUNCT
ejpam-4753	142	1	proof	proof	NOUN
ejpam-4753	142	2	.	.	PUNCT
ejpam-4753	143	1	put	put	VERB
ejpam-4753	143	2	s	s	PRON
ejpam-4753	143	3	the	the	DET
ejpam-4753	143	4	set	set	NOUN
ejpam-4753	143	5	of	of	ADP
ejpam-4753	143	6	all	all	DET
ejpam-4753	143	7	regular	regular	ADJ
ejpam-4753	143	8	homogeneous	homogeneous	ADJ
ejpam-4753	143	9	of	of	ADP
ejpam-4753	143	10	a\p	a\p	PROPN
ejpam-4753	143	11	,	,	PUNCT
ejpam-4753	143	12	then	then	ADV
ejpam-4753	143	13	sph	sph	PROPN
ejpam-4753	143	14	=	=	PUNCT
ejpam-4753	143	15	s.	s.	PROPN
ejpam-4753	143	16	3	3	NUM
ejpam-4753	143	17	.	.	PUNCT
ejpam-4753	144	1	functor	functor	PROPN
ejpam-4753	144	2	graduation	graduation	PROPN
ejpam-4753	144	3	s	s	PART
ejpam-4753	144	4	−1	−1	NOUN
ejpam-4753	144	5	h	h	NOUN
ejpam-4753	144	6	and	and	CCONJ
ejpam-4753	144	7	functorization	functorization	NOUN
ejpam-4753	144	8	of	of	ADP
ejpam-4753	144	9	graded	grade	VERB
ejpam-4753	144	10	modules	module	NOUN
ejpam-4753	144	11	theorem	theorem	VERB
ejpam-4753	144	12	3	3	X
ejpam-4753	144	13	.	.	PUNCT
ejpam-4753	145	1	let	let	VERB
ejpam-4753	145	2	a	a	DET
ejpam-4753	145	3	=	=	SYM
ejpam-4753	145	4	⊕	⊕	PROPN
ejpam-4753	145	5	n∈z	n∈z	VERB
ejpam-4753	145	6	an	an	DET
ejpam-4753	145	7	be	be	AUX
ejpam-4753	145	8	a	a	DET
ejpam-4753	145	9	graded	grade	VERB
ejpam-4753	145	10	ring	ring	NOUN
ejpam-4753	145	11	,	,	PUNCT
ejpam-4753	145	12	m	m	VERB
ejpam-4753	145	13	=	=	ADJ
ejpam-4753	145	14	⊕	⊕	PROPN
ejpam-4753	145	15	n∈z	n∈z	VERB
ejpam-4753	145	16	mn	mn	PROPN
ejpam-4753	145	17	and	and	CCONJ
ejpam-4753	145	18	n	n	PROPN
ejpam-4753	145	19	=	=	PROPN
ejpam-4753	145	20	⊕	⊕	PROPN
ejpam-4753	145	21	n∈z	n∈z	VERB
ejpam-4753	146	1	nn	nn	INTJ
ejpam-4753	146	2	be	be	AUX
ejpam-4753	146	3	a	a	DET
ejpam-4753	146	4	two	two	NUM
ejpam-4753	146	5	graded	grade	VERB
ejpam-4753	146	6	left	leave	VERB
ejpam-4753	146	7	a−modules	a−module	NOUN
ejpam-4753	146	8	and	and	CCONJ
ejpam-4753	146	9	s	s	VERB
ejpam-4753	146	10	is	be	AUX
ejpam-4753	146	11	a	a	DET
ejpam-4753	146	12	multiplicatively	multiplicatively	ADV
ejpam-4753	146	13	closed	close	VERB
ejpam-4753	146	14	subset	subset	NOUN
ejpam-4753	146	15	satisfying	satisfy	VERB
ejpam-4753	146	16	the	the	DET
ejpam-4753	146	17	left	left	ADJ
ejpam-4753	146	18	conditions	condition	NOUN
ejpam-4753	146	19	of	of	ADP
ejpam-4753	146	20	ore	ore	NOUN
ejpam-4753	146	21	formed	form	VERB
ejpam-4753	146	22	of	of	ADP
ejpam-4753	146	23	homogeneous	homogeneous	ADJ
ejpam-4753	146	24	elements	element	NOUN
ejpam-4753	146	25	of	of	ADP
ejpam-4753	146	26	a	a	DET
ejpam-4753	146	27	graded	grade	VERB
ejpam-4753	146	28	ring	ring	NOUN
ejpam-4753	146	29	a.	a.	NOUN
ejpam-4753	146	30	let	let	VERB
ejpam-4753	146	31	f	f	NOUN
ejpam-4753	146	32	:	:	PUNCT
ejpam-4753	146	33	m	m	VERB
ejpam-4753	146	34	−→	−→	ADJ
ejpam-4753	146	35	n	n	CCONJ
ejpam-4753	146	36	be	be	AUX
ejpam-4753	146	37	graded	grade	VERB
ejpam-4753	146	38	morphism	morphism	NOUN
ejpam-4753	146	39	of	of	ADP
ejpam-4753	146	40	degree	degree	NOUN
ejpam-4753	146	41	k	k	PROPN
ejpam-4753	146	42	∈	∈	PROPN
ejpam-4753	146	43	z	z	PROPN
ejpam-4753	146	44	of	of	ADP
ejpam-4753	146	45	graded	grade	VERB
ejpam-4753	146	46	left	leave	VERB
ejpam-4753	146	47	a−modules	a−module	NOUN
ejpam-4753	146	48	,	,	PUNCT
ejpam-4753	146	49	then	then	ADV
ejpam-4753	146	50	:	:	PUNCT
ejpam-4753	146	51	s−1(f	s−1(f	PROPN
ejpam-4753	146	52	)	)	PUNCT
ejpam-4753	146	53	:	:	PUNCT
ejpam-4753	147	1	s−1	s−1	PROPN
ejpam-4753	147	2	m	m	VERB
ejpam-4753	147	3	−→	−→	NOUN
ejpam-4753	147	4	s−1n	s−1n	NOUN
ejpam-4753	147	5	m	m	PROPN
ejpam-4753	147	6	s	s	PROPN
ejpam-4753	147	7	7−→	7−→	PROPN
ejpam-4753	147	8	s−1(f	s−1(f	PROPN
ejpam-4753	147	9	)	)	PUNCT
ejpam-4753	147	10	(	(	PUNCT
ejpam-4753	147	11	m	m	PROPN
ejpam-4753	147	12	s	s	PART
ejpam-4753	147	13	)	)	PUNCT
ejpam-4753	147	14	=	=	SYM
ejpam-4753	147	15	f(m	f(m	PROPN
ejpam-4753	147	16	)	)	PUNCT
ejpam-4753	147	17	s	s	VERB
ejpam-4753	147	18	is	be	AUX
ejpam-4753	147	19	a	a	DET
ejpam-4753	147	20	graded	grade	VERB
ejpam-4753	147	21	morphism	morphism	NOUN
ejpam-4753	147	22	of	of	ADP
ejpam-4753	147	23	degree	degree	NOUN
ejpam-4753	147	24	k	k	PROPN
ejpam-4753	147	25	∈	∈	PROPN
ejpam-4753	147	26	z	z	PROPN
ejpam-4753	147	27	of	of	ADP
ejpam-4753	147	28	graded	grade	VERB
ejpam-4753	147	29	left	left	ADJ
ejpam-4753	147	30	s−1a−module	s−1a−module	PROPN
ejpam-4753	147	31	.	.	PUNCT
ejpam-4753	148	1	proof	proof	NOUN
ejpam-4753	148	2	.	.	PUNCT
ejpam-4753	149	1	since	since	SCONJ
ejpam-4753	149	2	[	[	X
ejpam-4753	149	3	1	1	NUM
ejpam-4753	149	4	]	]	PUNCT
ejpam-4753	149	5	,	,	PUNCT
ejpam-4753	149	6	s−1(f	s−1(f	PROPN
ejpam-4753	149	7	)	)	PUNCT
ejpam-4753	149	8	is	be	AUX
ejpam-4753	149	9	a	a	DET
ejpam-4753	149	10	morphism	morphism	NOUN
ejpam-4753	149	11	of	of	ADP
ejpam-4753	149	12	left	left	ADJ
ejpam-4753	149	13	s−1a−module	s−1a−module	PROPN
ejpam-4753	149	14	.	.	PUNCT
ejpam-4753	150	1	show	show	VERB
ejpam-4753	150	2	that	that	SCONJ
ejpam-4753	150	3	s−1(f	s−1(f	PROPN
ejpam-4753	150	4	)	)	PUNCT
ejpam-4753	150	5	is	be	AUX
ejpam-4753	150	6	graded	grade	VERB
ejpam-4753	150	7	morphism	morphism	NOUN
ejpam-4753	150	8	of	of	ADP
ejpam-4753	150	9	degree	degree	NOUN
ejpam-4753	151	1	k	k	PROPN
ejpam-4753	151	2	∈	∈	PROPN
ejpam-4753	151	3	z	z	NOUN
ejpam-4753	151	4	,	,	PUNCT
ejpam-4753	151	5	let	let	VERB
ejpam-4753	151	6	m	m	PRON
ejpam-4753	151	7	∈	∈	VERB
ejpam-4753	151	8	m	m	NOUN
ejpam-4753	151	9	homogeneous	homogeneous	ADJ
ejpam-4753	151	10	such	such	ADJ
ejpam-4753	151	11	that	that	SCONJ
ejpam-4753	151	12	m	m	PROPN
ejpam-4753	151	13	s	s	NOUN
ejpam-4753	151	14	∈	∈	NOUN
ejpam-4753	151	15	s−1	s−1	PROPN
ejpam-4753	151	16	m	m	NOUN
ejpam-4753	151	17	is	be	AUX
ejpam-4753	151	18	of	of	ADP
ejpam-4753	151	19	degree	degree	NOUN
ejpam-4753	151	20	d	d	NOUN
ejpam-4753	151	21	,	,	PUNCT
ejpam-4753	151	22	then	then	ADV
ejpam-4753	151	23	d	d	PROPN
ejpam-4753	151	24	=	=	SYM
ejpam-4753	151	25	deg	deg	PROPN
ejpam-4753	151	26	(	(	PUNCT
ejpam-4753	151	27	m	m	PROPN
ejpam-4753	151	28	s	s	PART
ejpam-4753	151	29	)	)	PUNCT
ejpam-4753	151	30	=	=	SYM
ejpam-4753	151	31	deg(m)−	deg(m)−	NOUN
ejpam-4753	151	32	deg(s	deg(s	PROPN
ejpam-4753	151	33	)	)	PUNCT
ejpam-4753	151	34	,	,	PUNCT
ejpam-4753	151	35	on	on	ADP
ejpam-4753	151	36	the	the	DET
ejpam-4753	151	37	other	other	ADJ
ejpam-4753	151	38	hand	hand	NOUN
ejpam-4753	151	39	deg(s−1(f	deg(s−1(f	PROPN
ejpam-4753	151	40	)	)	PUNCT
ejpam-4753	151	41	(	(	PUNCT
ejpam-4753	151	42	m	m	PROPN
ejpam-4753	151	43	s	s	PART
ejpam-4753	151	44	)	)	PUNCT
ejpam-4753	151	45	)	)	PUNCT
ejpam-4753	151	46	=	=	SYM
ejpam-4753	151	47	deg	deg	PROPN
ejpam-4753	151	48	(	(	PUNCT
ejpam-4753	151	49	f(m	f(m	PROPN
ejpam-4753	151	50	)	)	PUNCT
ejpam-4753	151	51	s	s	PART
ejpam-4753	151	52	)	)	PUNCT
ejpam-4753	151	53	a.	a.	NOUN
ejpam-4753	151	54	o.	o.	PROPN
ejpam-4753	151	55	chbih	chbih	PROPN
ejpam-4753	151	56	,	,	PUNCT
ejpam-4753	151	57	m.	m.	PROPN
ejpam-4753	151	58	b.	b.	PROPN
ejpam-4753	151	59	maaouia	maaouia	PROPN
ejpam-4753	151	60	,	,	PUNCT
ejpam-4753	151	61	m.	m.	NOUN
ejpam-4753	151	62	sanghare	sanghare	PROPN
ejpam-4753	151	63	/	/	SYM
ejpam-4753	151	64	eur	eur	PROPN
ejpam-4753	151	65	.	.	PUNCT
ejpam-4753	152	1	j.	j.	PROPN
ejpam-4753	152	2	pure	pure	PROPN
ejpam-4753	152	3	appl	appl	PROPN
ejpam-4753	152	4	.	.	PROPN
ejpam-4753	152	5	math	math	PROPN
ejpam-4753	152	6	,	,	PUNCT
ejpam-4753	152	7	16	16	NUM
ejpam-4753	152	8	(	(	PUNCT
ejpam-4753	152	9	3	3	NUM
ejpam-4753	152	10	)	)	PUNCT
ejpam-4753	152	11	(	(	PUNCT
ejpam-4753	152	12	2023	2023	NUM
ejpam-4753	152	13	)	)	PUNCT
ejpam-4753	152	14	,	,	PUNCT
ejpam-4753	152	15	1913	1913	NUM
ejpam-4753	152	16	-	-	SYM
ejpam-4753	152	17	1939	1939	NUM
ejpam-4753	152	18	1920	1920	NUM
ejpam-4753	152	19	=	=	PUNCT
ejpam-4753	152	20	deg(f(m))−	deg(f(m))−	NOUN
ejpam-4753	152	21	deg(s	deg(s	PROPN
ejpam-4753	152	22	)	)	PUNCT
ejpam-4753	152	23	=	=	PUNCT
ejpam-4753	152	24	deg(f(m))−	deg(f(m))−	NOUN
ejpam-4753	152	25	deg(s	deg(s	PROPN
ejpam-4753	152	26	)	)	PUNCT
ejpam-4753	152	27	=	=	SYM
ejpam-4753	152	28	(	(	PUNCT
ejpam-4753	152	29	deg(m	deg(m	PROPN
ejpam-4753	152	30	)	)	PUNCT
ejpam-4753	152	31	+	+	CCONJ
ejpam-4753	152	32	k)−	k)−	PROPN
ejpam-4753	152	33	deg(s	deg(s	PROPN
ejpam-4753	152	34	)	)	PUNCT
ejpam-4753	152	35	=	=	PUNCT
ejpam-4753	153	1	d+	d+	NOUN
ejpam-4753	153	2	k	k	NOUN
ejpam-4753	153	3	because	because	SCONJ
ejpam-4753	153	4	f	f	PROPN
ejpam-4753	153	5	is	be	AUX
ejpam-4753	153	6	graded	grade	VERB
ejpam-4753	153	7	of	of	ADP
ejpam-4753	153	8	degree	degree	NOUN
ejpam-4753	153	9	k	k	PROPN
ejpam-4753	153	10	∈	∈	PROPN
ejpam-4753	153	11	z	z	PROPN
ejpam-4753	153	12	,	,	PUNCT
ejpam-4753	153	13	thus	thus	ADV
ejpam-4753	153	14	s−1(f	s−1(f	PROPN
ejpam-4753	153	15	)	)	PUNCT
ejpam-4753	153	16	has	have	AUX
ejpam-4753	153	17	degree	degree	NOUN
ejpam-4753	153	18	k	k	NOUN
ejpam-4753	153	19	,	,	PUNCT
ejpam-4753	153	20	hence	hence	ADV
ejpam-4753	153	21	s−1(f	s−1(f	PROPN
ejpam-4753	153	22	)	)	PUNCT
ejpam-4753	153	23	is	be	AUX
ejpam-4753	153	24	graded	grade	VERB
ejpam-4753	153	25	morphism	morphism	NOUN
ejpam-4753	153	26	of	of	ADP
ejpam-4753	153	27	degree	degree	NOUN
ejpam-4753	153	28	k	k	PROPN
ejpam-4753	153	29	of	of	ADP
ejpam-4753	153	30	graded	grade	VERB
ejpam-4753	153	31	left	leave	VERB
ejpam-4753	153	32	s−1a−module	s−1a−module	PROPN
ejpam-4753	153	33	.	.	PUNCT
ejpam-4753	154	1	proposition	proposition	NOUN
ejpam-4753	154	2	9	9	NUM
ejpam-4753	154	3	.	.	PUNCT
ejpam-4753	155	1	let	let	VERB
ejpam-4753	155	2	a	a	DET
ejpam-4753	155	3	=	=	SYM
ejpam-4753	155	4	⊕	⊕	PROPN
ejpam-4753	155	5	n∈z	n∈z	VERB
ejpam-4753	155	6	an	an	DET
ejpam-4753	155	7	be	be	AUX
ejpam-4753	155	8	a	a	DET
ejpam-4753	155	9	graded	grade	VERB
ejpam-4753	155	10	duo	duo	NOUN
ejpam-4753	155	11	-	-	PUNCT
ejpam-4753	155	12	ring	ring	NOUN
ejpam-4753	155	13	,	,	PUNCT
ejpam-4753	155	14	m	m	VERB
ejpam-4753	155	15	=	=	ADJ
ejpam-4753	155	16	⊕	⊕	PROPN
ejpam-4753	155	17	n∈z	n∈z	VERB
ejpam-4753	155	18	mn	mn	PROPN
ejpam-4753	155	19	and	and	CCONJ
ejpam-4753	155	20	n	n	PROPN
ejpam-4753	155	21	=	=	PROPN
ejpam-4753	155	22	⊕	⊕	PROPN
ejpam-4753	155	23	n∈z	n∈z	VERB
ejpam-4753	156	1	nn	nn	INTJ
ejpam-4753	156	2	be	be	AUX
ejpam-4753	156	3	a	a	DET
ejpam-4753	156	4	two	two	NUM
ejpam-4753	156	5	graded	grade	VERB
ejpam-4753	156	6	left	leave	VERB
ejpam-4753	156	7	a−modules	a−module	NOUN
ejpam-4753	156	8	and	and	CCONJ
ejpam-4753	156	9	sh	sh	INTJ
ejpam-4753	156	10	be	be	AUX
ejpam-4753	156	11	a	a	DET
ejpam-4753	156	12	part	part	NOUN
ejpam-4753	156	13	formed	form	VERB
ejpam-4753	156	14	of	of	ADP
ejpam-4753	156	15	regulars	regular	NOUN
ejpam-4753	156	16	homogeneous	homogeneous	ADJ
ejpam-4753	156	17	elements	element	NOUN
ejpam-4753	156	18	of	of	ADP
ejpam-4753	156	19	a.	a.	NOUN
ejpam-4753	156	20	let	let	VERB
ejpam-4753	156	21	f	f	NOUN
ejpam-4753	156	22	:	:	PUNCT
ejpam-4753	156	23	m	m	VERB
ejpam-4753	156	24	−→	−→	ADJ
ejpam-4753	156	25	n	n	CCONJ
ejpam-4753	156	26	be	be	AUX
ejpam-4753	156	27	graded	grade	VERB
ejpam-4753	156	28	morphism	morphism	NOUN
ejpam-4753	156	29	of	of	ADP
ejpam-4753	156	30	degree	degree	NOUN
ejpam-4753	156	31	k	k	PROPN
ejpam-4753	156	32	∈	∈	PROPN
ejpam-4753	156	33	z	z	PROPN
ejpam-4753	156	34	of	of	ADP
ejpam-4753	156	35	graded	grade	VERB
ejpam-4753	156	36	left	leave	VERB
ejpam-4753	156	37	a−modules	a−module	NOUN
ejpam-4753	156	38	,	,	PUNCT
ejpam-4753	156	39	then	then	ADV
ejpam-4753	156	40	:	:	PUNCT
ejpam-4753	156	41	s	s	X
ejpam-4753	156	42	−1	−1	NOUN
ejpam-4753	156	43	h	h	NOUN
ejpam-4753	156	44	(	(	PUNCT
ejpam-4753	156	45	f	f	X
ejpam-4753	156	46	)	)	PUNCT
ejpam-4753	156	47	:	:	PUNCT
ejpam-4753	156	48	s	s	AUX
ejpam-4753	156	49	−1	−1	NOUN
ejpam-4753	156	50	h	h	NOUN
ejpam-4753	156	51	m	m	VERB
ejpam-4753	156	52	−→	−→	NOUN
ejpam-4753	156	53	s	s	PART
ejpam-4753	156	54	−1	−1	NOUN
ejpam-4753	156	55	h	h	NOUN
ejpam-4753	156	56	n	n	NOUN
ejpam-4753	156	57	m	m	NOUN
ejpam-4753	156	58	s	s	PROPN
ejpam-4753	156	59	7−→	7−→	PROPN
ejpam-4753	156	60	s	s	PART
ejpam-4753	156	61	−1	−1	NOUN
ejpam-4753	156	62	h	h	NOUN
ejpam-4753	156	63	(	(	PUNCT
ejpam-4753	156	64	f	f	X
ejpam-4753	156	65	)	)	PUNCT
ejpam-4753	156	66	(	(	PUNCT
ejpam-4753	156	67	m	m	PROPN
ejpam-4753	156	68	s	s	PART
ejpam-4753	156	69	)	)	PUNCT
ejpam-4753	156	70	=	=	SYM
ejpam-4753	156	71	f(m	f(m	PROPN
ejpam-4753	156	72	)	)	PUNCT
ejpam-4753	156	73	s	s	VERB
ejpam-4753	156	74	is	be	AUX
ejpam-4753	156	75	a	a	DET
ejpam-4753	156	76	graded	grade	VERB
ejpam-4753	156	77	morphism	morphism	NOUN
ejpam-4753	156	78	of	of	ADP
ejpam-4753	156	79	degree	degree	NOUN
ejpam-4753	156	80	k	k	PROPN
ejpam-4753	156	81	∈	∈	PROPN
ejpam-4753	156	82	z	z	PROPN
ejpam-4753	156	83	of	of	ADP
ejpam-4753	156	84	graded	grade	VERB
ejpam-4753	156	85	left	leave	VERB
ejpam-4753	156	86	s	s	PRON
ejpam-4753	156	87	−1	−1	NOUN
ejpam-4753	156	88	h	h	NOUN
ejpam-4753	156	89	a−module	a−module	NOUN
ejpam-4753	156	90	.	.	PUNCT
ejpam-4753	156	91	proof	proof	NOUN
ejpam-4753	156	92	.	.	PUNCT
ejpam-4753	157	1	since	since	SCONJ
ejpam-4753	157	2	the	the	DET
ejpam-4753	157	3	proposition	proposition	NOUN
ejpam-4753	157	4	6	6	NUM
ejpam-4753	157	5	,	,	PUNCT
ejpam-4753	157	6	sh	sh	PROPN
ejpam-4753	157	7	is	be	AUX
ejpam-4753	157	8	a	a	DET
ejpam-4753	157	9	multiplicatively	multiplicatively	ADV
ejpam-4753	157	10	closed	close	VERB
ejpam-4753	157	11	subset	subset	NOUN
ejpam-4753	157	12	satisfying	satisfy	VERB
ejpam-4753	157	13	the	the	DET
ejpam-4753	157	14	left	left	ADJ
ejpam-4753	157	15	conditions	condition	NOUN
ejpam-4753	157	16	of	of	ADP
ejpam-4753	157	17	ore	ore	NOUN
ejpam-4753	157	18	formed	form	VERB
ejpam-4753	157	19	of	of	ADP
ejpam-4753	157	20	homogeneous	homogeneous	ADJ
ejpam-4753	157	21	elements	element	NOUN
ejpam-4753	157	22	of	of	ADP
ejpam-4753	157	23	a	a	PRON
ejpam-4753	157	24	and	and	CCONJ
ejpam-4753	157	25	from	from	ADP
ejpam-4753	157	26	9	9	NUM
ejpam-4753	157	27	,	,	PUNCT
ejpam-4753	157	28	s	s	VERB
ejpam-4753	157	29	−1	−1	NOUN
ejpam-4753	157	30	h	h	NOUN
ejpam-4753	157	31	(	(	PUNCT
ejpam-4753	157	32	f	f	X
ejpam-4753	157	33	)	)	PUNCT
ejpam-4753	157	34	is	be	AUX
ejpam-4753	157	35	graded	grade	VERB
ejpam-4753	157	36	morphism	morphism	NOUN
ejpam-4753	157	37	of	of	ADP
ejpam-4753	157	38	degree	degree	NOUN
ejpam-4753	157	39	k	k	PROPN
ejpam-4753	157	40	of	of	ADP
ejpam-4753	157	41	graded	grade	VERB
ejpam-4753	157	42	left	leave	VERB
ejpam-4753	157	43	s	s	PRON
ejpam-4753	157	44	−1	−1	NOUN
ejpam-4753	157	45	h	h	NOUN
ejpam-4753	157	46	a−module	a−module	PROPN
ejpam-4753	157	47	.	.	PUNCT
ejpam-4753	157	48	proposition	proposition	NOUN
ejpam-4753	157	49	10	10	NUM
ejpam-4753	157	50	.	.	PUNCT
ejpam-4753	158	1	let	let	VERB
ejpam-4753	158	2	a	a	DET
ejpam-4753	158	3	=	=	SYM
ejpam-4753	158	4	⊕	⊕	PROPN
ejpam-4753	158	5	n∈z	n∈z	VERB
ejpam-4753	158	6	an	an	DET
ejpam-4753	158	7	be	be	AUX
ejpam-4753	158	8	a	a	DET
ejpam-4753	158	9	graded	grade	VERB
ejpam-4753	158	10	ring	ring	NOUN
ejpam-4753	158	11	,	,	PUNCT
ejpam-4753	158	12	m	m	VERB
ejpam-4753	158	13	=	=	PROPN
ejpam-4753	158	14	⊕	⊕	PROPN
ejpam-4753	158	15	n∈z	n∈z	PROPN
ejpam-4753	158	16	mn	mn	PROPN
ejpam-4753	158	17	,	,	PUNCT
ejpam-4753	158	18	n	n	PROPN
ejpam-4753	158	19	=	=	PROPN
ejpam-4753	158	20	⊕	⊕	PROPN
ejpam-4753	158	21	n∈z	n∈z	VERB
ejpam-4753	159	1	nn	nn	NOUN
ejpam-4753	159	2	and	and	CCONJ
ejpam-4753	159	3	l	l	NOUN
ejpam-4753	159	4	=	=	PROPN
ejpam-4753	159	5	⊕	⊕	PROPN
ejpam-4753	159	6	n∈z	n∈z	PRON
ejpam-4753	159	7	ln	ln	ADV
ejpam-4753	159	8	be	be	AUX
ejpam-4753	159	9	three	three	NUM
ejpam-4753	159	10	graded	grade	VERB
ejpam-4753	159	11	left	leave	VERB
ejpam-4753	159	12	a−modules	a−module	NOUN
ejpam-4753	159	13	and	and	CCONJ
ejpam-4753	159	14	s	s	AUX
ejpam-4753	159	15	be	be	AUX
ejpam-4753	159	16	a	a	DET
ejpam-4753	159	17	multiplicatively	multiplicatively	ADV
ejpam-4753	159	18	closed	close	VERB
ejpam-4753	159	19	subset	subset	NOUN
ejpam-4753	159	20	satisfying	satisfy	VERB
ejpam-4753	159	21	the	the	DET
ejpam-4753	159	22	left	left	ADJ
ejpam-4753	159	23	conditions	condition	NOUN
ejpam-4753	159	24	of	of	ADP
ejpam-4753	159	25	ore	ore	NOUN
ejpam-4753	159	26	formed	form	VERB
ejpam-4753	159	27	of	of	ADP
ejpam-4753	159	28	homogeneous	homogeneous	ADJ
ejpam-4753	159	29	elements	element	NOUN
ejpam-4753	159	30	of	of	ADP
ejpam-4753	159	31	a	a	PRON
ejpam-4753	159	32	,	,	PUNCT
ejpam-4753	159	33	then	then	ADV
ejpam-4753	159	34	for	for	ADP
ejpam-4753	159	35	every	every	DET
ejpam-4753	159	36	short	short	ADJ
ejpam-4753	159	37	exact	exact	ADJ
ejpam-4753	159	38	sequences	sequence	NOUN
ejpam-4753	159	39	of	of	ADP
ejpam-4753	159	40	a	a	DET
ejpam-4753	159	41	graded	grade	VERB
ejpam-4753	159	42	morphisms	morphism	NOUN
ejpam-4753	159	43	of	of	ADP
ejpam-4753	159	44	degree	degree	NOUN
ejpam-4753	159	45	k	k	PROPN
ejpam-4753	159	46	∈	∈	PROPN
ejpam-4753	159	47	z	z	PROPN
ejpam-4753	159	48	of	of	ADP
ejpam-4753	159	49	a	a	DET
ejpam-4753	159	50	graded	grade	VERB
ejpam-4753	159	51	left	leave	VERB
ejpam-4753	159	52	a−module	a−module	ADP
ejpam-4753	159	53	0	0	NUM
ejpam-4753	159	54	−→	−→	NOUN
ejpam-4753	159	55	m	m	VERB
ejpam-4753	159	56	φ−→	φ−→	PROPN
ejpam-4753	159	57	n	n	NUM
ejpam-4753	159	58	ϕ−→	ϕ−→	NOUN
ejpam-4753	159	59	l	l	NOUN
ejpam-4753	159	60	−→	−→	NOUN
ejpam-4753	159	61	0	0	NUM
ejpam-4753	159	62	,	,	PUNCT
ejpam-4753	159	63	we	we	PRON
ejpam-4753	159	64	have	have	VERB
ejpam-4753	159	65	the	the	DET
ejpam-4753	159	66	following	follow	VERB
ejpam-4753	159	67	short	short	ADJ
ejpam-4753	159	68	exact	exact	ADJ
ejpam-4753	159	69	sequences	sequence	NOUN
ejpam-4753	159	70	of	of	ADP
ejpam-4753	159	71	a	a	DET
ejpam-4753	159	72	graded	grade	VERB
ejpam-4753	159	73	morphisms	morphism	NOUN
ejpam-4753	159	74	of	of	ADP
ejpam-4753	159	75	degree	degree	NOUN
ejpam-4753	159	76	k	k	PROPN
ejpam-4753	159	77	∈	∈	PROPN
ejpam-4753	159	78	z	z	PROPN
ejpam-4753	159	79	of	of	ADP
ejpam-4753	159	80	a	a	DET
ejpam-4753	159	81	graded	grade	VERB
ejpam-4753	159	82	left	leave	VERB
ejpam-4753	159	83	s−1a−modules	s−1a−module	NOUN
ejpam-4753	159	84	:	:	PUNCT
ejpam-4753	159	85	0	0	NUM
ejpam-4753	160	1	−→	−→	NOUN
ejpam-4753	160	2	s−1	s−1	PROPN
ejpam-4753	160	3	m	m	VERB
ejpam-4753	160	4	s−1(φ)−→	s−1(φ)−→	NOUN
ejpam-4753	160	5	s−1n	s−1n	NOUN
ejpam-4753	160	6	s−1(ϕ)−→	s−1(ϕ)−→	PROPN
ejpam-4753	160	7	s−1l	s−1l	NOUN
ejpam-4753	160	8	−→	−→	NOUN
ejpam-4753	160	9	0	0	NUM
ejpam-4753	160	10	.	.	PUNCT
ejpam-4753	161	1	proof	proof	NOUN
ejpam-4753	161	2	.	.	PUNCT
ejpam-4753	162	1	since	since	SCONJ
ejpam-4753	162	2	the	the	DET
ejpam-4753	162	3	theorem	theorem	NOUN
ejpam-4753	162	4	3.4	3.4	NUM
ejpam-4753	162	5	of	of	ADP
ejpam-4753	162	6	[	[	X
ejpam-4753	162	7	8	8	NUM
ejpam-4753	162	8	]	]	PUNCT
ejpam-4753	162	9	,	,	PUNCT
ejpam-4753	162	10	if	if	SCONJ
ejpam-4753	162	11	0	0	NUM
ejpam-4753	162	12	−→	−→	NOUN
ejpam-4753	162	13	m	m	VERB
ejpam-4753	162	14	φ−→	φ−→	PROPN
ejpam-4753	162	15	n	n	NUM
ejpam-4753	162	16	ϕ−→	ϕ−→	NOUN
ejpam-4753	162	17	l	l	NOUN
ejpam-4753	162	18	−→	−→	NOUN
ejpam-4753	162	19	0	0	NUM
ejpam-4753	162	20	is	be	AUX
ejpam-4753	162	21	a	a	DET
ejpam-4753	162	22	short	short	ADJ
ejpam-4753	162	23	exact	exact	ADJ
ejpam-4753	162	24	sequences	sequence	NOUN
ejpam-4753	162	25	of	of	ADP
ejpam-4753	162	26	a	a	DET
ejpam-4753	162	27	morphisms	morphism	NOUN
ejpam-4753	162	28	of	of	ADP
ejpam-4753	162	29	degree	degree	NOUN
ejpam-4753	162	30	k	k	PROPN
ejpam-4753	162	31	∈	∈	PROPN
ejpam-4753	162	32	z	z	PROPN
ejpam-4753	162	33	of	of	ADP
ejpam-4753	162	34	a	a	DET
ejpam-4753	162	35	left	left	ADJ
ejpam-4753	162	36	a−modules	a−module	NOUN
ejpam-4753	162	37	,	,	PUNCT
ejpam-4753	162	38	then	then	ADV
ejpam-4753	162	39	0	0	NUM
ejpam-4753	162	40	−→	−→	NOUN
ejpam-4753	162	41	s−1	s−1	PROPN
ejpam-4753	162	42	m	m	VERB
ejpam-4753	162	43	s−1(φ)−→	s−1(φ)−→	NOUN
ejpam-4753	162	44	s−1n	s−1n	NOUN
ejpam-4753	162	45	s−1(ϕ)−→	s−1(ϕ)−→	PROPN
ejpam-4753	162	46	s−1l	s−1l	NOUN
ejpam-4753	163	1	−→	−→	NOUN
ejpam-4753	163	2	0	0	NUM
ejpam-4753	163	3	a.	a.	NOUN
ejpam-4753	163	4	o.	o.	PROPN
ejpam-4753	163	5	chbih	chbih	PROPN
ejpam-4753	163	6	,	,	PUNCT
ejpam-4753	163	7	m.	m.	PROPN
ejpam-4753	163	8	b.	b.	PROPN
ejpam-4753	163	9	maaouia	maaouia	PROPN
ejpam-4753	163	10	,	,	PUNCT
ejpam-4753	163	11	m.	m.	NOUN
ejpam-4753	163	12	sanghare	sanghare	PROPN
ejpam-4753	163	13	/	/	SYM
ejpam-4753	163	14	eur	eur	PROPN
ejpam-4753	163	15	.	.	PUNCT
ejpam-4753	164	1	j.	j.	PROPN
ejpam-4753	164	2	pure	pure	PROPN
ejpam-4753	164	3	appl	appl	PROPN
ejpam-4753	164	4	.	.	PROPN
ejpam-4753	164	5	math	math	PROPN
ejpam-4753	164	6	,	,	PUNCT
ejpam-4753	164	7	16	16	NUM
ejpam-4753	164	8	(	(	PUNCT
ejpam-4753	164	9	3	3	NUM
ejpam-4753	164	10	)	)	PUNCT
ejpam-4753	164	11	(	(	PUNCT
ejpam-4753	164	12	2023	2023	NUM
ejpam-4753	164	13	)	)	PUNCT
ejpam-4753	164	14	,	,	PUNCT
ejpam-4753	164	15	1913	1913	NUM
ejpam-4753	164	16	-	-	SYM
ejpam-4753	164	17	1939	1939	NUM
ejpam-4753	164	18	1921	1921	NUM
ejpam-4753	164	19	is	be	AUX
ejpam-4753	164	20	a	a	DET
ejpam-4753	164	21	short	short	ADJ
ejpam-4753	164	22	exact	exact	ADJ
ejpam-4753	164	23	sequences	sequence	NOUN
ejpam-4753	164	24	of	of	ADP
ejpam-4753	164	25	a	a	DET
ejpam-4753	164	26	morphisms	morphism	NOUN
ejpam-4753	164	27	of	of	ADP
ejpam-4753	164	28	degree	degree	NOUN
ejpam-4753	164	29	k	k	PROPN
ejpam-4753	164	30	∈	∈	PROPN
ejpam-4753	164	31	z	z	PROPN
ejpam-4753	164	32	of	of	ADP
ejpam-4753	164	33	a	a	DET
ejpam-4753	164	34	left	left	ADJ
ejpam-4753	164	35	s−1a−modules	s−1a−module	NOUN
ejpam-4753	164	36	,	,	PUNCT
ejpam-4753	164	37	and	and	CCONJ
ejpam-4753	164	38	as	as	SCONJ
ejpam-4753	164	39	s	s	NOUN
ejpam-4753	164	40	is	be	AUX
ejpam-4753	164	41	a	a	DET
ejpam-4753	164	42	set	set	NOUN
ejpam-4753	164	43	formed	form	VERB
ejpam-4753	164	44	of	of	ADP
ejpam-4753	164	45	no	no	DET
ejpam-4753	164	46	null	null	ADJ
ejpam-4753	164	47	homogeneous	homogeneous	ADJ
ejpam-4753	164	48	elements	element	NOUN
ejpam-4753	164	49	of	of	ADP
ejpam-4753	164	50	a	a	PRON
ejpam-4753	164	51	and	and	CCONJ
ejpam-4753	164	52	s−1(−	s−1(−	NOUN
ejpam-4753	164	53	)	)	PUNCT
ejpam-4753	164	54	preserve	preserve	NOUN
ejpam-4753	164	55	degree	degree	NOUN
ejpam-4753	164	56	,	,	PUNCT
ejpam-4753	164	57	then	then	ADV
ejpam-4753	164	58	we	we	PRON
ejpam-4753	164	59	have	have	VERB
ejpam-4753	164	60	0	0	NUM
ejpam-4753	164	61	−→	−→	ADJ
ejpam-4753	164	62	s−1	s−1	PROPN
ejpam-4753	164	63	m	m	VERB
ejpam-4753	164	64	s−1(φ)−→	s−1(φ)−→	NOUN
ejpam-4753	164	65	s−1n	s−1n	NOUN
ejpam-4753	164	66	s−1(ϕ)−→	s−1(ϕ)−→	PROPN
ejpam-4753	164	67	s−1l	s−1l	NOUN
ejpam-4753	164	68	−→	−→	NOUN
ejpam-4753	164	69	0	0	NUM
ejpam-4753	164	70	is	be	AUX
ejpam-4753	164	71	a	a	DET
ejpam-4753	164	72	short	short	ADJ
ejpam-4753	164	73	exact	exact	ADJ
ejpam-4753	164	74	sequences	sequence	NOUN
ejpam-4753	164	75	of	of	ADP
ejpam-4753	164	76	a	a	DET
ejpam-4753	164	77	graded	grade	VERB
ejpam-4753	164	78	morphisms	morphism	NOUN
ejpam-4753	164	79	of	of	ADP
ejpam-4753	164	80	degree	degree	NOUN
ejpam-4753	164	81	k	k	PROPN
ejpam-4753	164	82	∈	∈	PROPN
ejpam-4753	164	83	z	z	PROPN
ejpam-4753	164	84	of	of	ADP
ejpam-4753	164	85	a	a	DET
ejpam-4753	164	86	graded	grade	VERB
ejpam-4753	164	87	left	leave	VERB
ejpam-4753	164	88	s−1a−modules	s−1a−module	NOUN
ejpam-4753	164	89	.	.	PUNCT
ejpam-4753	165	1	corollary	corollary	ADJ
ejpam-4753	165	2	3	3	X
ejpam-4753	165	3	.	.	PUNCT
ejpam-4753	166	1	let	let	VERB
ejpam-4753	166	2	a	a	DET
ejpam-4753	166	3	=	=	SYM
ejpam-4753	166	4	⊕	⊕	PROPN
ejpam-4753	166	5	n∈z	n∈z	VERB
ejpam-4753	166	6	an	an	DET
ejpam-4753	166	7	be	be	AUX
ejpam-4753	166	8	a	a	DET
ejpam-4753	166	9	graded	grade	VERB
ejpam-4753	166	10	duo	duo	NOUN
ejpam-4753	166	11	-	-	PUNCT
ejpam-4753	166	12	ring	ring	NOUN
ejpam-4753	166	13	,	,	PUNCT
ejpam-4753	166	14	m	m	VERB
ejpam-4753	166	15	=	=	PROPN
ejpam-4753	166	16	⊕	⊕	PROPN
ejpam-4753	166	17	n∈z	n∈z	PROPN
ejpam-4753	166	18	mn	mn	PROPN
ejpam-4753	166	19	,	,	PUNCT
ejpam-4753	166	20	n	n	PROPN
ejpam-4753	166	21	=	=	PROPN
ejpam-4753	166	22	⊕	⊕	PROPN
ejpam-4753	166	23	n∈z	n∈z	VERB
ejpam-4753	166	24	nn	nn	NOUN
ejpam-4753	166	25	and	and	CCONJ
ejpam-4753	167	1	l	l	NOUN
ejpam-4753	167	2	=	=	PROPN
ejpam-4753	167	3	⊕	⊕	PROPN
ejpam-4753	167	4	n∈z	n∈z	PRON
ejpam-4753	167	5	ln	ln	ADV
ejpam-4753	167	6	be	be	AUX
ejpam-4753	167	7	three	three	NUM
ejpam-4753	167	8	graded	grade	VERB
ejpam-4753	167	9	left	leave	VERB
ejpam-4753	167	10	a−modules	a−module	NOUN
ejpam-4753	167	11	and	and	CCONJ
ejpam-4753	167	12	sh	sh	PROPN
ejpam-4753	167	13	be	be	AUX
ejpam-4753	167	14	part	part	NOUN
ejpam-4753	167	15	formed	form	VERB
ejpam-4753	167	16	of	of	ADP
ejpam-4753	167	17	regulars	regular	NOUN
ejpam-4753	167	18	homogeneous	homogeneous	ADJ
ejpam-4753	167	19	elements	element	NOUN
ejpam-4753	167	20	of	of	ADP
ejpam-4753	167	21	a	a	PRON
ejpam-4753	167	22	,	,	PUNCT
ejpam-4753	167	23	then	then	ADV
ejpam-4753	167	24	for	for	ADP
ejpam-4753	167	25	every	every	DET
ejpam-4753	167	26	short	short	ADJ
ejpam-4753	167	27	exact	exact	ADJ
ejpam-4753	167	28	sequences	sequence	NOUN
ejpam-4753	167	29	of	of	ADP
ejpam-4753	167	30	a	a	DET
ejpam-4753	167	31	graded	grade	VERB
ejpam-4753	167	32	morphisms	morphism	NOUN
ejpam-4753	167	33	of	of	ADP
ejpam-4753	167	34	degree	degree	NOUN
ejpam-4753	167	35	k	k	PROPN
ejpam-4753	167	36	∈	∈	PROPN
ejpam-4753	167	37	z	z	PROPN
ejpam-4753	167	38	of	of	ADP
ejpam-4753	167	39	a	a	DET
ejpam-4753	167	40	graded	grade	VERB
ejpam-4753	167	41	left	leave	VERB
ejpam-4753	167	42	a−module	a−module	ADP
ejpam-4753	167	43	0	0	NUM
ejpam-4753	167	44	−→	−→	NOUN
ejpam-4753	167	45	m	m	VERB
ejpam-4753	168	1	φ−→	φ−→	PROPN
ejpam-4753	168	2	n	n	NUM
ejpam-4753	168	3	ϕ−→	ϕ−→	NOUN
ejpam-4753	168	4	l	l	NOUN
ejpam-4753	168	5	−→	−→	NOUN
ejpam-4753	168	6	0	0	NUM
ejpam-4753	169	1	we	we	PRON
ejpam-4753	169	2	have	have	VERB
ejpam-4753	169	3	the	the	DET
ejpam-4753	169	4	following	follow	VERB
ejpam-4753	169	5	short	short	ADJ
ejpam-4753	169	6	exact	exact	ADJ
ejpam-4753	169	7	sequences	sequence	NOUN
ejpam-4753	169	8	of	of	ADP
ejpam-4753	169	9	a	a	DET
ejpam-4753	169	10	graded	grade	VERB
ejpam-4753	169	11	morphisms	morphism	NOUN
ejpam-4753	169	12	of	of	ADP
ejpam-4753	169	13	degree	degree	NOUN
ejpam-4753	169	14	k	k	PROPN
ejpam-4753	169	15	∈	∈	PROPN
ejpam-4753	169	16	z	z	PROPN
ejpam-4753	169	17	of	of	ADP
ejpam-4753	169	18	a	a	DET
ejpam-4753	169	19	graded	grade	VERB
ejpam-4753	169	20	left	leave	VERB
ejpam-4753	169	21	s	s	PRON
ejpam-4753	169	22	−1	−1	NOUN
ejpam-4753	169	23	h	h	NOUN
ejpam-4753	169	24	a−modules	a−module	NOUN
ejpam-4753	169	25	:	:	PUNCT
ejpam-4753	169	26	0	0	NUM
ejpam-4753	169	27	−→	−→	NOUN
ejpam-4753	169	28	s	s	PART
ejpam-4753	169	29	−1	−1	NOUN
ejpam-4753	169	30	h	h	NOUN
ejpam-4753	169	31	m	m	PROPN
ejpam-4753	169	32	s	s	NOUN
ejpam-4753	169	33	−1	−1	NOUN
ejpam-4753	169	34	h	h	NOUN
ejpam-4753	169	35	(	(	PUNCT
ejpam-4753	169	36	φ)−→	φ)−→	NOUN
ejpam-4753	169	37	s	s	PART
ejpam-4753	169	38	−1	−1	NOUN
ejpam-4753	169	39	h	h	NOUN
ejpam-4753	169	40	n	n	NOUN
ejpam-4753	169	41	s	s	PART
ejpam-4753	169	42	−1	−1	NOUN
ejpam-4753	169	43	h	h	NOUN
ejpam-4753	169	44	(	(	PUNCT
ejpam-4753	169	45	ϕ)−→	ϕ)−→	PROPN
ejpam-4753	169	46	s	s	VERB
ejpam-4753	169	47	−1	−1	NOUN
ejpam-4753	169	48	h	h	NOUN
ejpam-4753	169	49	l	l	NOUN
ejpam-4753	170	1	−→	−→	NOUN
ejpam-4753	170	2	0	0	NUM
ejpam-4753	170	3	proof	proof	NOUN
ejpam-4753	170	4	.	.	PUNCT
ejpam-4753	171	1	since	since	SCONJ
ejpam-4753	171	2	the	the	DET
ejpam-4753	171	3	proposition	proposition	NOUN
ejpam-4753	171	4	6	6	NUM
ejpam-4753	171	5	,	,	PUNCT
ejpam-4753	171	6	sh	sh	PROPN
ejpam-4753	171	7	is	be	AUX
ejpam-4753	171	8	a	a	DET
ejpam-4753	171	9	multiplicatively	multiplicatively	ADV
ejpam-4753	171	10	closed	close	VERB
ejpam-4753	171	11	subset	subset	NOUN
ejpam-4753	171	12	satisfying	satisfy	VERB
ejpam-4753	171	13	the	the	DET
ejpam-4753	171	14	left	left	ADJ
ejpam-4753	171	15	conditions	condition	NOUN
ejpam-4753	171	16	of	of	ADP
ejpam-4753	171	17	ore	ore	NOUN
ejpam-4753	171	18	formed	form	VERB
ejpam-4753	171	19	of	of	ADP
ejpam-4753	171	20	homogeneous	homogeneous	ADJ
ejpam-4753	171	21	elements	element	NOUN
ejpam-4753	171	22	of	of	ADP
ejpam-4753	171	23	a.	a.	NOUN
ejpam-4753	171	24	corollary	corollary	NOUN
ejpam-4753	171	25	4	4	NUM
ejpam-4753	171	26	.	.	PUNCT
ejpam-4753	172	1	let	let	VERB
ejpam-4753	172	2	a	a	DET
ejpam-4753	172	3	=	=	SYM
ejpam-4753	172	4	⊕	⊕	PROPN
ejpam-4753	172	5	n∈z	n∈z	VERB
ejpam-4753	172	6	an	an	DET
ejpam-4753	172	7	be	be	AUX
ejpam-4753	172	8	a	a	DET
ejpam-4753	172	9	graded	grade	VERB
ejpam-4753	172	10	duo	duo	NOUN
ejpam-4753	172	11	-	-	PUNCT
ejpam-4753	172	12	ring	ring	NOUN
ejpam-4753	172	13	,	,	PUNCT
ejpam-4753	172	14	m	m	VERB
ejpam-4753	172	15	=	=	PROPN
ejpam-4753	172	16	⊕	⊕	PROPN
ejpam-4753	172	17	n∈z	n∈z	PROPN
ejpam-4753	172	18	mn	mn	PROPN
ejpam-4753	172	19	,	,	PUNCT
ejpam-4753	172	20	n	n	PROPN
ejpam-4753	172	21	=	=	PROPN
ejpam-4753	172	22	⊕	⊕	PROPN
ejpam-4753	172	23	n∈z	n∈z	VERB
ejpam-4753	172	24	nn	nn	NOUN
ejpam-4753	172	25	and	and	CCONJ
ejpam-4753	173	1	l	l	NOUN
ejpam-4753	173	2	=	=	PROPN
ejpam-4753	173	3	⊕	⊕	PROPN
ejpam-4753	173	4	n∈z	n∈z	PRON
ejpam-4753	173	5	ln	ln	ADV
ejpam-4753	173	6	be	be	AUX
ejpam-4753	173	7	three	three	NUM
ejpam-4753	173	8	graded	grade	VERB
ejpam-4753	173	9	left	leave	VERB
ejpam-4753	173	10	a−modules	a−module	NOUN
ejpam-4753	173	11	and	and	CCONJ
ejpam-4753	173	12	sh	sh	INTJ
ejpam-4753	173	13	the	the	DET
ejpam-4753	173	14	set	set	NOUN
ejpam-4753	173	15	of	of	ADP
ejpam-4753	173	16	all	all	DET
ejpam-4753	173	17	regular	regular	ADJ
ejpam-4753	173	18	homogeneous	homogeneous	ADJ
ejpam-4753	173	19	of	of	ADP
ejpam-4753	173	20	a	a	PRON
ejpam-4753	173	21	,	,	PUNCT
ejpam-4753	173	22	then	then	ADV
ejpam-4753	173	23	for	for	ADP
ejpam-4753	173	24	every	every	DET
ejpam-4753	173	25	short	short	ADJ
ejpam-4753	173	26	exact	exact	ADJ
ejpam-4753	173	27	sequences	sequence	NOUN
ejpam-4753	173	28	of	of	ADP
ejpam-4753	173	29	a	a	DET
ejpam-4753	173	30	graded	grade	VERB
ejpam-4753	173	31	morphisms	morphism	NOUN
ejpam-4753	173	32	of	of	ADP
ejpam-4753	173	33	degree	degree	NOUN
ejpam-4753	173	34	k	k	PROPN
ejpam-4753	173	35	∈	∈	PROPN
ejpam-4753	173	36	z	z	PROPN
ejpam-4753	173	37	of	of	ADP
ejpam-4753	173	38	a	a	DET
ejpam-4753	173	39	graded	grade	VERB
ejpam-4753	173	40	left	leave	VERB
ejpam-4753	173	41	a−module	a−module	ADP
ejpam-4753	173	42	0	0	NUM
ejpam-4753	173	43	−→	−→	NOUN
ejpam-4753	173	44	m	m	VERB
ejpam-4753	173	45	φ−→	φ−→	PROPN
ejpam-4753	173	46	n	n	NUM
ejpam-4753	173	47	ϕ−→	ϕ−→	NOUN
ejpam-4753	173	48	l	l	NOUN
ejpam-4753	173	49	−→	−→	NOUN
ejpam-4753	173	50	0	0	NUM
ejpam-4753	174	1	we	we	PRON
ejpam-4753	174	2	have	have	VERB
ejpam-4753	174	3	the	the	DET
ejpam-4753	174	4	following	follow	VERB
ejpam-4753	174	5	short	short	ADJ
ejpam-4753	174	6	exact	exact	ADJ
ejpam-4753	174	7	sequences	sequence	NOUN
ejpam-4753	174	8	of	of	ADP
ejpam-4753	174	9	a	a	DET
ejpam-4753	174	10	graded	grade	VERB
ejpam-4753	174	11	morphisms	morphism	NOUN
ejpam-4753	174	12	of	of	ADP
ejpam-4753	174	13	degree	degree	NOUN
ejpam-4753	174	14	k	k	PROPN
ejpam-4753	174	15	∈	∈	PROPN
ejpam-4753	174	16	z	z	PROPN
ejpam-4753	174	17	of	of	ADP
ejpam-4753	174	18	a	a	DET
ejpam-4753	174	19	graded	grade	VERB
ejpam-4753	174	20	left	leave	VERB
ejpam-4753	174	21	s−1	s−1	PROPN
ejpam-4753	174	22	h	h	NOUN
ejpam-4753	174	23	a−modules	a−module	NOUN
ejpam-4753	174	24	:	:	PUNCT
ejpam-4753	174	25	0	0	NUM
ejpam-4753	175	1	−→	−→	NOUN
ejpam-4753	175	2	s−1	s−1	PROPN
ejpam-4753	175	3	h	h	NOUN
ejpam-4753	175	4	m	m	VERB
ejpam-4753	175	5	s−1	s−1	ADJ
ejpam-4753	175	6	h	h	NOUN
ejpam-4753	175	7	(	(	PUNCT
ejpam-4753	175	8	φ	φ	NOUN
ejpam-4753	175	9	)	)	PUNCT
ejpam-4753	175	10	−→	−→	ADJ
ejpam-4753	175	11	s−1	s−1	PROPN
ejpam-4753	175	12	h	h	NOUN
ejpam-4753	175	13	n	n	PRON
ejpam-4753	175	14	s−1	s−1	PROPN
ejpam-4753	175	15	h	h	NOUN
ejpam-4753	175	16	(	(	PUNCT
ejpam-4753	175	17	ϕ	ϕ	NOUN
ejpam-4753	175	18	)	)	PUNCT
ejpam-4753	175	19	−→	−→	NOUN
ejpam-4753	176	1	s−1	s−1	PROPN
ejpam-4753	176	2	h	h	NOUN
ejpam-4753	176	3	l	l	NOUN
ejpam-4753	176	4	−→	−→	NOUN
ejpam-4753	176	5	0	0	NUM
ejpam-4753	176	6	proof	proof	NOUN
ejpam-4753	176	7	.	.	PUNCT
ejpam-4753	177	1	similarly	similarly	ADV
ejpam-4753	177	2	to	to	ADP
ejpam-4753	177	3	the	the	DET
ejpam-4753	177	4	proof	proof	NOUN
ejpam-4753	177	5	of	of	ADP
ejpam-4753	177	6	the	the	DET
ejpam-4753	177	7	corollary	corollary	ADJ
ejpam-4753	177	8	precedent	precedent	NOUN
ejpam-4753	177	9	3	3	NUM
ejpam-4753	177	10	with	with	ADP
ejpam-4753	177	11	sh	sh	PROPN
ejpam-4753	177	12	=	=	PUNCT
ejpam-4753	177	13	sh	sh	PROPN
ejpam-4753	177	14	a.	a.	PROPN
ejpam-4753	177	15	o.	o.	PROPN
ejpam-4753	177	16	chbih	chbih	PROPN
ejpam-4753	177	17	,	,	PUNCT
ejpam-4753	177	18	m.	m.	PROPN
ejpam-4753	177	19	b.	b.	PROPN
ejpam-4753	177	20	maaouia	maaouia	PROPN
ejpam-4753	177	21	,	,	PUNCT
ejpam-4753	177	22	m.	m.	NOUN
ejpam-4753	177	23	sanghare	sanghare	PROPN
ejpam-4753	177	24	/	/	SYM
ejpam-4753	177	25	eur	eur	PROPN
ejpam-4753	177	26	.	.	PUNCT
ejpam-4753	178	1	j.	j.	PROPN
ejpam-4753	178	2	pure	pure	PROPN
ejpam-4753	178	3	appl	appl	PROPN
ejpam-4753	178	4	.	.	PROPN
ejpam-4753	178	5	math	math	PROPN
ejpam-4753	178	6	,	,	PUNCT
ejpam-4753	178	7	16	16	NUM
ejpam-4753	178	8	(	(	PUNCT
ejpam-4753	178	9	3	3	NUM
ejpam-4753	178	10	)	)	PUNCT
ejpam-4753	178	11	(	(	PUNCT
ejpam-4753	178	12	2023	2023	NUM
ejpam-4753	178	13	)	)	PUNCT
ejpam-4753	178	14	,	,	PUNCT
ejpam-4753	178	15	1913	1913	NUM
ejpam-4753	178	16	-	-	SYM
ejpam-4753	178	17	1939	1939	NUM
ejpam-4753	178	18	1922	1922	NUM
ejpam-4753	178	19	theorem	theorem	NOUN
ejpam-4753	178	20	4	4	NUM
ejpam-4753	178	21	.	.	PUNCT
ejpam-4753	179	1	let	let	VERB
ejpam-4753	179	2	a	a	DET
ejpam-4753	179	3	=	=	SYM
ejpam-4753	179	4	⊕	⊕	PROPN
ejpam-4753	179	5	n∈z	n∈z	VERB
ejpam-4753	179	6	an	an	DET
ejpam-4753	179	7	be	be	AUX
ejpam-4753	179	8	a	a	DET
ejpam-4753	179	9	graded	grade	VERB
ejpam-4753	179	10	ring	ring	NOUN
ejpam-4753	179	11	and	and	CCONJ
ejpam-4753	179	12	s	s	AUX
ejpam-4753	179	13	be	be	AUX
ejpam-4753	179	14	a	a	DET
ejpam-4753	179	15	multiplicatively	multiplicatively	ADV
ejpam-4753	179	16	closed	close	VERB
ejpam-4753	179	17	subset	subset	NOUN
ejpam-4753	179	18	satisfying	satisfy	VERB
ejpam-4753	179	19	the	the	DET
ejpam-4753	179	20	left	left	ADJ
ejpam-4753	179	21	conditions	condition	NOUN
ejpam-4753	179	22	of	of	ADP
ejpam-4753	179	23	ore	ore	NOUN
ejpam-4753	179	24	formed	form	VERB
ejpam-4753	179	25	of	of	ADP
ejpam-4753	179	26	homogeneous	homogeneous	ADJ
ejpam-4753	179	27	elements	element	NOUN
ejpam-4753	179	28	of	of	ADP
ejpam-4753	179	29	a	a	PRON
ejpam-4753	179	30	,	,	PUNCT
ejpam-4753	179	31	then	then	ADV
ejpam-4753	179	32	the	the	DET
ejpam-4753	179	33	relation	relation	NOUN
ejpam-4753	179	34	s−1(−	s−1(−	PROPN
ejpam-4753	179	35	)	)	PUNCT
ejpam-4753	179	36	:	:	PUNCT
ejpam-4753	180	1	gr(a	gr(a	PUNCT
ejpam-4753	180	2	−	−	PROPN
ejpam-4753	180	3	mod	mod	ADJ
ejpam-4753	180	4	)	)	PUNCT
ejpam-4753	180	5	−→	−→	NOUN
ejpam-4753	180	6	gr(s	gr(s	NOUN
ejpam-4753	180	7	−1a	−1a	NOUN
ejpam-4753	180	8	−	−	NOUN
ejpam-4753	180	9	mod	mod	PROPN
ejpam-4753	180	10	)	)	PUNCT
ejpam-4753	180	11	which	which	PRON
ejpam-4753	180	12	that	that	SCONJ
ejpam-4753	180	13	for	for	ADP
ejpam-4753	180	14	any	any	DET
ejpam-4753	180	15	graded	grade	VERB
ejpam-4753	180	16	left	leave	VERB
ejpam-4753	180	17	a−module	a−module	ADP
ejpam-4753	180	18	m	m	VERB
ejpam-4753	180	19	we	we	PRON
ejpam-4753	180	20	correspond	correspond	VERB
ejpam-4753	180	21	s−1(m	s−1(m	PROPN
ejpam-4753	180	22	)	)	PUNCT
ejpam-4753	180	23	and	and	CCONJ
ejpam-4753	180	24	for	for	ADP
ejpam-4753	180	25	all	all	DET
ejpam-4753	180	26	graded	grade	VERB
ejpam-4753	180	27	morphism	morphism	NOUN
ejpam-4753	180	28	of	of	ADP
ejpam-4753	180	29	degree	degree	NOUN
ejpam-4753	180	30	k	k	PROPN
ejpam-4753	180	31	∈	∈	PROPN
ejpam-4753	180	32	z	z	PROPN
ejpam-4753	180	33	of	of	ADP
ejpam-4753	180	34	graded	grade	VERB
ejpam-4753	180	35	left	leave	VERB
ejpam-4753	180	36	a−modules	a−module	NOUN
ejpam-4753	180	37	f	f	X
ejpam-4753	180	38	:	:	PUNCT
ejpam-4753	180	39	m	m	VERB
ejpam-4753	180	40	−→	−→	ADJ
ejpam-4753	181	1	n	n	PRON
ejpam-4753	181	2	we	we	PRON
ejpam-4753	181	3	correspond	correspond	VERB
ejpam-4753	181	4	s−1(f	s−1(f	PROPN
ejpam-4753	181	5	)	)	PUNCT
ejpam-4753	181	6	of	of	ADP
ejpam-4753	181	7	degree	degree	NOUN
ejpam-4753	181	8	k	k	PROPN
ejpam-4753	181	9	∈	∈	PROPN
ejpam-4753	181	10	z	z	NOUN
ejpam-4753	181	11	is	be	AUX
ejpam-4753	181	12	a	a	DET
ejpam-4753	181	13	exact	exact	ADJ
ejpam-4753	181	14	additively	additively	ADV
ejpam-4753	181	15	covariant	covariant	ADJ
ejpam-4753	181	16	functor	functor	PROPN
ejpam-4753	181	17	.	.	PUNCT
ejpam-4753	181	18	proof	proof	NOUN
ejpam-4753	181	19	.	.	PUNCT
ejpam-4753	182	1	let	let	VERB
ejpam-4753	182	2	f	f	NOUN
ejpam-4753	182	3	:	:	PUNCT
ejpam-4753	182	4	m	m	VERB
ejpam-4753	182	5	−→	−→	ADJ
ejpam-4753	182	6	n	n	AUX
ejpam-4753	182	7	be	be	AUX
ejpam-4753	182	8	a	a	DET
ejpam-4753	182	9	graded	grade	VERB
ejpam-4753	182	10	morphism	morphism	NOUN
ejpam-4753	182	11	of	of	ADP
ejpam-4753	182	12	degree	degree	NOUN
ejpam-4753	182	13	k	k	PROPN
ejpam-4753	182	14	∈	∈	PROPN
ejpam-4753	182	15	z	z	PROPN
ejpam-4753	182	16	of	of	ADP
ejpam-4753	182	17	a	a	DET
ejpam-4753	182	18	graded	grade	VERB
ejpam-4753	182	19	left	leave	VERB
ejpam-4753	182	20	a−modules	a−module	NOUN
ejpam-4753	182	21	,	,	PUNCT
ejpam-4753	182	22	then	then	ADV
ejpam-4753	182	23	s−1(f	s−1(f	PROPN
ejpam-4753	182	24	)	)	PUNCT
ejpam-4753	182	25	:	:	PUNCT
ejpam-4753	183	1	s−1	s−1	PROPN
ejpam-4753	183	2	m	m	VERB
ejpam-4753	183	3	−→	−→	NOUN
ejpam-4753	183	4	s−1n	s−1n	NOUN
ejpam-4753	183	5	m	m	PROPN
ejpam-4753	183	6	s	s	PROPN
ejpam-4753	183	7	7−→	7−→	PROPN
ejpam-4753	183	8	f(m	f(m	PROPN
ejpam-4753	183	9	)	)	PUNCT
ejpam-4753	183	10	s	s	VERB
ejpam-4753	183	11	is	be	AUX
ejpam-4753	183	12	a	a	DET
ejpam-4753	183	13	morphism	morphism	NOUN
ejpam-4753	183	14	of	of	ADP
ejpam-4753	183	15	degree	degree	NOUN
ejpam-4753	183	16	k	k	PROPN
ejpam-4753	183	17	∈	∈	PROPN
ejpam-4753	183	18	z	z	NOUN
ejpam-4753	183	19	of	of	ADP
ejpam-4753	183	20	left	left	ADJ
ejpam-4753	183	21	s−1a−modules	s−1a−module	NOUN
ejpam-4753	183	22	.	.	PUNCT
ejpam-4753	184	1	so	so	ADV
ejpam-4753	184	2	(	(	PUNCT
ejpam-4753	184	3	i	i	NOUN
ejpam-4753	184	4	)	)	PUNCT
ejpam-4753	184	5	let	let	VERB
ejpam-4753	184	6	m	m	PRON
ejpam-4753	184	7	∈	∈	NOUN
ejpam-4753	184	8	gr(a	gr(a	NUM
ejpam-4753	184	9	−mod	−mod	NOUN
ejpam-4753	184	10	)	)	PUNCT
ejpam-4753	184	11	,	,	PUNCT
ejpam-4753	184	12	then	then	ADV
ejpam-4753	184	13	s−1	s−1	PROPN
ejpam-4753	184	14	m	m	NOUN
ejpam-4753	184	15	is	be	AUX
ejpam-4753	184	16	a	a	DET
ejpam-4753	184	17	graded	grade	VERB
ejpam-4753	184	18	left	leave	VERB
ejpam-4753	184	19	s−1a−module	s−1a−module	NOUN
ejpam-4753	184	20	,	,	PUNCT
ejpam-4753	184	21	thus	thus	ADV
ejpam-4753	184	22	s−1	s−1	PROPN
ejpam-4753	184	23	m	m	NOUN
ejpam-4753	184	24	∈	∈	NOUN
ejpam-4753	184	25	gr(s	gr(s	X
ejpam-4753	184	26	−1a−mod	−1a−mod	PROPN
ejpam-4753	184	27	)	)	PUNCT
ejpam-4753	184	28	.	.	PUNCT
ejpam-4753	185	1	(	(	PUNCT
ejpam-4753	185	2	ii	ii	X
ejpam-4753	185	3	)	)	PUNCT
ejpam-4753	185	4	let	let	VERB
ejpam-4753	185	5	f	f	PRON
ejpam-4753	185	6	:	:	PUNCT
ejpam-4753	185	7	m	m	VERB
ejpam-4753	185	8	−→	−→	ADJ
ejpam-4753	185	9	n	n	AUX
ejpam-4753	185	10	be	be	AUX
ejpam-4753	185	11	a	a	DET
ejpam-4753	185	12	graded	grade	VERB
ejpam-4753	185	13	morphism	morphism	NOUN
ejpam-4753	185	14	of	of	ADP
ejpam-4753	185	15	the	the	DET
ejpam-4753	185	16	graded	grade	VERB
ejpam-4753	185	17	left	leave	VERB
ejpam-4753	185	18	a−modules	a−module	NOUN
ejpam-4753	185	19	,	,	PUNCT
ejpam-4753	185	20	then	then	ADV
ejpam-4753	185	21	s−1(g	s−1(g	PROPN
ejpam-4753	185	22	◦	◦	NOUN
ejpam-4753	185	23	f	f	PROPN
ejpam-4753	185	24	)	)	PUNCT
ejpam-4753	185	25	:	:	PUNCT
ejpam-4753	186	1	s−1	s−1	PROPN
ejpam-4753	186	2	m	m	VERB
ejpam-4753	186	3	−→	−→	NOUN
ejpam-4753	186	4	s−1n	s−1n	NOUN
ejpam-4753	186	5	s−1(g	s−1(g	PROPN
ejpam-4753	186	6	◦	◦	NOUN
ejpam-4753	186	7	f)(m	f)(m	PROPN
ejpam-4753	186	8	s	s	PART
ejpam-4753	186	9	)	)	PUNCT
ejpam-4753	186	10	=	=	SYM
ejpam-4753	186	11	(	(	PUNCT
ejpam-4753	186	12	g	g	NOUN
ejpam-4753	186	13	◦	◦	NOUN
ejpam-4753	186	14	f)(m	f)(m	NOUN
ejpam-4753	186	15	)	)	PUNCT
ejpam-4753	186	16	s	s	PART
ejpam-4753	186	17	=	=	SYM
ejpam-4753	186	18	g(f(m	g(f(m	PROPN
ejpam-4753	186	19	)	)	PUNCT
ejpam-4753	186	20	)	)	PUNCT
ejpam-4753	187	1	s	s	VERB
ejpam-4753	187	2	=	=	SYM
ejpam-4753	187	3	g	g	PROPN
ejpam-4753	187	4	(	(	PUNCT
ejpam-4753	187	5	f(m	f(m	PROPN
ejpam-4753	187	6	)	)	PUNCT
ejpam-4753	187	7	s	s	PART
ejpam-4753	187	8	)	)	PUNCT
ejpam-4753	187	9	=	=	SYM
ejpam-4753	187	10	s−1(g	s−1(g	PROPN
ejpam-4753	187	11	)	)	PUNCT
ejpam-4753	187	12	(	(	PUNCT
ejpam-4753	187	13	f(m	f(m	PROPN
ejpam-4753	187	14	)	)	PUNCT
ejpam-4753	187	15	s	s	PART
ejpam-4753	187	16	)	)	PUNCT
ejpam-4753	187	17	=	=	SYM
ejpam-4753	187	18	s−1(g	s−1(g	PROPN
ejpam-4753	187	19	)	)	PUNCT
ejpam-4753	187	20	◦	◦	PROPN
ejpam-4753	187	21	s−1(f	s−1(f	PROPN
ejpam-4753	187	22	)	)	PUNCT
ejpam-4753	187	23	(	(	PUNCT
ejpam-4753	187	24	m	m	PROPN
ejpam-4753	187	25	s	s	PART
ejpam-4753	187	26	)	)	PUNCT
ejpam-4753	187	27	thus	thus	ADV
ejpam-4753	187	28	∀	∀	NUM
ejpam-4753	187	29	m	m	VERB
ejpam-4753	187	30	s	s	NOUN
ejpam-4753	187	31	∈	∈	NOUN
ejpam-4753	187	32	s−1	s−1	PROPN
ejpam-4753	187	33	m	m	PROPN
ejpam-4753	187	34	,	,	PUNCT
ejpam-4753	187	35	s−1(g	s−1(g	PROPN
ejpam-4753	188	1	◦	◦	NOUN
ejpam-4753	188	2	f	f	X
ejpam-4753	188	3	)	)	PUNCT
ejpam-4753	188	4	=	=	SYM
ejpam-4753	188	5	s−1(g	s−1(g	PROPN
ejpam-4753	188	6	)	)	PUNCT
ejpam-4753	188	7	◦	◦	PROPN
ejpam-4753	188	8	s−1(f	s−1(f	PROPN
ejpam-4753	188	9	)	)	PUNCT
ejpam-4753	188	10	.	.	PUNCT
ejpam-4753	189	1	s−1(1	s−1(1	PROPN
ejpam-4753	189	2	m	m	VERB
ejpam-4753	189	3	)	)	PUNCT
ejpam-4753	190	1	:	:	PUNCT
ejpam-4753	191	1	s−1	s−1	PROPN
ejpam-4753	191	2	m	m	VERB
ejpam-4753	191	3	−→	−→	ADJ
ejpam-4753	191	4	s−1	s−1	PROPN
ejpam-4753	191	5	m	m	NOUN
ejpam-4753	191	6	m	m	NOUN
ejpam-4753	191	7	s	s	NOUN
ejpam-4753	191	8	7−→	7−→	PROPN
ejpam-4753	191	9	1	1	NUM
ejpam-4753	191	10	m	m	NOUN
ejpam-4753	191	11	(	(	PUNCT
ejpam-4753	191	12	m	m	NOUN
ejpam-4753	191	13	)	)	PUNCT
ejpam-4753	191	14	s	s	PART
ejpam-4753	191	15	=	=	PUNCT
ejpam-4753	191	16	m	m	PROPN
ejpam-4753	191	17	s	s	PART
ejpam-4753	191	18	=	=	ADJ
ejpam-4753	191	19	1s−1	1s−1	NUM
ejpam-4753	191	20	m	m	NOUN
ejpam-4753	191	21	(	(	PUNCT
ejpam-4753	191	22	m	m	PROPN
ejpam-4753	191	23	s	s	PART
ejpam-4753	191	24	)	)	PUNCT
ejpam-4753	192	1	so	so	SCONJ
ejpam-4753	192	2	∀m	∀m	PROPN
ejpam-4753	192	3	s	s	X
ejpam-4753	192	4	∈	∈	PROPN
ejpam-4753	192	5	s−1	s−1	PROPN
ejpam-4753	192	6	m	m	VERB
ejpam-4753	192	7	we	we	PRON
ejpam-4753	192	8	have	have	VERB
ejpam-4753	192	9	s−1(1	s−1(1	ADJ
ejpam-4753	192	10	m	m	NOUN
ejpam-4753	192	11	)	)	PUNCT
ejpam-4753	193	1	=	=	PUNCT
ejpam-4753	193	2	1s−1	1s−1	NUM
ejpam-4753	193	3	m	m	NOUN
ejpam-4753	193	4	,	,	PUNCT
ejpam-4753	193	5	so	so	ADV
ejpam-4753	193	6	s−1(−	s−1(−	PROPN
ejpam-4753	193	7	)	)	PUNCT
ejpam-4753	193	8	:	:	PUNCT
ejpam-4753	193	9	gr(a−mod	gr(a−mod	X
ejpam-4753	193	10	)	)	PUNCT
ejpam-4753	193	11	−→	−→	NOUN
ejpam-4753	193	12	gr(s	gr(s	PUNCT
ejpam-4753	193	13	−1a−mod	−1a−mod	PROPN
ejpam-4753	193	14	)	)	PUNCT
ejpam-4753	193	15	is	be	AUX
ejpam-4753	193	16	a	a	DET
ejpam-4753	193	17	covariant	covariant	ADJ
ejpam-4753	193	18	functor	functor	NOUN
ejpam-4753	193	19	.	.	PUNCT
ejpam-4753	193	20	a.	a.	PROPN
ejpam-4753	193	21	o.	o.	PROPN
ejpam-4753	193	22	chbih	chbih	PROPN
ejpam-4753	193	23	,	,	PUNCT
ejpam-4753	193	24	m.	m.	PROPN
ejpam-4753	193	25	b.	b.	PROPN
ejpam-4753	193	26	maaouia	maaouia	PROPN
ejpam-4753	193	27	,	,	PUNCT
ejpam-4753	193	28	m.	m.	NOUN
ejpam-4753	193	29	sanghare	sanghare	PROPN
ejpam-4753	193	30	/	/	SYM
ejpam-4753	193	31	eur	eur	PROPN
ejpam-4753	193	32	.	.	PUNCT
ejpam-4753	194	1	j.	j.	PROPN
ejpam-4753	194	2	pure	pure	PROPN
ejpam-4753	194	3	appl	appl	PROPN
ejpam-4753	194	4	.	.	PROPN
ejpam-4753	194	5	math	math	PROPN
ejpam-4753	194	6	,	,	PUNCT
ejpam-4753	194	7	16	16	NUM
ejpam-4753	194	8	(	(	PUNCT
ejpam-4753	194	9	3	3	NUM
ejpam-4753	194	10	)	)	PUNCT
ejpam-4753	194	11	(	(	PUNCT
ejpam-4753	194	12	2023	2023	NUM
ejpam-4753	194	13	)	)	PUNCT
ejpam-4753	194	14	,	,	PUNCT
ejpam-4753	194	15	1913	1913	NUM
ejpam-4753	194	16	-	-	SYM
ejpam-4753	194	17	1939	1939	NUM
ejpam-4753	194	18	1923	1923	NUM
ejpam-4753	194	19	furthermore	furthermore	ADV
ejpam-4753	194	20	deg	deg	PROPN
ejpam-4753	194	21	(	(	PUNCT
ejpam-4753	194	22	m	m	PROPN
ejpam-4753	194	23	s	s	PART
ejpam-4753	194	24	)	)	PUNCT
ejpam-4753	194	25	=	=	PUNCT
ejpam-4753	194	26	deg(m)−deg(s	deg(m)−deg(s	NOUN
ejpam-4753	194	27	)	)	PUNCT
ejpam-4753	194	28	or	or	CCONJ
ejpam-4753	194	29	f	f	PROPN
ejpam-4753	194	30	is	be	AUX
ejpam-4753	194	31	graded	grade	VERB
ejpam-4753	194	32	of	of	ADP
ejpam-4753	194	33	degree	degree	NOUN
ejpam-4753	195	1	k	k	PROPN
ejpam-4753	195	2	∈	∈	PROPN
ejpam-4753	195	3	z	z	PROPN
ejpam-4753	195	4	,	,	PUNCT
ejpam-4753	195	5	then	then	ADV
ejpam-4753	195	6	deg(m)+k	deg(m)+k	PUNCT
ejpam-4753	195	7	=	=	SYM
ejpam-4753	195	8	deg(f(m	deg(f(m	NOUN
ejpam-4753	195	9	)	)	PUNCT
ejpam-4753	195	10	)	)	PUNCT
ejpam-4753	196	1	so	so	ADV
ejpam-4753	196	2	deg(s−1(f	deg(s−1(f	PROPN
ejpam-4753	196	3	)	)	PUNCT
ejpam-4753	196	4	(	(	PUNCT
ejpam-4753	196	5	m	m	PROPN
ejpam-4753	196	6	s	s	PART
ejpam-4753	196	7	)	)	PUNCT
ejpam-4753	196	8	)	)	PUNCT
ejpam-4753	197	1	=	=	SYM
ejpam-4753	197	2	deg	deg	PROPN
ejpam-4753	197	3	(	(	PUNCT
ejpam-4753	197	4	f(m	f(m	PROPN
ejpam-4753	197	5	)	)	PUNCT
ejpam-4753	197	6	s	s	PART
ejpam-4753	197	7	)	)	PUNCT
ejpam-4753	197	8	=	=	PUNCT
ejpam-4753	197	9	deg(f(m))−	deg(f(m))−	NOUN
ejpam-4753	197	10	deg(s	deg(s	PROPN
ejpam-4753	197	11	)	)	PUNCT
ejpam-4753	197	12	=	=	SYM
ejpam-4753	197	13	(	(	PUNCT
ejpam-4753	197	14	deg(m	deg(m	PROPN
ejpam-4753	197	15	)	)	PUNCT
ejpam-4753	197	16	+	+	CCONJ
ejpam-4753	197	17	k)−	k)−	PROPN
ejpam-4753	197	18	deg(s	deg(s	PROPN
ejpam-4753	197	19	)	)	PUNCT
ejpam-4753	197	20	=	=	SYM
ejpam-4753	197	21	deg	deg	PROPN
ejpam-4753	197	22	(	(	PUNCT
ejpam-4753	197	23	m	m	PROPN
ejpam-4753	197	24	s	s	PART
ejpam-4753	197	25	)	)	PUNCT
ejpam-4753	198	1	+	+	CCONJ
ejpam-4753	198	2	k.	k.	NOUN
ejpam-4753	198	3	thus	thus	ADV
ejpam-4753	198	4	s−1(−	s−1(−	PROPN
ejpam-4753	198	5	)	)	PUNCT
ejpam-4753	198	6	is	be	AUX
ejpam-4753	198	7	additively	additively	ADV
ejpam-4753	198	8	exact	exact	ADJ
ejpam-4753	198	9	covariant	covariant	PROPN
ejpam-4753	198	10	functor	functor	PROPN
ejpam-4753	198	11	.	.	PROPN
ejpam-4753	198	12	or	or	CCONJ
ejpam-4753	198	13	s−1(−	s−1(−	NOUN
ejpam-4753	198	14	)	)	PUNCT
ejpam-4753	198	15	is	be	AUX
ejpam-4753	198	16	exact	exact	ADJ
ejpam-4753	198	17	then	then	ADV
ejpam-4753	198	18	additively	additively	ADV
ejpam-4753	198	19	exact	exact	ADJ
ejpam-4753	198	20	covariant	covariant	PROPN
ejpam-4753	198	21	functor	functor	PROPN
ejpam-4753	198	22	.	.	PUNCT
ejpam-4753	198	23	proposition	proposition	NOUN
ejpam-4753	198	24	11	11	NUM
ejpam-4753	198	25	.	.	PUNCT
ejpam-4753	199	1	let	let	VERB
ejpam-4753	199	2	a	a	DET
ejpam-4753	199	3	=	=	SYM
ejpam-4753	199	4	⊕	⊕	PROPN
ejpam-4753	199	5	n∈z	n∈z	VERB
ejpam-4753	199	6	an	an	DET
ejpam-4753	199	7	be	be	AUX
ejpam-4753	199	8	a	a	DET
ejpam-4753	199	9	graded	grade	VERB
ejpam-4753	199	10	duo	duo	NOUN
ejpam-4753	199	11	-	-	PUNCT
ejpam-4753	199	12	ring	ring	NOUN
ejpam-4753	199	13	and	and	CCONJ
ejpam-4753	199	14	sh	sh	INTJ
ejpam-4753	199	15	be	be	AUX
ejpam-4753	199	16	the	the	DET
ejpam-4753	199	17	part	part	NOUN
ejpam-4753	199	18	formed	form	VERB
ejpam-4753	199	19	of	of	ADP
ejpam-4753	199	20	all	all	DET
ejpam-4753	199	21	regulars	regular	NOUN
ejpam-4753	199	22	homogeneous	homogeneous	ADJ
ejpam-4753	199	23	elements	element	NOUN
ejpam-4753	199	24	of	of	ADP
ejpam-4753	199	25	a	a	PRON
ejpam-4753	199	26	,	,	PUNCT
ejpam-4753	199	27	then	then	ADV
ejpam-4753	199	28	the	the	DET
ejpam-4753	199	29	relation	relation	NOUN
ejpam-4753	199	30	s	s	PART
ejpam-4753	199	31	−1	−1	NOUN
ejpam-4753	199	32	h	h	NOUN
ejpam-4753	199	33	(	(	PUNCT
ejpam-4753	199	34	−	−	PROPN
ejpam-4753	199	35	)	)	PUNCT
ejpam-4753	199	36	:	:	PUNCT
ejpam-4753	199	37	gr(a	gr(a	PUNCT
ejpam-4753	199	38	−	−	PROPN
ejpam-4753	199	39	mod	mod	ADJ
ejpam-4753	199	40	)	)	PUNCT
ejpam-4753	199	41	−→	−→	NOUN
ejpam-4753	199	42	gr(s	gr(s	PUNCT
ejpam-4753	199	43	−1	−1	NOUN
ejpam-4753	199	44	h	h	NOUN
ejpam-4753	199	45	a	a	DET
ejpam-4753	199	46	−	−	PROPN
ejpam-4753	199	47	mod	mod	PROPN
ejpam-4753	199	48	)	)	PUNCT
ejpam-4753	199	49	which	which	PRON
ejpam-4753	199	50	that	that	SCONJ
ejpam-4753	199	51	for	for	ADP
ejpam-4753	199	52	any	any	DET
ejpam-4753	199	53	graded	grade	VERB
ejpam-4753	199	54	left	leave	VERB
ejpam-4753	199	55	a−module	a−module	ADP
ejpam-4753	199	56	m	m	VERB
ejpam-4753	199	57	we	we	PRON
ejpam-4753	199	58	correspond	correspond	VERB
ejpam-4753	199	59	s	s	PRON
ejpam-4753	199	60	−1	−1	NOUN
ejpam-4753	199	61	h	h	NOUN
ejpam-4753	199	62	(	(	PUNCT
ejpam-4753	199	63	m	m	NOUN
ejpam-4753	199	64	)	)	PUNCT
ejpam-4753	199	65	and	and	CCONJ
ejpam-4753	199	66	for	for	ADP
ejpam-4753	199	67	all	all	DET
ejpam-4753	199	68	graded	grade	VERB
ejpam-4753	199	69	morphism	morphism	NOUN
ejpam-4753	199	70	of	of	ADP
ejpam-4753	199	71	degree	degree	NOUN
ejpam-4753	199	72	k	k	PROPN
ejpam-4753	199	73	∈	∈	PROPN
ejpam-4753	199	74	z	z	PROPN
ejpam-4753	199	75	of	of	ADP
ejpam-4753	199	76	graded	grade	VERB
ejpam-4753	199	77	left	leave	VERB
ejpam-4753	199	78	a−modules	a−module	NOUN
ejpam-4753	199	79	f	f	X
ejpam-4753	199	80	:	:	PUNCT
ejpam-4753	199	81	m	m	VERB
ejpam-4753	199	82	−→	−→	ADJ
ejpam-4753	200	1	n	n	INTJ
ejpam-4753	200	2	we	we	PRON
ejpam-4753	200	3	correspond	correspond	VERB
ejpam-4753	200	4	s	s	PRON
ejpam-4753	200	5	−1	−1	NOUN
ejpam-4753	200	6	h	h	NOUN
ejpam-4753	200	7	(	(	PUNCT
ejpam-4753	200	8	f	f	X
ejpam-4753	200	9	)	)	PUNCT
ejpam-4753	200	10	of	of	ADP
ejpam-4753	200	11	degree	degree	NOUN
ejpam-4753	200	12	k	k	PROPN
ejpam-4753	200	13	∈	∈	PROPN
ejpam-4753	200	14	z	z	NOUN
ejpam-4753	200	15	is	be	AUX
ejpam-4753	200	16	a	a	DET
ejpam-4753	200	17	exact	exact	ADJ
ejpam-4753	200	18	additively	additively	ADV
ejpam-4753	200	19	covariant	covariant	ADJ
ejpam-4753	200	20	functor	functor	PROPN
ejpam-4753	200	21	.	.	PUNCT
ejpam-4753	200	22	proof	proof	NOUN
ejpam-4753	200	23	.	.	PUNCT
ejpam-4753	201	1	similarly	similarly	ADV
ejpam-4753	201	2	to	to	ADP
ejpam-4753	201	3	the	the	DET
ejpam-4753	201	4	proof	proof	NOUN
ejpam-4753	201	5	of	of	ADP
ejpam-4753	201	6	the	the	DET
ejpam-4753	201	7	theorem	theorem	ADJ
ejpam-4753	201	8	precedent	precedent	NOUN
ejpam-4753	201	9	4	4	NUM
ejpam-4753	201	10	.	.	PUNCT
ejpam-4753	201	11	corollary	corollary	ADJ
ejpam-4753	201	12	5	5	NUM
ejpam-4753	201	13	.	.	PUNCT
ejpam-4753	202	1	let	let	VERB
ejpam-4753	202	2	a	a	DET
ejpam-4753	202	3	=	=	SYM
ejpam-4753	202	4	⊕	⊕	PROPN
ejpam-4753	202	5	n∈z	n∈z	VERB
ejpam-4753	202	6	an	an	DET
ejpam-4753	202	7	be	be	AUX
ejpam-4753	202	8	a	a	DET
ejpam-4753	202	9	graded	grade	VERB
ejpam-4753	202	10	duo	duo	NOUN
ejpam-4753	202	11	-	-	PUNCT
ejpam-4753	202	12	ring	ring	NOUN
ejpam-4753	202	13	and	and	CCONJ
ejpam-4753	202	14	sh	sh	INTJ
ejpam-4753	202	15	the	the	DET
ejpam-4753	202	16	set	set	NOUN
ejpam-4753	202	17	of	of	ADP
ejpam-4753	202	18	all	all	DET
ejpam-4753	202	19	regular	regular	ADJ
ejpam-4753	202	20	homogeneous	homogeneous	ADJ
ejpam-4753	202	21	of	of	ADP
ejpam-4753	202	22	a	a	PRON
ejpam-4753	202	23	,	,	PUNCT
ejpam-4753	202	24	then	then	ADV
ejpam-4753	202	25	the	the	DET
ejpam-4753	202	26	relation	relation	NOUN
ejpam-4753	202	27	s−1	s−1	PROPN
ejpam-4753	202	28	h	h	NOUN
ejpam-4753	202	29	(	(	PUNCT
ejpam-4753	202	30	−	−	PROPN
ejpam-4753	202	31	)	)	PUNCT
ejpam-4753	202	32	:	:	PUNCT
ejpam-4753	202	33	gr(a−mod	gr(a−mod	X
ejpam-4753	202	34	)	)	PUNCT
ejpam-4753	202	35	−→	−→	NOUN
ejpam-4753	202	36	s−1	s−1	PROPN
ejpam-4753	202	37	h	h	NOUN
ejpam-4753	202	38	a−mod	a−mod	NOUN
ejpam-4753	202	39	which	which	PRON
ejpam-4753	202	40	that	that	SCONJ
ejpam-4753	202	41	for	for	ADP
ejpam-4753	202	42	any	any	DET
ejpam-4753	202	43	graded	grade	VERB
ejpam-4753	202	44	left	leave	VERB
ejpam-4753	202	45	a−module	a−module	ADP
ejpam-4753	202	46	m	m	VERB
ejpam-4753	202	47	we	we	PRON
ejpam-4753	202	48	correspond	correspond	VERB
ejpam-4753	202	49	s−1	s−1	PROPN
ejpam-4753	202	50	h	h	NOUN
ejpam-4753	202	51	(	(	PUNCT
ejpam-4753	202	52	m	m	NOUN
ejpam-4753	202	53	)	)	PUNCT
ejpam-4753	202	54	and	and	CCONJ
ejpam-4753	202	55	for	for	ADP
ejpam-4753	202	56	all	all	DET
ejpam-4753	202	57	graded	grade	VERB
ejpam-4753	202	58	morphism	morphism	NOUN
ejpam-4753	202	59	of	of	ADP
ejpam-4753	202	60	degree	degree	NOUN
ejpam-4753	202	61	k	k	PROPN
ejpam-4753	202	62	∈	∈	PROPN
ejpam-4753	202	63	z	z	PROPN
ejpam-4753	202	64	of	of	ADP
ejpam-4753	202	65	graded	grade	VERB
ejpam-4753	202	66	left	leave	VERB
ejpam-4753	202	67	a−modules	a−module	NOUN
ejpam-4753	202	68	f	f	X
ejpam-4753	202	69	:	:	PUNCT
ejpam-4753	202	70	m	m	VERB
ejpam-4753	202	71	−→	−→	ADJ
ejpam-4753	203	1	n	n	CCONJ
ejpam-4753	203	2	we	we	PRON
ejpam-4753	203	3	correspond	correspond	VERB
ejpam-4753	203	4	s−1	s−1	PROPN
ejpam-4753	203	5	h	h	NOUN
ejpam-4753	203	6	(	(	PUNCT
ejpam-4753	203	7	f	f	X
ejpam-4753	203	8	)	)	PUNCT
ejpam-4753	203	9	is	be	AUX
ejpam-4753	203	10	additively	additively	ADV
ejpam-4753	203	11	exact	exact	ADJ
ejpam-4753	203	12	covariant	covariant	ADJ
ejpam-4753	203	13	functor	functor	PROPN
ejpam-4753	203	14	.	.	PUNCT
ejpam-4753	203	15	proof	proof	NOUN
ejpam-4753	203	16	.	.	PUNCT
ejpam-4753	204	1	sh	sh	PROPN
ejpam-4753	204	2	is	be	AUX
ejpam-4753	204	3	the	the	DET
ejpam-4753	204	4	set	set	NOUN
ejpam-4753	204	5	of	of	ADP
ejpam-4753	204	6	regular	regular	ADJ
ejpam-4753	204	7	homogeneous	homogeneous	ADJ
ejpam-4753	204	8	of	of	ADP
ejpam-4753	204	9	a	a	PRON
ejpam-4753	204	10	then	then	ADV
ejpam-4753	204	11	sh	sh	PROPN
ejpam-4753	204	12	is	be	AUX
ejpam-4753	204	13	homogeneous	homogeneous	ADJ
ejpam-4753	204	14	multiplicatively	multiplicatively	ADV
ejpam-4753	204	15	closed	close	VERB
ejpam-4753	204	16	subset	subset	NOUN
ejpam-4753	205	1	so	so	SCONJ
ejpam-4753	205	2	sh	sh	PROPN
ejpam-4753	205	3	=	=	NOUN
ejpam-4753	206	1	sh	sh	NOUN
ejpam-4753	206	2	then	then	ADV
ejpam-4753	206	3	according	accord	VERB
ejpam-4753	206	4	to	to	ADP
ejpam-4753	206	5	proposition	proposition	NOUN
ejpam-4753	206	6	precedent	precedent	NOUN
ejpam-4753	206	7	11	11	NUM
ejpam-4753	206	8	.	.	PUNCT
ejpam-4753	207	1	proposition	proposition	NOUN
ejpam-4753	207	2	12	12	NUM
ejpam-4753	207	3	.	.	PUNCT
ejpam-4753	208	1	let	let	VERB
ejpam-4753	208	2	a	a	DET
ejpam-4753	208	3	=	=	SYM
ejpam-4753	208	4	⊕	⊕	PROPN
ejpam-4753	208	5	n∈z	n∈z	VERB
ejpam-4753	208	6	an	an	DET
ejpam-4753	208	7	be	be	AUX
ejpam-4753	208	8	a	a	DET
ejpam-4753	208	9	graded	grade	VERB
ejpam-4753	208	10	duo	duo	NOUN
ejpam-4753	208	11	-	-	PUNCT
ejpam-4753	208	12	ring	ring	NOUN
ejpam-4753	208	13	,	,	PUNCT
ejpam-4753	208	14	p	p	X
ejpam-4753	208	15	be	be	AUX
ejpam-4753	208	16	a	a	DET
ejpam-4753	208	17	prime	prime	ADJ
ejpam-4753	208	18	ideal	ideal	NOUN
ejpam-4753	208	19	of	of	ADP
ejpam-4753	208	20	a	a	PRON
ejpam-4753	208	21	and	and	CCONJ
ejpam-4753	208	22	sph	sph	PROPN
ejpam-4753	208	23	be	be	AUX
ejpam-4753	208	24	a	a	DET
ejpam-4753	208	25	set	set	NOUN
ejpam-4753	208	26	formed	form	VERB
ejpam-4753	208	27	of	of	ADP
ejpam-4753	208	28	homogeneous	homogeneous	ADJ
ejpam-4753	208	29	regular	regular	ADJ
ejpam-4753	208	30	elements	element	NOUN
ejpam-4753	208	31	of	of	ADP
ejpam-4753	208	32	a\p	a\p	PROPN
ejpam-4753	208	33	,	,	PUNCT
ejpam-4753	208	34	then	then	ADV
ejpam-4753	208	35	the	the	DET
ejpam-4753	208	36	relation	relation	NOUN
ejpam-4753	208	37	s	s	PART
ejpam-4753	208	38	−1	−1	NOUN
ejpam-4753	208	39	ph	ph	NOUN
ejpam-4753	208	40	(	(	PUNCT
ejpam-4753	208	41	−	−	NOUN
ejpam-4753	208	42	)	)	PUNCT
ejpam-4753	208	43	:	:	PUNCT
ejpam-4753	208	44	gr(a	gr(a	NUM
ejpam-4753	208	45	−mod	−mod	ADJ
ejpam-4753	208	46	)	)	PUNCT
ejpam-4753	208	47	−→	−→	NOUN
ejpam-4753	208	48	gr(s	gr(s	PUNCT
ejpam-4753	208	49	−1	−1	NOUN
ejpam-4753	208	50	ph	ph	NOUN
ejpam-4753	208	51	a	a	DET
ejpam-4753	208	52	−mod	−mod	NOUN
ejpam-4753	208	53	)	)	PUNCT
ejpam-4753	208	54	which	which	PRON
ejpam-4753	208	55	that	that	SCONJ
ejpam-4753	208	56	for	for	ADP
ejpam-4753	208	57	any	any	DET
ejpam-4753	208	58	graded	grade	VERB
ejpam-4753	208	59	left	leave	VERB
ejpam-4753	208	60	a−module	a−module	ADP
ejpam-4753	208	61	m	m	VERB
ejpam-4753	208	62	we	we	PRON
ejpam-4753	208	63	correspond	correspond	VERB
ejpam-4753	208	64	s	s	PRON
ejpam-4753	208	65	−1	−1	NOUN
ejpam-4753	208	66	ph	ph	NOUN
ejpam-4753	208	67	(	(	PUNCT
ejpam-4753	208	68	m	m	NOUN
ejpam-4753	208	69	)	)	PUNCT
ejpam-4753	208	70	and	and	CCONJ
ejpam-4753	208	71	for	for	ADP
ejpam-4753	208	72	all	all	DET
ejpam-4753	208	73	graded	grade	VERB
ejpam-4753	208	74	morphism	morphism	NOUN
ejpam-4753	208	75	of	of	ADP
ejpam-4753	208	76	degree	degree	NOUN
ejpam-4753	208	77	k	k	PROPN
ejpam-4753	208	78	∈	∈	PROPN
ejpam-4753	208	79	z	z	PROPN
ejpam-4753	208	80	of	of	ADP
ejpam-4753	208	81	graded	grade	VERB
ejpam-4753	208	82	left	leave	VERB
ejpam-4753	208	83	a−modules	a−module	NOUN
ejpam-4753	208	84	f	f	X
ejpam-4753	208	85	:	:	PUNCT
ejpam-4753	208	86	m	m	VERB
ejpam-4753	208	87	−→	−→	ADJ
ejpam-4753	209	1	n	n	INTJ
ejpam-4753	209	2	we	we	PRON
ejpam-4753	209	3	correspond	correspond	VERB
ejpam-4753	209	4	s	s	PRON
ejpam-4753	209	5	−1	−1	NOUN
ejpam-4753	209	6	ph	ph	NOUN
ejpam-4753	209	7	(	(	PUNCT
ejpam-4753	209	8	f	f	NOUN
ejpam-4753	209	9	)	)	PUNCT
ejpam-4753	209	10	of	of	ADP
ejpam-4753	209	11	degree	degree	NOUN
ejpam-4753	209	12	k	k	PROPN
ejpam-4753	209	13	∈	∈	PROPN
ejpam-4753	209	14	z	z	NOUN
ejpam-4753	209	15	is	be	AUX
ejpam-4753	209	16	additively	additively	ADV
ejpam-4753	209	17	exact	exact	ADJ
ejpam-4753	209	18	covariant	covariant	ADJ
ejpam-4753	209	19	functor	functor	PROPN
ejpam-4753	209	20	.	.	PUNCT
ejpam-4753	209	21	proof	proof	NOUN
ejpam-4753	209	22	.	.	PUNCT
ejpam-4753	210	1	since	since	SCONJ
ejpam-4753	210	2	the	the	DET
ejpam-4753	210	3	proposition	proposition	NOUN
ejpam-4753	210	4	8	8	NUM
ejpam-4753	210	5	sph	sph	NOUN
ejpam-4753	210	6	homogeneous	homogeneous	ADJ
ejpam-4753	210	7	multiplicatively	multiplicatively	ADV
ejpam-4753	210	8	closed	close	VERB
ejpam-4753	210	9	subset	subset	NOUN
ejpam-4753	210	10	,	,	PUNCT
ejpam-4753	210	11	so	so	SCONJ
ejpam-4753	210	12	s	s	NOUN
ejpam-4753	210	13	−1	−1	NOUN
ejpam-4753	210	14	ph	ph	NOUN
ejpam-4753	210	15	(	(	PUNCT
ejpam-4753	210	16	−	−	NOUN
ejpam-4753	210	17	)	)	PUNCT
ejpam-4753	210	18	:	:	PUNCT
ejpam-4753	210	19	gr(a	gr(a	PUNCT
ejpam-4753	210	20	−	−	PROPN
ejpam-4753	210	21	mod	mod	ADJ
ejpam-4753	210	22	)	)	PUNCT
ejpam-4753	210	23	−→	−→	NOUN
ejpam-4753	210	24	gr(s	gr(s	PUNCT
ejpam-4753	210	25	−1	−1	NOUN
ejpam-4753	210	26	ph	ph	NOUN
ejpam-4753	210	27	a	a	DET
ejpam-4753	210	28	−	−	NOUN
ejpam-4753	210	29	mod	mod	NOUN
ejpam-4753	210	30	)	)	PUNCT
ejpam-4753	210	31	is	be	AUX
ejpam-4753	210	32	a	a	DET
ejpam-4753	210	33	covariant	covariant	ADJ
ejpam-4753	210	34	functor	functor	NOUN
ejpam-4753	210	35	indeed	indeed	ADV
ejpam-4753	210	36	.	.	PUNCT
ejpam-4753	211	1	let	let	VERB
ejpam-4753	211	2	m	m	PRON
ejpam-4753	211	3	,	,	PUNCT
ejpam-4753	211	4	n	n	DET
ejpam-4753	211	5	two	two	NUM
ejpam-4753	211	6	graded	grade	VERB
ejpam-4753	211	7	left	leave	VERB
ejpam-4753	211	8	a−modules	a−module	NOUN
ejpam-4753	211	9	and	and	CCONJ
ejpam-4753	211	10	f	f	X
ejpam-4753	211	11	:	:	PUNCT
ejpam-4753	211	12	m	m	VERB
ejpam-4753	211	13	−→	−→	ADJ
ejpam-4753	212	1	n	n	NOUN
ejpam-4753	212	2	is	be	AUX
ejpam-4753	212	3	a	a	DET
ejpam-4753	212	4	graded	grade	VERB
ejpam-4753	212	5	morphism	morphism	NOUN
ejpam-4753	212	6	of	of	ADP
ejpam-4753	212	7	degree	degree	NOUN
ejpam-4753	213	1	k	k	PROPN
ejpam-4753	213	2	∈	∈	PROPN
ejpam-4753	213	3	z	z	PROPN
ejpam-4753	213	4	,	,	PUNCT
ejpam-4753	213	5	then	then	ADV
ejpam-4753	213	6	s	s	VERB
ejpam-4753	213	7	−1	−1	NOUN
ejpam-4753	213	8	ph	ph	NOUN
ejpam-4753	213	9	(	(	PUNCT
ejpam-4753	213	10	−)(f	−)(f	ADV
ejpam-4753	213	11	)	)	PUNCT
ejpam-4753	213	12	:	:	PUNCT
ejpam-4753	213	13	s	s	AUX
ejpam-4753	213	14	−1	−1	NOUN
ejpam-4753	213	15	ph	ph	ADJ
ejpam-4753	213	16	m	m	PROPN
ejpam-4753	213	17	−→	−→	NOUN
ejpam-4753	213	18	s	s	PART
ejpam-4753	213	19	−1	−1	NOUN
ejpam-4753	213	20	ph	ph	NOUN
ejpam-4753	213	21	n	n	PROPN
ejpam-4753	213	22	a.	a.	NOUN
ejpam-4753	213	23	o.	o.	PROPN
ejpam-4753	213	24	chbih	chbih	PROPN
ejpam-4753	213	25	,	,	PUNCT
ejpam-4753	213	26	m.	m.	PROPN
ejpam-4753	213	27	b.	b.	PROPN
ejpam-4753	213	28	maaouia	maaouia	PROPN
ejpam-4753	213	29	,	,	PUNCT
ejpam-4753	213	30	m.	m.	NOUN
ejpam-4753	213	31	sanghare	sanghare	PROPN
ejpam-4753	213	32	/	/	SYM
ejpam-4753	213	33	eur	eur	PROPN
ejpam-4753	213	34	.	.	PUNCT
ejpam-4753	214	1	j.	j.	PROPN
ejpam-4753	214	2	pure	pure	PROPN
ejpam-4753	214	3	appl	appl	PROPN
ejpam-4753	214	4	.	.	PROPN
ejpam-4753	214	5	math	math	PROPN
ejpam-4753	214	6	,	,	PUNCT
ejpam-4753	214	7	16	16	NUM
ejpam-4753	214	8	(	(	PUNCT
ejpam-4753	214	9	3	3	NUM
ejpam-4753	214	10	)	)	PUNCT
ejpam-4753	214	11	(	(	PUNCT
ejpam-4753	214	12	2023	2023	NUM
ejpam-4753	214	13	)	)	PUNCT
ejpam-4753	214	14	,	,	PUNCT
ejpam-4753	214	15	1913	1913	NUM
ejpam-4753	214	16	-	-	SYM
ejpam-4753	214	17	1939	1939	NUM
ejpam-4753	214	18	1924	1924	NUM
ejpam-4753	214	19	m	m	PROPN
ejpam-4753	214	20	s	s	PROPN
ejpam-4753	214	21	7−→	7−→	PROPN
ejpam-4753	214	22	f(m	f(m	PROPN
ejpam-4753	214	23	)	)	PUNCT
ejpam-4753	214	24	s	s	VERB
ejpam-4753	214	25	is	be	AUX
ejpam-4753	214	26	a	a	DET
ejpam-4753	214	27	graded	grade	VERB
ejpam-4753	214	28	morphism	morphism	NOUN
ejpam-4753	214	29	of	of	ADP
ejpam-4753	214	30	degree	degree	NOUN
ejpam-4753	215	1	k	k	PROPN
ejpam-4753	215	2	∈	∈	PROPN
ejpam-4753	215	3	z	z	X
ejpam-4753	215	4	of	of	ADP
ejpam-4753	215	5	s	s	NUM
ejpam-4753	215	6	−1	−1	NOUN
ejpam-4753	215	7	ph	ph	ADJ
ejpam-4753	215	8	a−modules	a−module	NOUN
ejpam-4753	215	9	and	and	CCONJ
ejpam-4753	215	10	for	for	ADP
ejpam-4753	215	11	any	any	DET
ejpam-4753	215	12	graded	grade	VERB
ejpam-4753	215	13	left	leave	VERB
ejpam-4753	215	14	a−module	a−module	ADP
ejpam-4753	215	15	m	m	NOUN
ejpam-4753	215	16	,	,	PUNCT
ejpam-4753	215	17	s−1	s−1	PROPN
ejpam-4753	215	18	ph	ph	NOUN
ejpam-4753	215	19	(	(	PUNCT
ejpam-4753	215	20	−)(m	−)(m	NOUN
ejpam-4753	215	21	)	)	PUNCT
ejpam-4753	215	22	=	=	SYM
ejpam-4753	215	23	s	s	PART
ejpam-4753	215	24	−1	−1	NOUN
ejpam-4753	215	25	ph	ph	NOUN
ejpam-4753	215	26	m	m	VERB
ejpam-4753	215	27	is	be	AUX
ejpam-4753	215	28	a	a	DET
ejpam-4753	215	29	graded	grade	VERB
ejpam-4753	215	30	left	leave	VERB
ejpam-4753	215	31	s	s	PRON
ejpam-4753	215	32	−1	−1	NOUN
ejpam-4753	215	33	ph	ph	ADJ
ejpam-4753	215	34	a−module	a−module	ADP
ejpam-4753	215	35	,	,	PUNCT
ejpam-4753	215	36	so	so	ADV
ejpam-4753	215	37	s−1	s−1	PROPN
ejpam-4753	215	38	ph	ph	NOUN
ejpam-4753	215	39	(	(	PUNCT
ejpam-4753	215	40	−	−	NOUN
ejpam-4753	215	41	)	)	PUNCT
ejpam-4753	215	42	is	be	AUX
ejpam-4753	215	43	a	a	DET
ejpam-4753	215	44	functor	functor	PROPN
ejpam-4753	215	45	covariant	covariant	NOUN
ejpam-4753	215	46	for	for	ADP
ejpam-4753	215	47	the	the	DET
ejpam-4753	215	48	category	category	NOUN
ejpam-4753	215	49	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	215	50	)	)	PUNCT
ejpam-4753	215	51	to	to	ADP
ejpam-4753	215	52	the	the	DET
ejpam-4753	215	53	category	category	NOUN
ejpam-4753	215	54	gr(s	gr(s	PUNCT
ejpam-4753	215	55	−1	−1	NOUN
ejpam-4753	215	56	ph	ph	NOUN
ejpam-4753	215	57	a−mod	a−mod	NOUN
ejpam-4753	215	58	)	)	PUNCT
ejpam-4753	215	59	.	.	PUNCT
ejpam-4753	216	1	furthermore	furthermore	ADV
ejpam-4753	216	2	s	s	AUX
ejpam-4753	216	3	−1	−1	NOUN
ejpam-4753	216	4	ph	ph	NOUN
ejpam-4753	216	5	(	(	PUNCT
ejpam-4753	216	6	−	−	NOUN
ejpam-4753	216	7	)	)	PUNCT
ejpam-4753	216	8	is	be	AUX
ejpam-4753	216	9	of	of	ADP
ejpam-4753	216	10	degree	degree	NOUN
ejpam-4753	216	11	k	k	PROPN
ejpam-4753	216	12	∈	∈	PROPN
ejpam-4753	216	13	z	z	PROPN
ejpam-4753	216	14	,	,	PUNCT
ejpam-4753	216	15	indeed	indeed	ADV
ejpam-4753	216	16	let	let	VERB
ejpam-4753	216	17	(	(	PUNCT
ejpam-4753	216	18	s	s	X
ejpam-4753	216	19	,	,	PUNCT
ejpam-4753	216	20	m	m	NOUN
ejpam-4753	216	21	)	)	PUNCT
ejpam-4753	216	22	∈	∈	PROPN
ejpam-4753	216	23	s	s	PART
ejpam-4753	216	24	×m	×m	NOUN
ejpam-4753	216	25	such	such	ADJ
ejpam-4753	216	26	that	that	DET
ejpam-4753	216	27	deg	deg	PROPN
ejpam-4753	216	28	(	(	PUNCT
ejpam-4753	216	29	m	m	PROPN
ejpam-4753	216	30	s	s	PART
ejpam-4753	216	31	)	)	PUNCT
ejpam-4753	216	32	=	=	SYM
ejpam-4753	216	33	deg(m)−	deg(m)−	NOUN
ejpam-4753	216	34	deg(s	deg(s	PROPN
ejpam-4753	216	35	)	)	PUNCT
ejpam-4753	217	1	=	=	SYM
ejpam-4753	217	2	d1	d1	PROPN
ejpam-4753	217	3	so	so	ADV
ejpam-4753	217	4	s	s	PART
ejpam-4753	217	5	−1	−1	NOUN
ejpam-4753	217	6	ph	ph	NOUN
ejpam-4753	217	7	(	(	PUNCT
ejpam-4753	217	8	−)(f	−)(f	ADV
ejpam-4753	217	9	)	)	PUNCT
ejpam-4753	217	10	(	(	PUNCT
ejpam-4753	217	11	m	m	PROPN
ejpam-4753	217	12	s	s	PART
ejpam-4753	217	13	)	)	PUNCT
ejpam-4753	217	14	=	=	SYM
ejpam-4753	217	15	f(m	f(m	PROPN
ejpam-4753	217	16	)	)	PUNCT
ejpam-4753	217	17	s	s	PART
ejpam-4753	217	18	,	,	PUNCT
ejpam-4753	217	19	and	and	CCONJ
ejpam-4753	217	20	deg(s	deg(s	PRON
ejpam-4753	217	21	−1	−1	NOUN
ejpam-4753	217	22	ph	ph	NOUN
ejpam-4753	217	23	(	(	PUNCT
ejpam-4753	217	24	f	f	NOUN
ejpam-4753	217	25	)	)	PUNCT
ejpam-4753	217	26	(	(	PUNCT
ejpam-4753	217	27	m	m	PROPN
ejpam-4753	217	28	s	s	PART
ejpam-4753	217	29	)	)	PUNCT
ejpam-4753	217	30	)	)	PUNCT
ejpam-4753	217	31	=	=	SYM
ejpam-4753	217	32	deg	deg	PROPN
ejpam-4753	217	33	(	(	PUNCT
ejpam-4753	217	34	f(m	f(m	PROPN
ejpam-4753	217	35	)	)	PUNCT
ejpam-4753	217	36	s	s	PART
ejpam-4753	217	37	)	)	PUNCT
ejpam-4753	217	38	=	=	PUNCT
ejpam-4753	217	39	deg(f(m))−	deg(f(m))−	NOUN
ejpam-4753	217	40	deg(s	deg(s	PROPN
ejpam-4753	217	41	)	)	PUNCT
ejpam-4753	217	42	=	=	SYM
ejpam-4753	217	43	(	(	PUNCT
ejpam-4753	217	44	deg(m	deg(m	PROPN
ejpam-4753	217	45	)	)	PUNCT
ejpam-4753	217	46	+	+	CCONJ
ejpam-4753	217	47	k)−	k)−	PROPN
ejpam-4753	217	48	deg(s	deg(s	PROPN
ejpam-4753	217	49	)	)	PUNCT
ejpam-4753	217	50	=	=	SYM
ejpam-4753	217	51	deg	deg	PROPN
ejpam-4753	217	52	(	(	PUNCT
ejpam-4753	217	53	m	m	PROPN
ejpam-4753	217	54	s	s	PART
ejpam-4753	217	55	)	)	PUNCT
ejpam-4753	218	1	+	+	CCONJ
ejpam-4753	218	2	k	k	X
ejpam-4753	218	3	=	=	SYM
ejpam-4753	218	4	d1	d1	PROPN
ejpam-4753	218	5	+	+	CCONJ
ejpam-4753	218	6	k.	k.	PROPN
ejpam-4753	218	7	thus	thus	ADV
ejpam-4753	218	8	s	s	VERB
ejpam-4753	218	9	−1	−1	NOUN
ejpam-4753	218	10	ph	ph	NOUN
ejpam-4753	218	11	(	(	PUNCT
ejpam-4753	218	12	−	−	NOUN
ejpam-4753	218	13	)	)	PUNCT
ejpam-4753	218	14	is	be	AUX
ejpam-4753	218	15	additively	additively	ADV
ejpam-4753	218	16	exact	exact	ADJ
ejpam-4753	218	17	covariant	covariant	ADJ
ejpam-4753	218	18	functor	functor	NOUN
ejpam-4753	218	19	,	,	PUNCT
ejpam-4753	218	20	since	since	SCONJ
ejpam-4753	218	21	s	s	PRON
ejpam-4753	218	22	−1	−1	NOUN
ejpam-4753	218	23	ph	ph	NOUN
ejpam-4753	218	24	(	(	PUNCT
ejpam-4753	218	25	−	−	NOUN
ejpam-4753	218	26	)	)	PUNCT
ejpam-4753	218	27	preserve	preserve	VERB
ejpam-4753	218	28	the	the	DET
ejpam-4753	218	29	exactness	exactness	NOUN
ejpam-4753	218	30	.	.	PUNCT
ejpam-4753	219	1	definition	definition	NOUN
ejpam-4753	219	2	10	10	NUM
ejpam-4753	219	3	.	.	PUNCT
ejpam-4753	220	1	let	let	VERB
ejpam-4753	220	2	a	a	DET
ejpam-4753	220	3	=	=	SYM
ejpam-4753	220	4	⊕	⊕	PROPN
ejpam-4753	220	5	n∈z	n∈z	VERB
ejpam-4753	220	6	an	an	PRON
ejpam-4753	220	7	is	be	AUX
ejpam-4753	220	8	a	a	DET
ejpam-4753	220	9	graded	grade	VERB
ejpam-4753	220	10	duo	duo	NOUN
ejpam-4753	220	11	-	-	PUNCT
ejpam-4753	220	12	ring	ring	NOUN
ejpam-4753	220	13	,	,	PUNCT
ejpam-4753	220	14	m	m	VERB
ejpam-4753	220	15	=	=	ADJ
ejpam-4753	220	16	⊕	⊕	PROPN
ejpam-4753	221	1	n∈z	n∈z	ADJ
ejpam-4753	221	2	mn	mn	PROPN
ejpam-4753	221	3	be	be	AUX
ejpam-4753	221	4	a	a	DET
ejpam-4753	221	5	left	leave	VERB
ejpam-4753	221	6	graded	grade	VERB
ejpam-4753	221	7	a−module	a−module	ADP
ejpam-4753	221	8	,	,	PUNCT
ejpam-4753	221	9	p	p	PRON
ejpam-4753	221	10	is	be	AUX
ejpam-4753	221	11	a	a	DET
ejpam-4753	221	12	prime	prime	ADJ
ejpam-4753	221	13	ideal	ideal	NOUN
ejpam-4753	221	14	of	of	ADP
ejpam-4753	221	15	a	a	PRON
ejpam-4753	221	16	and	and	CCONJ
ejpam-4753	221	17	sh	sh	INTJ
ejpam-4753	221	18	be	be	AUX
ejpam-4753	221	19	the	the	DET
ejpam-4753	221	20	set	set	NOUN
ejpam-4753	221	21	of	of	ADP
ejpam-4753	221	22	homogeneous	homogeneous	ADJ
ejpam-4753	221	23	regular	regular	ADJ
ejpam-4753	221	24	elements	element	NOUN
ejpam-4753	221	25	of	of	ADP
ejpam-4753	221	26	a\p	a\p	NOUN
ejpam-4753	221	27	then	then	ADV
ejpam-4753	221	28	:	:	PUNCT
ejpam-4753	221	29	(	(	PUNCT
ejpam-4753	221	30	i	i	NOUN
ejpam-4753	221	31	)	)	PUNCT
ejpam-4753	221	32	s−1	s−1	NOUN
ejpam-4753	221	33	ph	ph	VERB
ejpam-4753	221	34	a	a	PRON
ejpam-4753	221	35	is	be	AUX
ejpam-4753	221	36	called	call	VERB
ejpam-4753	221	37	homogeneous	homogeneous	ADJ
ejpam-4753	221	38	localized	localize	VERB
ejpam-4753	221	39	to	to	ADP
ejpam-4753	221	40	a	a	PRON
ejpam-4753	221	41	in	in	ADP
ejpam-4753	221	42	p	p	NOUN
ejpam-4753	221	43	.	.	PUNCT
ejpam-4753	222	1	and	and	CCONJ
ejpam-4753	222	2	denoted	denote	VERB
ejpam-4753	222	3	by	by	ADP
ejpam-4753	222	4	aph	aph	PROPN
ejpam-4753	222	5	;	;	PUNCT
ejpam-4753	222	6	(	(	PUNCT
ejpam-4753	222	7	ii	ii	X
ejpam-4753	222	8	)	)	PUNCT
ejpam-4753	222	9	s−1	s−1	PROPN
ejpam-4753	222	10	ph	ph	ADJ
ejpam-4753	222	11	m	m	VERB
ejpam-4753	222	12	is	be	AUX
ejpam-4753	222	13	called	call	VERB
ejpam-4753	222	14	homogeneous	homogeneous	ADJ
ejpam-4753	222	15	localized	localize	VERB
ejpam-4753	222	16	to	to	ADP
ejpam-4753	222	17	m	m	PRON
ejpam-4753	222	18	in	in	ADP
ejpam-4753	222	19	p	p	PROPN
ejpam-4753	222	20	.	.	PUNCT
ejpam-4753	223	1	and	and	CCONJ
ejpam-4753	223	2	denoted	denote	VERB
ejpam-4753	223	3	by	by	ADP
ejpam-4753	223	4	mph	mph	NOUN
ejpam-4753	223	5	.	.	PUNCT
ejpam-4753	224	1	corollary	corollary	ADJ
ejpam-4753	224	2	6	6	NUM
ejpam-4753	224	3	.	.	PUNCT
ejpam-4753	225	1	let	let	VERB
ejpam-4753	225	2	a	a	DET
ejpam-4753	225	3	=	=	SYM
ejpam-4753	225	4	⊕	⊕	PROPN
ejpam-4753	225	5	n∈z	n∈z	VERB
ejpam-4753	225	6	an	an	DET
ejpam-4753	225	7	be	be	AUX
ejpam-4753	225	8	a	a	DET
ejpam-4753	225	9	graded	grade	VERB
ejpam-4753	225	10	duo	duo	NOUN
ejpam-4753	225	11	-	-	PUNCT
ejpam-4753	225	12	ring	ring	NOUN
ejpam-4753	225	13	,	,	PUNCT
ejpam-4753	225	14	p	p	X
ejpam-4753	225	15	be	be	AUX
ejpam-4753	225	16	a	a	DET
ejpam-4753	225	17	prime	prime	ADJ
ejpam-4753	225	18	ideal	ideal	NOUN
ejpam-4753	225	19	of	of	ADP
ejpam-4753	225	20	a	a	PRON
ejpam-4753	225	21	and	and	CCONJ
ejpam-4753	225	22	sph	sph	PROPN
ejpam-4753	225	23	be	be	AUX
ejpam-4753	225	24	the	the	DET
ejpam-4753	225	25	set	set	NOUN
ejpam-4753	225	26	of	of	ADP
ejpam-4753	225	27	all	all	DET
ejpam-4753	225	28	homogeneous	homogeneous	ADJ
ejpam-4753	225	29	regular	regular	ADJ
ejpam-4753	225	30	elements	element	NOUN
ejpam-4753	225	31	of	of	ADP
ejpam-4753	225	32	a\p	a\p	PROPN
ejpam-4753	225	33	,	,	PUNCT
ejpam-4753	225	34	then	then	ADV
ejpam-4753	225	35	the	the	DET
ejpam-4753	225	36	relation	relation	NOUN
ejpam-4753	225	37	s−1	s−1	PROPN
ejpam-4753	225	38	ph	ph	NOUN
ejpam-4753	225	39	(	(	PUNCT
ejpam-4753	225	40	−	−	NOUN
ejpam-4753	225	41	)	)	PUNCT
ejpam-4753	225	42	:	:	PUNCT
ejpam-4753	225	43	gr(a−mod	gr(a−mod	X
ejpam-4753	225	44	)	)	PUNCT
ejpam-4753	225	45	−→	−→	NOUN
ejpam-4753	225	46	aph	aph	PROPN
ejpam-4753	225	47	−mod	−mod	NOUN
ejpam-4753	226	1	which	which	DET
ejpam-4753	226	2	that	that	SCONJ
ejpam-4753	226	3	for	for	ADP
ejpam-4753	226	4	any	any	DET
ejpam-4753	226	5	graded	grade	VERB
ejpam-4753	226	6	left	leave	VERB
ejpam-4753	226	7	a−module	a−module	ADP
ejpam-4753	226	8	m	m	VERB
ejpam-4753	226	9	we	we	PRON
ejpam-4753	226	10	correspond	correspond	VERB
ejpam-4753	226	11	mph	mph	NOUN
ejpam-4753	226	12	and	and	CCONJ
ejpam-4753	226	13	for	for	ADP
ejpam-4753	226	14	all	all	DET
ejpam-4753	226	15	graded	grade	VERB
ejpam-4753	226	16	morphism	morphism	NOUN
ejpam-4753	226	17	of	of	ADP
ejpam-4753	226	18	degree	degree	NOUN
ejpam-4753	226	19	k	k	PROPN
ejpam-4753	226	20	∈	∈	PROPN
ejpam-4753	226	21	z	z	PROPN
ejpam-4753	226	22	of	of	ADP
ejpam-4753	226	23	graded	grade	VERB
ejpam-4753	226	24	left	leave	VERB
ejpam-4753	226	25	a−modules	a−module	NOUN
ejpam-4753	226	26	f	f	X
ejpam-4753	226	27	:	:	PUNCT
ejpam-4753	226	28	m	m	VERB
ejpam-4753	226	29	−→	−→	ADJ
ejpam-4753	227	1	n	n	CCONJ
ejpam-4753	227	2	we	we	PRON
ejpam-4753	227	3	correspond	correspond	VERB
ejpam-4753	227	4	s−1	s−1	ADJ
ejpam-4753	227	5	ph	ph	NOUN
ejpam-4753	227	6	(	(	PUNCT
ejpam-4753	227	7	f	f	NOUN
ejpam-4753	227	8	)	)	PUNCT
ejpam-4753	227	9	of	of	ADP
ejpam-4753	227	10	degree	degree	NOUN
ejpam-4753	228	1	k	k	PROPN
ejpam-4753	228	2	∈	∈	PROPN
ejpam-4753	229	1	z	z	NOUN
ejpam-4753	229	2	is	be	AUX
ejpam-4753	229	3	additively	additively	ADV
ejpam-4753	229	4	exact	exact	ADJ
ejpam-4753	229	5	covariant	covariant	ADJ
ejpam-4753	229	6	functor	functor	PROPN
ejpam-4753	229	7	.	.	PUNCT
ejpam-4753	229	8	proof	proof	NOUN
ejpam-4753	229	9	.	.	PUNCT
ejpam-4753	230	1	it	it	PRON
ejpam-4753	230	2	is	be	AUX
ejpam-4753	230	3	enough	enough	ADJ
ejpam-4753	230	4	to	to	PART
ejpam-4753	230	5	note	note	VERB
ejpam-4753	230	6	that	that	DET
ejpam-4753	230	7	sph	sph	PROPN
ejpam-4753	230	8	=	=	PUNCT
ejpam-4753	230	9	sph	sph	PROPN
ejpam-4753	230	10	since	since	SCONJ
ejpam-4753	230	11	the	the	DET
ejpam-4753	230	12	corollary	corollary	ADJ
ejpam-4753	230	13	2	2	NUM
ejpam-4753	230	14	.	.	NOUN
ejpam-4753	230	15	4	4	NUM
ejpam-4753	230	16	.	.	X
ejpam-4753	230	17	localization	localization	NOUN
ejpam-4753	230	18	of	of	ADP
ejpam-4753	230	19	complex	complex	NOUN
ejpam-4753	230	20	in	in	ADP
ejpam-4753	230	21	comp	comp	NOUN
ejpam-4753	230	22	(	(	PUNCT
ejpam-4753	230	23	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	230	24	)	)	PUNCT
ejpam-4753	230	25	)	)	PUNCT
ejpam-4753	230	26	over	over	ADP
ejpam-4753	230	27	a	a	DET
ejpam-4753	230	28	duo	duo	NOUN
ejpam-4753	230	29	-	-	PUNCT
ejpam-4753	230	30	ring	ring	NOUN
ejpam-4753	230	31	proposition	proposition	NOUN
ejpam-4753	230	32	13	13	NUM
ejpam-4753	230	33	.	.	PUNCT
ejpam-4753	231	1	let	let	VERB
ejpam-4753	231	2	a	a	DET
ejpam-4753	231	3	=	=	SYM
ejpam-4753	231	4	⊕	⊕	PROPN
ejpam-4753	231	5	n∈z	n∈z	VERB
ejpam-4753	231	6	an	an	DET
ejpam-4753	231	7	be	be	AUX
ejpam-4753	231	8	a	a	DET
ejpam-4753	231	9	graded	grade	VERB
ejpam-4753	231	10	ring	ring	NOUN
ejpam-4753	231	11	,	,	PUNCT
ejpam-4753	231	12	m	m	VERB
ejpam-4753	231	13	=	=	ADJ
ejpam-4753	231	14	⊕	⊕	PROPN
ejpam-4753	231	15	n∈z	n∈z	VERB
ejpam-4753	231	16	mn	mn	PROPN
ejpam-4753	231	17	and	and	CCONJ
ejpam-4753	231	18	n	n	PROPN
ejpam-4753	231	19	=	=	PROPN
ejpam-4753	231	20	⊕	⊕	PROPN
ejpam-4753	231	21	n∈z	n∈z	ADJ
ejpam-4753	231	22	nn	nn	PROPN
ejpam-4753	231	23	are	be	AUX
ejpam-4753	231	24	two	two	NUM
ejpam-4753	231	25	graded	grade	VERB
ejpam-4753	231	26	left	leave	VERB
ejpam-4753	231	27	a−module	a−module	ADP
ejpam-4753	231	28	f	f	X
ejpam-4753	231	29	:	:	PUNCT
ejpam-4753	231	30	m	m	VERB
ejpam-4753	231	31	−→	−→	ADJ
ejpam-4753	232	1	n	n	NOUN
ejpam-4753	232	2	is	be	AUX
ejpam-4753	232	3	a	a	DET
ejpam-4753	232	4	graded	grade	VERB
ejpam-4753	232	5	morphism	morphism	NOUN
ejpam-4753	232	6	of	of	ADP
ejpam-4753	232	7	degree	degree	NOUN
ejpam-4753	232	8	k	k	PROPN
ejpam-4753	232	9	∈	∈	PROPN
ejpam-4753	232	10	z	z	PROPN
ejpam-4753	232	11	of	of	ADP
ejpam-4753	232	12	a	a	DET
ejpam-4753	232	13	graded	grade	VERB
ejpam-4753	232	14	left	leave	VERB
ejpam-4753	232	15	a−modules	a−module	NOUN
ejpam-4753	232	16	,	,	PUNCT
ejpam-4753	232	17	then	then	ADV
ejpam-4753	232	18	for	for	ADP
ejpam-4753	232	19	all	all	DET
ejpam-4753	232	20	n	n	PRON
ejpam-4753	232	21	∈	∈	PROPN
ejpam-4753	232	22	z	z	NOUN
ejpam-4753	232	23	fk(n	fk(n	PROPN
ejpam-4753	232	24	)	)	PUNCT
ejpam-4753	232	25	:	:	PUNCT
ejpam-4753	233	1	m(n	m(n	PROPN
ejpam-4753	233	2	)	)	PUNCT
ejpam-4753	233	3	−→	−→	ADJ
ejpam-4753	233	4	n(n	n(n	NUM
ejpam-4753	233	5	)	)	PUNCT
ejpam-4753	233	6	a.	a.	NOUN
ejpam-4753	233	7	o.	o.	PROPN
ejpam-4753	233	8	chbih	chbih	PROPN
ejpam-4753	233	9	,	,	PUNCT
ejpam-4753	233	10	m.	m.	PROPN
ejpam-4753	233	11	b.	b.	PROPN
ejpam-4753	233	12	maaouia	maaouia	PROPN
ejpam-4753	233	13	,	,	PUNCT
ejpam-4753	233	14	m.	m.	NOUN
ejpam-4753	233	15	sanghare	sanghare	PROPN
ejpam-4753	233	16	/	/	SYM
ejpam-4753	233	17	eur	eur	PROPN
ejpam-4753	233	18	.	.	PUNCT
ejpam-4753	234	1	j.	j.	PROPN
ejpam-4753	234	2	pure	pure	PROPN
ejpam-4753	234	3	appl	appl	PROPN
ejpam-4753	234	4	.	.	PROPN
ejpam-4753	234	5	math	math	PROPN
ejpam-4753	234	6	,	,	PUNCT
ejpam-4753	234	7	16	16	NUM
ejpam-4753	234	8	(	(	PUNCT
ejpam-4753	234	9	3	3	NUM
ejpam-4753	234	10	)	)	PUNCT
ejpam-4753	234	11	(	(	PUNCT
ejpam-4753	234	12	2023	2023	NUM
ejpam-4753	234	13	)	)	PUNCT
ejpam-4753	234	14	,	,	PUNCT
ejpam-4753	234	15	1913	1913	NUM
ejpam-4753	234	16	-	-	SYM
ejpam-4753	234	17	1939	1939	NUM
ejpam-4753	234	18	1925	1925	NUM
ejpam-4753	234	19	m	m	NOUN
ejpam-4753	234	20	7−→	7−→	NOUN
ejpam-4753	234	21	fk(n)(m	fk(n)(m	NOUN
ejpam-4753	234	22	)	)	PUNCT
ejpam-4753	234	23	=	=	SYM
ejpam-4753	235	1	f(m	f(m	PROPN
ejpam-4753	235	2	)	)	PUNCT
ejpam-4753	235	3	is	be	AUX
ejpam-4753	235	4	graded	grade	VERB
ejpam-4753	235	5	morphism	morphism	NOUN
ejpam-4753	235	6	of	of	ADP
ejpam-4753	235	7	degree	degree	NOUN
ejpam-4753	235	8	k	k	PROPN
ejpam-4753	235	9	∈	∈	PROPN
ejpam-4753	235	10	z	z	PROPN
ejpam-4753	235	11	of	of	ADP
ejpam-4753	235	12	graded	grade	VERB
ejpam-4753	235	13	left	leave	VERB
ejpam-4753	235	14	a−modules	a−module	NOUN
ejpam-4753	235	15	.	.	PUNCT
ejpam-4753	236	1	proof	proof	NOUN
ejpam-4753	236	2	.	.	PUNCT
ejpam-4753	237	1	we	we	PRON
ejpam-4753	237	2	have	have	VERB
ejpam-4753	237	3	f	f	X
ejpam-4753	237	4	:	:	PUNCT
ejpam-4753	237	5	m	m	VERB
ejpam-4753	237	6	−→	−→	ADJ
ejpam-4753	237	7	n	n	NOUN
ejpam-4753	237	8	is	be	AUX
ejpam-4753	237	9	graded	grade	VERB
ejpam-4753	237	10	morphism	morphism	NOUN
ejpam-4753	237	11	of	of	ADP
ejpam-4753	237	12	degree	degree	NOUN
ejpam-4753	237	13	k	k	PROPN
ejpam-4753	237	14	∈	∈	PROPN
ejpam-4753	237	15	z	z	PROPN
ejpam-4753	237	16	of	of	ADP
ejpam-4753	237	17	graded	grade	VERB
ejpam-4753	237	18	left	leave	VERB
ejpam-4753	237	19	a−modules	a−module	NOUN
ejpam-4753	237	20	,	,	PUNCT
ejpam-4753	237	21	and	and	CCONJ
ejpam-4753	237	22	m(n	m(n	PROPN
ejpam-4753	237	23	)	)	PUNCT
ejpam-4753	237	24	is	be	AUX
ejpam-4753	237	25	a	a	DET
ejpam-4753	237	26	sub	sub	NOUN
ejpam-4753	237	27	-	-	NOUN
ejpam-4753	237	28	module	module	NOUN
ejpam-4753	237	29	of	of	ADP
ejpam-4753	237	30	graded	grade	VERB
ejpam-4753	237	31	left	leave	VERB
ejpam-4753	237	32	a−module	a−module	ADP
ejpam-4753	237	33	m	m	AUX
ejpam-4753	237	34	then	then	ADV
ejpam-4753	237	35	let	let	VERB
ejpam-4753	237	36	m	m	PRON
ejpam-4753	237	37	∈	∈	VERB
ejpam-4753	237	38	m(n	m(n	NOUN
ejpam-4753	237	39	)	)	PUNCT
ejpam-4753	237	40	,	,	PUNCT
ejpam-4753	237	41	so	so	ADV
ejpam-4753	237	42	m	m	ADV
ejpam-4753	237	43	=	=	VERB
ejpam-4753	237	44	∑	∑	PUNCT
ejpam-4753	237	45	i∈z	i∈z	PROPN
ejpam-4753	237	46	mi+n	mi+n	PROPN
ejpam-4753	237	47	=	=	AUX
ejpam-4753	237	48	⇒	⇒	NOUN
ejpam-4753	237	49	fk(n)(m	fk(n)(m	VERB
ejpam-4753	237	50	)	)	PUNCT
ejpam-4753	237	51	=	=	SYM
ejpam-4753	237	52	f(m	f(m	PROPN
ejpam-4753	237	53	)	)	PUNCT
ejpam-4753	237	54	=	=	SYM
ejpam-4753	238	1	f	f	X
ejpam-4753	238	2	(	(	PUNCT
ejpam-4753	238	3	∑	∑	ADV
ejpam-4753	238	4	i∈z	i∈z	PROPN
ejpam-4753	238	5	mi+n	mi+n	PROPN
ejpam-4753	238	6	)	)	PUNCT
ejpam-4753	238	7	=	=	PUNCT
ejpam-4753	238	8	∑	∑	PUNCT
ejpam-4753	238	9	i∈z	i∈z	PROPN
ejpam-4753	238	10	f(mi+n	f(mi+n	NOUN
ejpam-4753	238	11	)	)	PUNCT
ejpam-4753	238	12	or	or	CCONJ
ejpam-4753	238	13	f(mi+n	f(mi+n	NOUN
ejpam-4753	238	14	)	)	PUNCT
ejpam-4753	238	15	∈	∈	NOUN
ejpam-4753	238	16	ni+n+k	ni+n+k	NOUN
ejpam-4753	238	17	=	=	PUNCT
ejpam-4753	238	18	(	(	PUNCT
ejpam-4753	238	19	n(n))i+k	n(n))i+k	INTJ
ejpam-4753	238	20	thus	thus	ADV
ejpam-4753	238	21	f	f	X
ejpam-4753	238	22	is	be	AUX
ejpam-4753	238	23	graded	grade	VERB
ejpam-4753	238	24	morphism	morphism	NOUN
ejpam-4753	238	25	of	of	ADP
ejpam-4753	238	26	degree	degree	NOUN
ejpam-4753	238	27	k	k	PROPN
ejpam-4753	238	28	∈	∈	PROPN
ejpam-4753	238	29	z	z	PROPN
ejpam-4753	238	30	of	of	ADP
ejpam-4753	238	31	a	a	DET
ejpam-4753	238	32	graded	grade	VERB
ejpam-4753	238	33	left	leave	VERB
ejpam-4753	238	34	a−modules	a−module	NOUN
ejpam-4753	238	35	.	.	PUNCT
ejpam-4753	239	1	corollary	corollary	ADJ
ejpam-4753	239	2	7	7	NUM
ejpam-4753	239	3	.	.	PUNCT
ejpam-4753	240	1	let	let	VERB
ejpam-4753	240	2	a	a	DET
ejpam-4753	240	3	=	=	SYM
ejpam-4753	240	4	⊕	⊕	PROPN
ejpam-4753	240	5	n∈z	n∈z	VERB
ejpam-4753	240	6	an	an	DET
ejpam-4753	240	7	be	be	AUX
ejpam-4753	240	8	a	a	DET
ejpam-4753	240	9	graded	grade	VERB
ejpam-4753	240	10	ring	ring	NOUN
ejpam-4753	240	11	,	,	PUNCT
ejpam-4753	240	12	m	m	VERB
ejpam-4753	240	13	=	=	ADJ
ejpam-4753	240	14	⊕	⊕	PROPN
ejpam-4753	240	15	n∈z	n∈z	VERB
ejpam-4753	240	16	mn	mn	PROPN
ejpam-4753	240	17	and	and	CCONJ
ejpam-4753	240	18	n	n	PROPN
ejpam-4753	240	19	=	=	PROPN
ejpam-4753	240	20	⊕	⊕	PROPN
ejpam-4753	240	21	n∈z	n∈z	ADJ
ejpam-4753	240	22	nn	nn	PROPN
ejpam-4753	240	23	are	be	AUX
ejpam-4753	240	24	two	two	NUM
ejpam-4753	240	25	graded	grade	VERB
ejpam-4753	240	26	left	leave	VERB
ejpam-4753	240	27	a−module	a−module	ADP
ejpam-4753	240	28	f	f	X
ejpam-4753	240	29	:	:	PUNCT
ejpam-4753	240	30	m	m	VERB
ejpam-4753	240	31	−→	−→	ADJ
ejpam-4753	241	1	n	n	NOUN
ejpam-4753	241	2	is	be	AUX
ejpam-4753	241	3	a	a	DET
ejpam-4753	241	4	graded	grade	VERB
ejpam-4753	241	5	morphism	morphism	NOUN
ejpam-4753	241	6	of	of	ADP
ejpam-4753	241	7	degree	degree	NOUN
ejpam-4753	241	8	k	k	PROPN
ejpam-4753	241	9	∈	∈	PROPN
ejpam-4753	241	10	z	z	PROPN
ejpam-4753	241	11	of	of	ADP
ejpam-4753	241	12	a	a	DET
ejpam-4753	241	13	graded	grade	VERB
ejpam-4753	241	14	left	leave	VERB
ejpam-4753	241	15	a−modules	a−module	NOUN
ejpam-4753	241	16	,	,	PUNCT
ejpam-4753	241	17	then	then	ADV
ejpam-4753	241	18	f	f	X
ejpam-4753	241	19	:	:	PUNCT
ejpam-4753	241	20	m	m	VERB
ejpam-4753	241	21	−→	−→	ADJ
ejpam-4753	241	22	n(k	n(k	PROPN
ejpam-4753	241	23	)	)	PUNCT
ejpam-4753	241	24	is	be	AUX
ejpam-4753	241	25	graded	grade	VERB
ejpam-4753	241	26	morphism	morphism	NOUN
ejpam-4753	241	27	of	of	ADP
ejpam-4753	241	28	graded	grade	VERB
ejpam-4753	241	29	left	leave	VERB
ejpam-4753	241	30	a−modules	a−module	NOUN
ejpam-4753	241	31	.	.	PUNCT
ejpam-4753	242	1	proof	proof	NOUN
ejpam-4753	242	2	.	.	PUNCT
ejpam-4753	243	1	we	we	PRON
ejpam-4753	243	2	have	have	VERB
ejpam-4753	243	3	f	f	X
ejpam-4753	243	4	:	:	PUNCT
ejpam-4753	243	5	m	m	VERB
ejpam-4753	243	6	−→	−→	ADJ
ejpam-4753	243	7	n	n	NOUN
ejpam-4753	243	8	is	be	AUX
ejpam-4753	243	9	graded	grade	VERB
ejpam-4753	243	10	morphism	morphism	NOUN
ejpam-4753	243	11	of	of	ADP
ejpam-4753	243	12	degree	degree	NOUN
ejpam-4753	243	13	k	k	PROPN
ejpam-4753	243	14	∈	∈	PROPN
ejpam-4753	243	15	z	z	PROPN
ejpam-4753	243	16	of	of	ADP
ejpam-4753	243	17	graded	grade	VERB
ejpam-4753	243	18	left	leave	VERB
ejpam-4753	243	19	a−modules	a−module	NOUN
ejpam-4753	243	20	,	,	PUNCT
ejpam-4753	243	21	and	and	CCONJ
ejpam-4753	243	22	n(k	n(k	PROPN
ejpam-4753	243	23	)	)	PUNCT
ejpam-4753	243	24	is	be	AUX
ejpam-4753	243	25	a	a	DET
ejpam-4753	243	26	sub	sub	NOUN
ejpam-4753	243	27	-	-	NOUN
ejpam-4753	243	28	module	module	NOUN
ejpam-4753	243	29	of	of	ADP
ejpam-4753	243	30	graded	grade	VERB
ejpam-4753	243	31	left	leave	VERB
ejpam-4753	243	32	a−module	a−module	ADP
ejpam-4753	243	33	n	n	NOUN
ejpam-4753	243	34	then	then	ADV
ejpam-4753	243	35	let	let	VERB
ejpam-4753	243	36	m	m	PRON
ejpam-4753	243	37	∈	∈	VERB
ejpam-4753	243	38	m	m	NOUN
ejpam-4753	243	39	,	,	PUNCT
ejpam-4753	243	40	so	so	ADV
ejpam-4753	243	41	m	m	ADV
ejpam-4753	243	42	=	=	VERB
ejpam-4753	243	43	∑	∑	PUNCT
ejpam-4753	243	44	i∈z	i∈z	PROPN
ejpam-4753	243	45	mi	mi	PROPN
ejpam-4753	243	46	=	=	NOUN
ejpam-4753	243	47	⇒	⇒	PROPN
ejpam-4753	243	48	f(m	f(m	PROPN
ejpam-4753	243	49	)	)	PUNCT
ejpam-4753	243	50	=	=	SYM
ejpam-4753	244	1	f	f	X
ejpam-4753	244	2	(	(	PUNCT
ejpam-4753	244	3	∑	∑	PROPN
ejpam-4753	244	4	i∈z	i∈z	PROPN
ejpam-4753	244	5	mi	mi	NOUN
ejpam-4753	244	6	)	)	PUNCT
ejpam-4753	244	7	=	=	PUNCT
ejpam-4753	244	8	∑	∑	PUNCT
ejpam-4753	244	9	i∈z	i∈z	PROPN
ejpam-4753	244	10	f(mi	f(mi	NOUN
ejpam-4753	244	11	)	)	PUNCT
ejpam-4753	244	12	.	.	PUNCT
ejpam-4753	245	1	or	or	CCONJ
ejpam-4753	245	2	f(mi	f(mi	PROPN
ejpam-4753	245	3	)	)	PUNCT
ejpam-4753	245	4	∈	∈	PROPN
ejpam-4753	245	5	ni+k	ni+k	PROPN
ejpam-4753	245	6	=	=	PUNCT
ejpam-4753	245	7	(	(	PUNCT
ejpam-4753	245	8	n(k))i	n(k))i	PROPN
ejpam-4753	245	9	thus	thus	ADV
ejpam-4753	245	10	f	f	PROPN
ejpam-4753	245	11	is	be	AUX
ejpam-4753	245	12	graded	grade	VERB
ejpam-4753	245	13	morphism	morphism	NOUN
ejpam-4753	245	14	of	of	ADP
ejpam-4753	245	15	a	a	DET
ejpam-4753	245	16	graded	grade	VERB
ejpam-4753	245	17	left	leave	VERB
ejpam-4753	245	18	a−modules	a−module	NOUN
ejpam-4753	245	19	.	.	PUNCT
ejpam-4753	246	1	theorem	theorem	NOUN
ejpam-4753	246	2	5	5	NUM
ejpam-4753	246	3	.	.	PUNCT
ejpam-4753	247	1	let	let	VERB
ejpam-4753	247	2	a	a	DET
ejpam-4753	247	3	be	be	AUX
ejpam-4753	247	4	a	a	DET
ejpam-4753	247	5	graded	grade	VERB
ejpam-4753	247	6	ring	ring	NOUN
ejpam-4753	247	7	and	and	CCONJ
ejpam-4753	247	8	gr(a	gr(a	NOUN
ejpam-4753	247	9	−	−	PROPN
ejpam-4753	247	10	mod	mod	PROPN
ejpam-4753	247	11	)	)	PUNCT
ejpam-4753	247	12	the	the	DET
ejpam-4753	247	13	category	category	NOUN
ejpam-4753	247	14	of	of	ADP
ejpam-4753	247	15	a	a	DET
ejpam-4753	247	16	graded	grade	VERB
ejpam-4753	247	17	left	leave	VERB
ejpam-4753	247	18	a−modules	a−module	NOUN
ejpam-4753	247	19	,	,	PUNCT
ejpam-4753	247	20	then	then	ADV
ejpam-4753	247	21	for	for	ADP
ejpam-4753	247	22	all	all	DET
ejpam-4753	247	23	n	n	PRON
ejpam-4753	247	24	∈	∈	PROPN
ejpam-4753	247	25	z	z	NOUN
ejpam-4753	247	26	the	the	DET
ejpam-4753	247	27	relation	relation	NOUN
ejpam-4753	247	28	(	(	PUNCT
ejpam-4753	247	29	−)(n	−)(n	NOUN
ejpam-4753	247	30	)	)	PUNCT
ejpam-4753	247	31	:	:	PUNCT
ejpam-4753	247	32	gr(a	gr(a	PUNCT
ejpam-4753	248	1	−	−	PROPN
ejpam-4753	248	2	mod	mod	PROPN
ejpam-4753	248	3	)	)	PUNCT
ejpam-4753	248	4	−→	−→	NOUN
ejpam-4753	248	5	gr(a	gr(a	ADJ
ejpam-4753	248	6	−	−	PROPN
ejpam-4753	248	7	mod	mod	PROPN
ejpam-4753	248	8	)	)	PUNCT
ejpam-4753	248	9	which	which	PRON
ejpam-4753	248	10	that	that	SCONJ
ejpam-4753	248	11	for	for	ADP
ejpam-4753	248	12	any	any	DET
ejpam-4753	248	13	m	m	NOUN
ejpam-4753	248	14	∈	∈	NOUN
ejpam-4753	248	15	gr(a	gr(a	PUNCT
ejpam-4753	248	16	−	−	PROPN
ejpam-4753	248	17	mod	mod	PROPN
ejpam-4753	248	18	)	)	PUNCT
ejpam-4753	248	19	we	we	PRON
ejpam-4753	248	20	made	make	VERB
ejpam-4753	248	21	to	to	PART
ejpam-4753	248	22	correspond	correspond	VERB
ejpam-4753	248	23	m(n	m(n	NOUN
ejpam-4753	248	24	)	)	PUNCT
ejpam-4753	248	25	and	and	CCONJ
ejpam-4753	248	26	for	for	ADP
ejpam-4753	248	27	all	all	DET
ejpam-4753	248	28	graded	grade	VERB
ejpam-4753	248	29	morphism	morphism	NOUN
ejpam-4753	248	30	of	of	ADP
ejpam-4753	248	31	degree	degree	NOUN
ejpam-4753	248	32	k	k	PROPN
ejpam-4753	248	33	∈	∈	PROPN
ejpam-4753	248	34	z	z	PROPN
ejpam-4753	248	35	of	of	ADP
ejpam-4753	248	36	a	a	DET
ejpam-4753	248	37	graded	grade	VERB
ejpam-4753	248	38	left	leave	VERB
ejpam-4753	248	39	a−modules	a−module	NOUN
ejpam-4753	248	40	f	f	X
ejpam-4753	248	41	:	:	PUNCT
ejpam-4753	248	42	m	m	VERB
ejpam-4753	248	43	−→	−→	ADJ
ejpam-4753	249	1	n	n	PRON
ejpam-4753	249	2	we	we	PRON
ejpam-4753	249	3	correspond	correspond	VERB
ejpam-4753	249	4	fk(n	fk(n	NOUN
ejpam-4753	249	5	)	)	PUNCT
ejpam-4753	249	6	is	be	AUX
ejpam-4753	249	7	a	a	DET
ejpam-4753	249	8	additively	additively	ADV
ejpam-4753	249	9	exact	exact	ADJ
ejpam-4753	249	10	covariant	covariant	PROPN
ejpam-4753	249	11	functor	functor	PROPN
ejpam-4753	249	12	.	.	PUNCT
ejpam-4753	249	13	proof	proof	NOUN
ejpam-4753	249	14	.	.	PUNCT
ejpam-4753	250	1	let	let	VERB
ejpam-4753	250	2	f	f	NOUN
ejpam-4753	250	3	:	:	PUNCT
ejpam-4753	250	4	m	m	VERB
ejpam-4753	250	5	−→	−→	ADJ
ejpam-4753	250	6	n	n	AUX
ejpam-4753	250	7	be	be	AUX
ejpam-4753	250	8	a	a	DET
ejpam-4753	250	9	graded	grade	VERB
ejpam-4753	250	10	morphism	morphism	NOUN
ejpam-4753	250	11	of	of	ADP
ejpam-4753	250	12	degree	degree	NOUN
ejpam-4753	250	13	k	k	PROPN
ejpam-4753	250	14	∈	∈	PROPN
ejpam-4753	250	15	z	z	PROPN
ejpam-4753	250	16	of	of	ADP
ejpam-4753	250	17	a	a	DET
ejpam-4753	250	18	graded	grade	VERB
ejpam-4753	250	19	left	leave	VERB
ejpam-4753	250	20	a−modules	a−module	NOUN
ejpam-4753	250	21	,	,	PUNCT
ejpam-4753	250	22	we	we	PRON
ejpam-4753	250	23	denote	denote	VERB
ejpam-4753	250	24	by	by	ADP
ejpam-4753	250	25	(	(	PUNCT
ejpam-4753	250	26	−)(n)(f	−)(n)(f	PROPN
ejpam-4753	250	27	)	)	PUNCT
ejpam-4753	250	28	=	=	SYM
ejpam-4753	250	29	fk(n	fk(n	NOUN
ejpam-4753	250	30	)	)	PUNCT
ejpam-4753	250	31	the	the	DET
ejpam-4753	250	32	morphism	morphism	NOUN
ejpam-4753	250	33	of	of	ADP
ejpam-4753	250	34	left	left	ADJ
ejpam-4753	250	35	a−modules	a−module	NOUN
ejpam-4753	250	36	of	of	ADP
ejpam-4753	250	37	m(n	m(n	NOUN
ejpam-4753	250	38	)	)	PUNCT
ejpam-4753	250	39	to	to	ADP
ejpam-4753	250	40	n(n	n(n	NUM
ejpam-4753	250	41	)	)	PUNCT
ejpam-4753	250	42	thus	thus	ADV
ejpam-4753	250	43	(	(	PUNCT
ejpam-4753	250	44	−)(n)(m	−)(n)(m	PROPN
ejpam-4753	250	45	)	)	PUNCT
ejpam-4753	250	46	=	=	SYM
ejpam-4753	251	1	m(n	m(n	PROPN
ejpam-4753	251	2	)	)	PUNCT
ejpam-4753	251	3	is	be	AUX
ejpam-4753	251	4	in	in	ADP
ejpam-4753	251	5	a	a	DET
ejpam-4753	251	6	−mod	−mod	NOUN
ejpam-4753	251	7	,	,	PUNCT
ejpam-4753	251	8	furthermore	furthermore	ADV
ejpam-4753	251	9	m(n	m(n	NOUN
ejpam-4753	251	10	)	)	PUNCT
ejpam-4753	251	11	and	and	CCONJ
ejpam-4753	251	12	n(n	n(n	NUM
ejpam-4753	251	13	)	)	PUNCT
ejpam-4753	251	14	are	be	AUX
ejpam-4753	251	15	both	both	PRON
ejpam-4753	251	16	graded	grade	VERB
ejpam-4753	251	17	left	leave	VERB
ejpam-4753	251	18	a−module	a−module	ADP
ejpam-4753	251	19	then	then	ADV
ejpam-4753	251	20	m(n	m(n	PROPN
ejpam-4753	251	21	)	)	PUNCT
ejpam-4753	251	22	,	,	PUNCT
ejpam-4753	251	23	n(n	n(n	NUM
ejpam-4753	251	24	)	)	PUNCT
ejpam-4753	251	25	∈	∈	PROPN
ejpam-4753	251	26	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	251	27	)	)	PUNCT
ejpam-4753	251	28	.	.	PUNCT
ejpam-4753	252	1	thus	thus	ADV
ejpam-4753	252	2	(	(	PUNCT
ejpam-4753	252	3	−)(n	−)(n	NOUN
ejpam-4753	252	4	)	)	PUNCT
ejpam-4753	252	5	:	:	PUNCT
ejpam-4753	253	1	m(n	m(n	PROPN
ejpam-4753	253	2	)	)	PUNCT
ejpam-4753	253	3	−→	−→	PROPN
ejpam-4753	253	4	n(n	n(n	NOUN
ejpam-4753	253	5	)	)	PUNCT
ejpam-4753	253	6	has	have	VERB
ejpam-4753	253	7	a	a	DET
ejpam-4753	253	8	sense	sense	NOUN
ejpam-4753	253	9	.	.	PUNCT
ejpam-4753	254	1	(	(	PUNCT
ejpam-4753	254	2	i	i	NOUN
ejpam-4753	254	3	)	)	PUNCT
ejpam-4753	254	4	let	let	VERB
ejpam-4753	254	5	f	f	PRON
ejpam-4753	254	6	:	:	PUNCT
ejpam-4753	254	7	m	m	VERB
ejpam-4753	254	8	−→	−→	ADJ
ejpam-4753	254	9	n	n	NOUN
ejpam-4753	254	10	is	be	AUX
ejpam-4753	254	11	graded	grade	VERB
ejpam-4753	254	12	morphism	morphism	NOUN
ejpam-4753	254	13	of	of	ADP
ejpam-4753	254	14	degree	degree	NOUN
ejpam-4753	254	15	k	k	PROPN
ejpam-4753	254	16	∈	∈	PROPN
ejpam-4753	254	17	z	z	PROPN
ejpam-4753	254	18	of	of	ADP
ejpam-4753	254	19	graded	grade	VERB
ejpam-4753	254	20	left	leave	VERB
ejpam-4753	254	21	a−modules	a−module	NOUN
ejpam-4753	254	22	,	,	PUNCT
ejpam-4753	254	23	then	then	ADV
ejpam-4753	254	24	(	(	PUNCT
ejpam-4753	254	25	−)(n)(f	−)(n)(f	PROPN
ejpam-4753	254	26	)	)	PUNCT
ejpam-4753	254	27	:	:	PUNCT
ejpam-4753	255	1	m(n	m(n	PROPN
ejpam-4753	255	2	)	)	PUNCT
ejpam-4753	255	3	−→	−→	ADJ
ejpam-4753	255	4	n(n	n(n	NUM
ejpam-4753	255	5	)	)	PUNCT
ejpam-4753	255	6	fk(n	fk(n	NOUN
ejpam-4753	255	7	)	)	PUNCT
ejpam-4753	255	8	:	:	PUNCT
ejpam-4753	255	9	m(n	m(n	PROPN
ejpam-4753	255	10	)	)	PUNCT
ejpam-4753	255	11	−→	−→	ADJ
ejpam-4753	255	12	n(n	n(n	NUM
ejpam-4753	255	13	)	)	PUNCT
ejpam-4753	255	14	m	m	PROPN
ejpam-4753	255	15	7−→	7−→	NOUN
ejpam-4753	255	16	fk(n)(m	fk(n)(m	NOUN
ejpam-4753	255	17	)	)	PUNCT
ejpam-4753	256	1	=	=	SYM
ejpam-4753	256	2	f(m	f(m	PROPN
ejpam-4753	256	3	)	)	PUNCT
ejpam-4753	256	4	is	be	AUX
ejpam-4753	256	5	a	a	DET
ejpam-4753	256	6	graded	grade	VERB
ejpam-4753	256	7	morphism	morphism	NOUN
ejpam-4753	256	8	of	of	ADP
ejpam-4753	256	9	a	a	DET
ejpam-4753	256	10	graded	grade	VERB
ejpam-4753	256	11	left	leave	VERB
ejpam-4753	256	12	a−modules	a−module	NOUN
ejpam-4753	256	13	.	.	PUNCT
ejpam-4753	257	1	furthermore	furthermore	ADV
ejpam-4753	257	2	(	(	PUNCT
ejpam-4753	257	3	−)(n)(g	−)(n)(g	ADP
ejpam-4753	257	4	◦	◦	NOUN
ejpam-4753	257	5	f)(m	f)(m	NOUN
ejpam-4753	257	6	)	)	PUNCT
ejpam-4753	257	7	=	=	SYM
ejpam-4753	257	8	(	(	PUNCT
ejpam-4753	257	9	g	g	NOUN
ejpam-4753	257	10	◦	◦	NOUN
ejpam-4753	257	11	f)k(n)(m	f)k(n)(m	NOUN
ejpam-4753	257	12	)	)	PUNCT
ejpam-4753	257	13	=	=	SYM
ejpam-4753	257	14	(	(	PUNCT
ejpam-4753	257	15	g	g	NOUN
ejpam-4753	257	16	◦	◦	NOUN
ejpam-4753	257	17	f)(m	f)(m	NOUN
ejpam-4753	257	18	)	)	PUNCT
ejpam-4753	257	19	=	=	PUNCT
ejpam-4753	257	20	g[f(m	g[f(m	X
ejpam-4753	257	21	)	)	PUNCT
ejpam-4753	257	22	]	]	PUNCT
ejpam-4753	258	1	=	=	PUNCT
ejpam-4753	258	2	g[fk(n)(m	g[fk(n)(m	NOUN
ejpam-4753	258	3	)	)	PUNCT
ejpam-4753	258	4	]	]	PUNCT
ejpam-4753	258	5	a.	a.	PROPN
ejpam-4753	258	6	o.	o.	PROPN
ejpam-4753	258	7	chbih	chbih	PROPN
ejpam-4753	258	8	,	,	PUNCT
ejpam-4753	258	9	m.	m.	PROPN
ejpam-4753	258	10	b.	b.	PROPN
ejpam-4753	258	11	maaouia	maaouia	PROPN
ejpam-4753	258	12	,	,	PUNCT
ejpam-4753	258	13	m.	m.	NOUN
ejpam-4753	258	14	sanghare	sanghare	PROPN
ejpam-4753	258	15	/	/	SYM
ejpam-4753	258	16	eur	eur	PROPN
ejpam-4753	258	17	.	.	PUNCT
ejpam-4753	259	1	j.	j.	PROPN
ejpam-4753	259	2	pure	pure	PROPN
ejpam-4753	259	3	appl	appl	PROPN
ejpam-4753	259	4	.	.	PROPN
ejpam-4753	259	5	math	math	PROPN
ejpam-4753	259	6	,	,	PUNCT
ejpam-4753	259	7	16	16	NUM
ejpam-4753	259	8	(	(	PUNCT
ejpam-4753	259	9	3	3	NUM
ejpam-4753	259	10	)	)	PUNCT
ejpam-4753	259	11	(	(	PUNCT
ejpam-4753	259	12	2023	2023	NUM
ejpam-4753	259	13	)	)	PUNCT
ejpam-4753	259	14	,	,	PUNCT
ejpam-4753	259	15	1913	1913	NUM
ejpam-4753	259	16	-	-	SYM
ejpam-4753	259	17	1939	1939	NUM
ejpam-4753	259	18	1926	1926	NUM
ejpam-4753	259	19	=	=	SYM
ejpam-4753	259	20	gk(n)[fk(n)(m	gk(n)[fk(n)(m	VERB
ejpam-4753	259	21	)	)	PUNCT
ejpam-4753	259	22	]	]	PUNCT
ejpam-4753	260	1	=	=	PUNCT
ejpam-4753	260	2	(	(	PUNCT
ejpam-4753	260	3	gk(n	gk(n	X
ejpam-4753	260	4	)	)	PUNCT
ejpam-4753	260	5	◦	◦	NOUN
ejpam-4753	260	6	fk(n))(m	fk(n))(m	NUM
ejpam-4753	260	7	)	)	PUNCT
ejpam-4753	260	8	=	=	SYM
ejpam-4753	260	9	(	(	PUNCT
ejpam-4753	260	10	−)(n)(g	−)(n)(g	ADJ
ejpam-4753	260	11	)	)	PUNCT
ejpam-4753	260	12	◦	◦	NOUN
ejpam-4753	260	13	(	(	PUNCT
ejpam-4753	260	14	−)(n)(f)(m	−)(n)(f)(m	NUM
ejpam-4753	260	15	)	)	PUNCT
ejpam-4753	260	16	.	.	PUNCT
ejpam-4753	261	1	so	so	ADV
ejpam-4753	261	2	(	(	PUNCT
ejpam-4753	261	3	−)(n)(g	−)(n)(g	ADP
ejpam-4753	261	4	◦	◦	NOUN
ejpam-4753	261	5	f)(m	f)(m	NOUN
ejpam-4753	261	6	)	)	PUNCT
ejpam-4753	261	7	=	=	SYM
ejpam-4753	261	8	(	(	PUNCT
ejpam-4753	261	9	−)(n)(g	−)(n)(g	ADJ
ejpam-4753	261	10	)	)	PUNCT
ejpam-4753	261	11	◦	◦	NOUN
ejpam-4753	261	12	(	(	PUNCT
ejpam-4753	261	13	−)(n)(f)(m	−)(n)(f)(m	NUM
ejpam-4753	261	14	)	)	PUNCT
ejpam-4753	261	15	∀	∀	NOUN
ejpam-4753	262	1	m	m	VERB
ejpam-4753	262	2	∈	∈	ADJ
ejpam-4753	262	3	m(n	m(n	NOUN
ejpam-4753	262	4	)	)	PUNCT
ejpam-4753	262	5	.	.	PUNCT
ejpam-4753	263	1	thus	thus	ADV
ejpam-4753	263	2	(	(	PUNCT
ejpam-4753	263	3	−)(n)(g	−)(n)(g	ADP
ejpam-4753	263	4	◦	◦	NOUN
ejpam-4753	263	5	f	f	X
ejpam-4753	263	6	)	)	PUNCT
ejpam-4753	263	7	=	=	SYM
ejpam-4753	263	8	(	(	PUNCT
ejpam-4753	263	9	−)(n)(g	−)(n)(g	ADJ
ejpam-4753	263	10	)	)	PUNCT
ejpam-4753	263	11	◦	◦	NOUN
ejpam-4753	263	12	(	(	PUNCT
ejpam-4753	263	13	−)(n)(f	−)(n)(f	PROPN
ejpam-4753	263	14	)	)	PUNCT
ejpam-4753	263	15	.	.	PUNCT
ejpam-4753	264	1	on	on	ADP
ejpam-4753	264	2	the	the	DET
ejpam-4753	264	3	other	other	ADJ
ejpam-4753	264	4	hand	hand	NOUN
ejpam-4753	264	5	(	(	PUNCT
ejpam-4753	264	6	−)(n)(1m(n	−)(n)(1m(n	NOUN
ejpam-4753	264	7	)	)	PUNCT
ejpam-4753	264	8	)	)	PUNCT
ejpam-4753	264	9	:	:	PUNCT
ejpam-4753	265	1	m(n	m(n	X
ejpam-4753	265	2	)	)	PUNCT
ejpam-4753	265	3	−→	−→	ADJ
ejpam-4753	265	4	m(n	m(n	NOUN
ejpam-4753	265	5	)	)	PUNCT
ejpam-4753	265	6	1	1	NUM
ejpam-4753	265	7	m	m	NOUN
ejpam-4753	265	8	(	(	PUNCT
ejpam-4753	265	9	n	n	CCONJ
ejpam-4753	265	10	)	)	PUNCT
ejpam-4753	265	11	:	:	PUNCT
ejpam-4753	265	12	m(n	m(n	X
ejpam-4753	265	13	)	)	PUNCT
ejpam-4753	265	14	−→	−→	ADJ
ejpam-4753	265	15	m(n	m(n	NOUN
ejpam-4753	265	16	)	)	PUNCT
ejpam-4753	265	17	m	m	PROPN
ejpam-4753	265	18	7→	7→	NUM
ejpam-4753	265	19	1m(n)(n)(m	1m(n)(n)(m	NUM
ejpam-4753	265	20	)	)	PUNCT
ejpam-4753	265	21	=	=	SYM
ejpam-4753	265	22	1m(n)(m	1m(n)(m	NUM
ejpam-4753	265	23	)	)	PUNCT
ejpam-4753	265	24	=	=	PUNCT
ejpam-4753	266	1	m	m	PUNCT
ejpam-4753	266	2	=	=	NOUN
ejpam-4753	266	3	1(−)(n)(m)(m	1(−)(n)(m)(m	NUM
ejpam-4753	266	4	)	)	PUNCT
ejpam-4753	266	5	,	,	PUNCT
ejpam-4753	266	6	so	so	CCONJ
ejpam-4753	266	7	(	(	PUNCT
ejpam-4753	266	8	−)(n)(1m(n	−)(n)(1m(n	NOUN
ejpam-4753	266	9	)	)	PUNCT
ejpam-4753	266	10	)	)	PUNCT
ejpam-4753	267	1	=	=	PUNCT
ejpam-4753	267	2	1(−)(n)(m(n	1(−)(n)(m(n	NUM
ejpam-4753	267	3	)	)	PUNCT
ejpam-4753	267	4	)	)	PUNCT
ejpam-4753	267	5	,	,	PUNCT
ejpam-4753	267	6	∀m	∀m	PROPN
ejpam-4753	267	7	∈	∈	PROPN
ejpam-4753	267	8	m(n	m(n	PROPN
ejpam-4753	267	9	)	)	PUNCT
ejpam-4753	267	10	,	,	PUNCT
ejpam-4753	267	11	so	so	CCONJ
ejpam-4753	267	12	(	(	PUNCT
ejpam-4753	267	13	−)(n	−)(n	NOUN
ejpam-4753	267	14	)	)	PUNCT
ejpam-4753	267	15	is	be	AUX
ejpam-4753	267	16	a	a	DET
ejpam-4753	267	17	functor	functor	NOUN
ejpam-4753	267	18	of	of	ADP
ejpam-4753	267	19	gr(a	gr(a	NOUN
ejpam-4753	267	20	−mod	−mod	NOUN
ejpam-4753	267	21	)	)	PUNCT
ejpam-4753	267	22	to	to	ADP
ejpam-4753	267	23	gr(a−mod	gr(a−mod	NUM
ejpam-4753	267	24	)	)	PUNCT
ejpam-4753	267	25	.	.	PUNCT
ejpam-4753	268	1	thus	thus	ADV
ejpam-4753	268	2	(	(	PUNCT
ejpam-4753	268	3	−)(n	−)(n	NOUN
ejpam-4753	268	4	)	)	PUNCT
ejpam-4753	268	5	:	:	PUNCT
ejpam-4753	268	6	gr(a−mod	gr(a−mod	X
ejpam-4753	268	7	)	)	PUNCT
ejpam-4753	268	8	−→	−→	NOUN
ejpam-4753	268	9	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	268	10	)	)	PUNCT
ejpam-4753	268	11	is	be	AUX
ejpam-4753	268	12	a	a	DET
ejpam-4753	268	13	functor	functor	PROPN
ejpam-4753	268	14	covariant	covariant	NOUN
ejpam-4753	268	15	.	.	PUNCT
ejpam-4753	269	1	let	let	VERB
ejpam-4753	269	2	m	m	PRON
ejpam-4753	269	3	∈	∈	VERB
ejpam-4753	269	4	m	m	AUX
ejpam-4753	269	5	be	be	AUX
ejpam-4753	269	6	homogeneous	homogeneous	ADJ
ejpam-4753	269	7	of	of	ADP
ejpam-4753	269	8	degree	degree	NOUN
ejpam-4753	269	9	d	d	NOUN
ejpam-4753	269	10	,	,	PUNCT
ejpam-4753	269	11	then	then	ADV
ejpam-4753	269	12	(	(	PUNCT
ejpam-4753	269	13	−)(n)(f)(m	−)(n)(f)(m	X
ejpam-4753	269	14	)	)	PUNCT
ejpam-4753	269	15	=	=	PUNCT
ejpam-4753	269	16	fk(n)(m	fk(n)(m	PROPN
ejpam-4753	269	17	)	)	PUNCT
ejpam-4753	270	1	=	=	SYM
ejpam-4753	271	1	f(m	f(m	PROPN
ejpam-4753	271	2	)	)	PUNCT
ejpam-4753	271	3	is	be	AUX
ejpam-4753	271	4	of	of	ADP
ejpam-4753	271	5	degree	degree	NOUN
ejpam-4753	271	6	k	k	PROPN
ejpam-4753	272	1	+	+	CCONJ
ejpam-4753	272	2	n	n	CCONJ
ejpam-4753	272	3	thus	thus	ADV
ejpam-4753	272	4	(	(	PUNCT
ejpam-4753	272	5	−)(n	−)(n	NOUN
ejpam-4753	272	6	)	)	PUNCT
ejpam-4753	272	7	is	be	AUX
ejpam-4753	272	8	a	a	DET
ejpam-4753	272	9	additively	additively	ADV
ejpam-4753	272	10	exact	exact	ADJ
ejpam-4753	272	11	covariant	covariant	ADJ
ejpam-4753	272	12	functor	functor	NOUN
ejpam-4753	272	13	of	of	ADP
ejpam-4753	272	14	degree	degree	NOUN
ejpam-4753	272	15	k	k	PROPN
ejpam-4753	272	16	∈	∈	PROPN
ejpam-4753	272	17	z.	z.	PROPN
ejpam-4753	272	18	proposition	proposition	NOUN
ejpam-4753	272	19	14	14	NUM
ejpam-4753	272	20	.	.	PUNCT
ejpam-4753	273	1	let	let	VERB
ejpam-4753	273	2	a	a	DET
ejpam-4753	273	3	=	=	SYM
ejpam-4753	273	4	⊕	⊕	PROPN
ejpam-4753	273	5	n∈z	n∈z	VERB
ejpam-4753	273	6	an	an	DET
ejpam-4753	273	7	be	be	AUX
ejpam-4753	273	8	a	a	DET
ejpam-4753	273	9	graded	grade	VERB
ejpam-4753	273	10	ring	ring	NOUN
ejpam-4753	273	11	and	and	CCONJ
ejpam-4753	273	12	m	m	NOUN
ejpam-4753	273	13	=	=	PROPN
ejpam-4753	273	14	⊕	⊕	PROPN
ejpam-4753	273	15	n∈z	n∈z	ADJ
ejpam-4753	273	16	mn	mn	PROPN
ejpam-4753	273	17	be	be	AUX
ejpam-4753	273	18	a	a	DET
ejpam-4753	273	19	graded	grade	VERB
ejpam-4753	273	20	left	leave	VERB
ejpam-4753	273	21	a−module	a−module	ADP
ejpam-4753	273	22	,	,	PUNCT
ejpam-4753	273	23	then	then	ADV
ejpam-4753	273	24	we	we	PRON
ejpam-4753	273	25	have	have	VERB
ejpam-4753	273	26	the	the	DET
ejpam-4753	273	27	following	follow	VERB
ejpam-4753	273	28	associate	associate	ADJ
ejpam-4753	273	29	complex	complex	ADJ
ejpam-4753	273	30	sequence	sequence	NOUN
ejpam-4753	273	31	m∗	m∗	NOUN
ejpam-4753	273	32	of	of	ADP
ejpam-4753	273	33	a	a	DET
ejpam-4753	273	34	graded	grade	VERB
ejpam-4753	273	35	a−module	a−module	ADP
ejpam-4753	273	36	m	m	PROPN
ejpam-4753	273	37	=	=	PROPN
ejpam-4753	273	38	⊕	⊕	PROPN
ejpam-4753	274	1	n∈z	n∈z	ADJ
ejpam-4753	274	2	mn	mn	PROPN
ejpam-4753	274	3	:	:	PUNCT
ejpam-4753	274	4	m∗	m∗	VERB
ejpam-4753	274	5	:	:	PUNCT
ejpam-4753	274	6	·	·	PUNCT
ejpam-4753	274	7	·	·	PUNCT
ejpam-4753	274	8	·	·	PUNCT
ejpam-4753	275	1	→	→	SYM
ejpam-4753	275	2	m(n+	m(n+	NOUN
ejpam-4753	275	3	1	1	NUM
ejpam-4753	275	4	)	)	PUNCT
ejpam-4753	275	5	dn+1→	dn+1→	NOUN
ejpam-4753	275	6	m(n	m(n	PROPN
ejpam-4753	275	7	)	)	PUNCT
ejpam-4753	276	1	dn→	dn→	NOUN
ejpam-4753	276	2	m(n−	m(n−	NOUN
ejpam-4753	276	3	1	1	NUM
ejpam-4753	276	4	)	)	PUNCT
ejpam-4753	276	5	→	→	SYM
ejpam-4753	276	6	·	·	PUNCT
ejpam-4753	276	7	·	·	PUNCT
ejpam-4753	276	8	·	·	PUNCT
ejpam-4753	276	9	with	with	ADP
ejpam-4753	276	10	m(n	m(n	NOUN
ejpam-4753	276	11	)	)	PUNCT
ejpam-4753	276	12	=	=	PUNCT
ejpam-4753	277	1	⊕	⊕	PROPN
ejpam-4753	277	2	k∈z	k∈z	VERB
ejpam-4753	277	3	mn+k	mn+k	PROPN
ejpam-4753	277	4	and	and	CCONJ
ejpam-4753	277	5	dn	dn	INTJ
ejpam-4753	277	6	:	:	PUNCT
ejpam-4753	277	7	m(n	m(n	PROPN
ejpam-4753	277	8	)	)	PUNCT
ejpam-4753	277	9	−→	−→	NOUN
ejpam-4753	277	10	m(n−	m(n−	NOUN
ejpam-4753	277	11	1	1	NUM
ejpam-4753	277	12	)	)	PUNCT
ejpam-4753	277	13	x	x	X
ejpam-4753	278	1	=	=	PUNCT
ejpam-4753	278	2	y	y	PROPN
ejpam-4753	278	3	+	+	NOUN
ejpam-4753	278	4	z	z	PROPN
ejpam-4753	278	5	7−→	7−→	NUM
ejpam-4753	278	6	y	y	NOUN
ejpam-4753	278	7	with	with	ADP
ejpam-4753	278	8	(	(	PUNCT
ejpam-4753	278	9	y	y	PROPN
ejpam-4753	278	10	,	,	PUNCT
ejpam-4753	278	11	z	z	NOUN
ejpam-4753	278	12	)	)	PUNCT
ejpam-4753	278	13	∈	∈	PROPN
ejpam-4753	278	14	mn	mn	PROPN
ejpam-4753	278	15	×m(n+	×m(n+	PROPN
ejpam-4753	278	16	1	1	NUM
ejpam-4753	278	17	)	)	PUNCT
ejpam-4753	278	18	.	.	PUNCT
ejpam-4753	279	1	proof	proof	NOUN
ejpam-4753	279	2	.	.	PUNCT
ejpam-4753	280	1	we	we	PRON
ejpam-4753	280	2	have	have	VERB
ejpam-4753	280	3	m(n	m(n	NOUN
ejpam-4753	280	4	)	)	PUNCT
ejpam-4753	281	1	=	=	PUNCT
ejpam-4753	281	2	⊕	⊕	PROPN
ejpam-4753	281	3	k∈z	k∈z	VERB
ejpam-4753	281	4	mn+k	mn+k	PROPN
ejpam-4753	281	5	=	=	PROPN
ejpam-4753	281	6	⊕	⊕	PROPN
ejpam-4753	281	7	k≥n	k≥n	PROPN
ejpam-4753	281	8	mk	mk	PROPN
ejpam-4753	281	9	=	=	PROPN
ejpam-4753	281	10	mn	mn	PROPN
ejpam-4753	281	11	⊕	⊕	PROPN
ejpam-4753	281	12	mn+1	mn+1	PROPN
ejpam-4753	281	13	and	and	CCONJ
ejpam-4753	281	14	m(n−	m(n−	NOUN
ejpam-4753	281	15	1	1	NUM
ejpam-4753	281	16	)	)	PUNCT
ejpam-4753	281	17	=	=	SYM
ejpam-4753	281	18	mn−1	mn−1	PROPN
ejpam-4753	281	19	⊕	⊕	PROPN
ejpam-4753	281	20	m(n	m(n	PROPN
ejpam-4753	281	21	)	)	PUNCT
ejpam-4753	282	1	=	=	SYM
ejpam-4753	282	2	mn−1	mn−1	PROPN
ejpam-4753	282	3	⊕	⊕	PROPN
ejpam-4753	282	4	mn	mn	PROPN
ejpam-4753	282	5	⊕	⊕	PROPN
ejpam-4753	282	6	m(n+	m(n+	PROPN
ejpam-4753	282	7	1	1	NUM
ejpam-4753	282	8	)	)	PUNCT
ejpam-4753	282	9	.	.	PUNCT
ejpam-4753	283	1	let	let	VERB
ejpam-4753	283	2	x	x	PUNCT
ejpam-4753	283	3	∈	∈	PROPN
ejpam-4753	283	4	m(n	m(n	PROPN
ejpam-4753	283	5	)	)	PUNCT
ejpam-4753	283	6	,	,	PUNCT
ejpam-4753	283	7	then	then	ADV
ejpam-4753	283	8	it	it	PRON
ejpam-4753	283	9	is	be	AUX
ejpam-4753	283	10	exist	exist	VERB
ejpam-4753	284	1	a	a	DET
ejpam-4753	284	2	unique	unique	ADJ
ejpam-4753	284	3	(	(	PUNCT
ejpam-4753	284	4	y	y	NOUN
ejpam-4753	284	5	,	,	PUNCT
ejpam-4753	284	6	z	z	NOUN
ejpam-4753	284	7	)	)	PUNCT
ejpam-4753	284	8	∈	∈	PROPN
ejpam-4753	284	9	mn	mn	PROPN
ejpam-4753	284	10	×m(n+	×m(n+	PROPN
ejpam-4753	284	11	1	1	NUM
ejpam-4753	284	12	)	)	PUNCT
ejpam-4753	284	13	such	such	ADJ
ejpam-4753	284	14	that	that	SCONJ
ejpam-4753	284	15	x	x	X
ejpam-4753	284	16	=	=	SYM
ejpam-4753	284	17	y	y	PROPN
ejpam-4753	284	18	+	+	PROPN
ejpam-4753	284	19	z.	z.	PROPN
ejpam-4753	284	20	put	put	VERB
ejpam-4753	284	21	dn	dn	INTJ
ejpam-4753	284	22	:	:	PUNCT
ejpam-4753	284	23	m(n	m(n	PROPN
ejpam-4753	284	24	)	)	PUNCT
ejpam-4753	284	25	−→	−→	NOUN
ejpam-4753	284	26	m(n−	m(n−	NOUN
ejpam-4753	284	27	1	1	NUM
ejpam-4753	284	28	)	)	PUNCT
ejpam-4753	284	29	x	x	X
ejpam-4753	285	1	=	=	PUNCT
ejpam-4753	285	2	y	y	PROPN
ejpam-4753	285	3	+	+	NOUN
ejpam-4753	285	4	z	z	PROPN
ejpam-4753	285	5	7−→	7−→	PROPN
ejpam-4753	285	6	y	y	PROPN
ejpam-4753	285	7	,	,	PUNCT
ejpam-4753	285	8	a.	a.	PROPN
ejpam-4753	285	9	o.	o.	PROPN
ejpam-4753	285	10	chbih	chbih	PROPN
ejpam-4753	285	11	,	,	PUNCT
ejpam-4753	285	12	m.	m.	PROPN
ejpam-4753	285	13	b.	b.	PROPN
ejpam-4753	285	14	maaouia	maaouia	PROPN
ejpam-4753	285	15	,	,	PUNCT
ejpam-4753	285	16	m.	m.	NOUN
ejpam-4753	285	17	sanghare	sanghare	PROPN
ejpam-4753	285	18	/	/	SYM
ejpam-4753	285	19	eur	eur	PROPN
ejpam-4753	285	20	.	.	PUNCT
ejpam-4753	286	1	j.	j.	PROPN
ejpam-4753	286	2	pure	pure	PROPN
ejpam-4753	286	3	appl	appl	PROPN
ejpam-4753	286	4	.	.	PROPN
ejpam-4753	286	5	math	math	PROPN
ejpam-4753	286	6	,	,	PUNCT
ejpam-4753	286	7	16	16	NUM
ejpam-4753	286	8	(	(	PUNCT
ejpam-4753	286	9	3	3	NUM
ejpam-4753	286	10	)	)	PUNCT
ejpam-4753	286	11	(	(	PUNCT
ejpam-4753	286	12	2023	2023	NUM
ejpam-4753	286	13	)	)	PUNCT
ejpam-4753	286	14	,	,	PUNCT
ejpam-4753	286	15	1913	1913	NUM
ejpam-4753	286	16	-	-	SYM
ejpam-4753	286	17	1939	1939	NUM
ejpam-4753	286	18	1927	1927	NUM
ejpam-4753	286	19	so	so	SCONJ
ejpam-4753	286	20	im(dn	im(dn	PROPN
ejpam-4753	286	21	)	)	PUNCT
ejpam-4753	286	22	=	=	SYM
ejpam-4753	286	23	mn	mn	PROPN
ejpam-4753	286	24	;	;	PUNCT
ejpam-4753	286	25	on	on	ADP
ejpam-4753	286	26	the	the	DET
ejpam-4753	286	27	other	other	ADJ
ejpam-4753	286	28	hand	hand	NOUN
ejpam-4753	286	29	dn−1	dn−1	NOUN
ejpam-4753	286	30	:	:	PUNCT
ejpam-4753	286	31	m(n−	m(n−	NOUN
ejpam-4753	286	32	1	1	NUM
ejpam-4753	286	33	)	)	PUNCT
ejpam-4753	287	1	−→	−→	NOUN
ejpam-4753	287	2	m(n−	m(n−	NOUN
ejpam-4753	287	3	2	2	NUM
ejpam-4753	287	4	)	)	PUNCT
ejpam-4753	287	5	w	w	NOUN
ejpam-4753	288	1	=	=	PUNCT
ejpam-4753	288	2	u+	u+	NUM
ejpam-4753	288	3	v	v	NUM
ejpam-4753	288	4	7−→	7−→	PROPN
ejpam-4753	288	5	v	v	NOUN
ejpam-4753	288	6	with	with	ADP
ejpam-4753	288	7	(	(	PUNCT
ejpam-4753	288	8	u	u	NOUN
ejpam-4753	288	9	,	,	PUNCT
ejpam-4753	288	10	v	v	NOUN
ejpam-4753	288	11	)	)	PUNCT
ejpam-4753	288	12	∈	∈	PROPN
ejpam-4753	288	13	mn−1	mn−1	PROPN
ejpam-4753	288	14	×m(n	×m(n	PROPN
ejpam-4753	288	15	)	)	PUNCT
ejpam-4753	288	16	,	,	PUNCT
ejpam-4753	288	17	so	so	ADV
ejpam-4753	288	18	ker(dn−1	ker(dn−1	PROPN
ejpam-4753	288	19	)	)	PUNCT
ejpam-4753	288	20	=	=	SYM
ejpam-4753	288	21	m(n	m(n	PROPN
ejpam-4753	288	22	)	)	PUNCT
ejpam-4753	288	23	so	so	ADV
ejpam-4753	288	24	im(dn	im(dn	PROPN
ejpam-4753	288	25	)	)	PUNCT
ejpam-4753	289	1	⊂	⊂	PROPN
ejpam-4753	289	2	ker(dn−1	ker(dn−1	PROPN
ejpam-4753	289	3	)	)	PUNCT
ejpam-4753	289	4	,	,	PUNCT
ejpam-4753	289	5	so	so	ADV
ejpam-4753	289	6	dn−1	dn−1	ADJ
ejpam-4753	289	7	◦	◦	NOUN
ejpam-4753	289	8	dn	dn	PROPN
ejpam-4753	289	9	=	=	NOUN
ejpam-4753	289	10	0	0	NUM
ejpam-4753	289	11	,	,	PUNCT
ejpam-4753	289	12	thus	thus	ADV
ejpam-4753	289	13	m∗	m∗	VERB
ejpam-4753	289	14	:	:	PUNCT
ejpam-4753	289	15	·	·	PUNCT
ejpam-4753	289	16	·	·	PUNCT
ejpam-4753	289	17	·	·	PUNCT
ejpam-4753	290	1	→	→	SYM
ejpam-4753	290	2	m(n+	m(n+	NOUN
ejpam-4753	290	3	1	1	NUM
ejpam-4753	290	4	)	)	PUNCT
ejpam-4753	290	5	dn+1→	dn+1→	NOUN
ejpam-4753	290	6	m(n	m(n	PROPN
ejpam-4753	290	7	)	)	PUNCT
ejpam-4753	291	1	dn→	dn→	NOUN
ejpam-4753	291	2	m(n−	m(n−	NOUN
ejpam-4753	291	3	1	1	NUM
ejpam-4753	291	4	)	)	PUNCT
ejpam-4753	291	5	→	→	SYM
ejpam-4753	291	6	·	·	PUNCT
ejpam-4753	291	7	·	·	PUNCT
ejpam-4753	291	8	·	·	PUNCT
ejpam-4753	291	9	is	be	AUX
ejpam-4753	291	10	a	a	DET
ejpam-4753	291	11	complex	complex	ADJ
ejpam-4753	291	12	sequence	sequence	NOUN
ejpam-4753	291	13	.	.	PUNCT
ejpam-4753	292	1	proposition	proposition	NOUN
ejpam-4753	292	2	15	15	NUM
ejpam-4753	292	3	.	.	PUNCT
ejpam-4753	293	1	let	let	VERB
ejpam-4753	293	2	a	a	DET
ejpam-4753	293	3	=	=	SYM
ejpam-4753	293	4	⊕	⊕	PROPN
ejpam-4753	293	5	n∈z	n∈z	VERB
ejpam-4753	293	6	an	an	DET
ejpam-4753	293	7	be	be	AUX
ejpam-4753	293	8	a	a	DET
ejpam-4753	293	9	graded	grade	VERB
ejpam-4753	293	10	ring	ring	NOUN
ejpam-4753	293	11	,	,	PUNCT
ejpam-4753	293	12	m	m	VERB
ejpam-4753	293	13	=	=	PROPN
ejpam-4753	293	14	⊕	⊕	PROPN
ejpam-4753	293	15	n∈z	n∈z	PROPN
ejpam-4753	293	16	mn	mn	PROPN
ejpam-4753	293	17	,	,	PUNCT
ejpam-4753	293	18	n	n	PROPN
ejpam-4753	293	19	=	=	PROPN
ejpam-4753	293	20	⊕	⊕	PROPN
ejpam-4753	293	21	n∈z	n∈z	ADJ
ejpam-4753	293	22	nn	nn	PROPN
ejpam-4753	293	23	are	be	AUX
ejpam-4753	293	24	two	two	NUM
ejpam-4753	293	25	graded	grade	VERB
ejpam-4753	293	26	left	leave	VERB
ejpam-4753	293	27	a−modules	a−module	NOUN
ejpam-4753	293	28	and	and	CCONJ
ejpam-4753	293	29	f	f	X
ejpam-4753	293	30	:	:	PUNCT
ejpam-4753	294	1	m	m	VERB
ejpam-4753	294	2	=	=	SYM
ejpam-4753	294	3	⊕	⊕	PROPN
ejpam-4753	294	4	n∈z	n∈z	VERB
ejpam-4753	294	5	mn	mn	PROPN
ejpam-4753	295	1	−→	−→	NOUN
ejpam-4753	296	1	n	n	PROPN
ejpam-4753	296	2	=	=	SYM
ejpam-4753	296	3	⊕	⊕	PROPN
ejpam-4753	296	4	n∈z	n∈z	VERB
ejpam-4753	296	5	nn	nn	PROPN
ejpam-4753	296	6	is	be	AUX
ejpam-4753	296	7	a	a	DET
ejpam-4753	296	8	graded	grade	VERB
ejpam-4753	296	9	morphism	morphism	NOUN
ejpam-4753	296	10	of	of	ADP
ejpam-4753	296	11	degree	degree	NOUN
ejpam-4753	296	12	k	k	PROPN
ejpam-4753	296	13	∈	∈	PROPN
ejpam-4753	296	14	z	z	PROPN
ejpam-4753	296	15	of	of	ADP
ejpam-4753	296	16	a	a	DET
ejpam-4753	296	17	graded	grade	VERB
ejpam-4753	296	18	a−modules	a−module	NOUN
ejpam-4753	296	19	,	,	PUNCT
ejpam-4753	296	20	then	then	ADV
ejpam-4753	296	21	we	we	PRON
ejpam-4753	296	22	have	have	VERB
ejpam-4753	296	23	the	the	DET
ejpam-4753	296	24	following	follow	VERB
ejpam-4753	296	25	associate	associate	NOUN
ejpam-4753	296	26	complex	complex	NOUN
ejpam-4753	296	27	fk	fk	INTJ
ejpam-4753	296	28	∗	∗	NOUN
ejpam-4753	296	29	of	of	ADP
ejpam-4753	296	30	graded	grade	VERB
ejpam-4753	296	31	morphism	morphism	NOUN
ejpam-4753	296	32	f	f	PROPN
ejpam-4753	296	33	:	:	PUNCT
ejpam-4753	296	34	m	m	VERB
ejpam-4753	296	35	=	=	SYM
ejpam-4753	296	36	⊕	⊕	PROPN
ejpam-4753	296	37	n∈z	n∈z	VERB
ejpam-4753	296	38	mn	mn	PROPN
ejpam-4753	297	1	−→	−→	NOUN
ejpam-4753	298	1	n	n	PROPN
ejpam-4753	298	2	=	=	SYM
ejpam-4753	298	3	⊕	⊕	PROPN
ejpam-4753	298	4	n∈z	n∈z	NOUN
ejpam-4753	298	5	nn	nn	PROPN
ejpam-4753	298	6	of	of	ADP
ejpam-4753	298	7	a	a	DET
ejpam-4753	298	8	graded	grade	VERB
ejpam-4753	298	9	a−modules	a−module	NOUN
ejpam-4753	298	10	:	:	PUNCT
ejpam-4753	298	11	m∗	m∗	NOUN
ejpam-4753	298	12	:	:	PUNCT
ejpam-4753	298	13	·	·	PUNCT
ejpam-4753	298	14	·	·	PUNCT
ejpam-4753	298	15	·	·	PUNCT
ejpam-4753	299	1	//	//	PUNCT
ejpam-4753	299	2	fk	fk	INTJ
ejpam-4753	299	3	∗	∗	X
ejpam-4753	299	4	�	�	PROPN
ejpam-4753	299	5	�	�	PROPN
ejpam-4753	299	6	m(n+	m(n+	PROPN
ejpam-4753	299	7	1	1	NUM
ejpam-4753	299	8	)	)	PUNCT
ejpam-4753	299	9	dn+1	dn+1	PROPN
ejpam-4753	299	10	//	//	X
ejpam-4753	299	11	fk(n+1	fk(n+1	PROPN
ejpam-4753	299	12	)	)	PUNCT
ejpam-4753	299	13	�	�	PROPN
ejpam-4753	299	14	�	�	PROPN
ejpam-4753	299	15	m(n	m(n	PROPN
ejpam-4753	299	16	)	)	PUNCT
ejpam-4753	299	17	dn	dn	PROPN
ejpam-4753	299	18	//	//	PROPN
ejpam-4753	299	19	fk(n	fk(n	NUM
ejpam-4753	299	20	)	)	PUNCT
ejpam-4753	299	21	�	�	PROPN
ejpam-4753	299	22	�	�	PROPN
ejpam-4753	299	23	m(n−	m(n−	PROPN
ejpam-4753	299	24	1	1	NUM
ejpam-4753	299	25	)	)	PUNCT
ejpam-4753	299	26	//	//	X
ejpam-4753	299	27	fk(n−1	fk(n−1	PROPN
ejpam-4753	299	28	)	)	PUNCT
ejpam-4753	299	29	�	�	PROPN
ejpam-4753	299	30	�	�	PROPN
ejpam-4753	299	31	.	.	PUNCT
ejpam-4753	299	32	.	.	PUNCT
ejpam-4753	299	33	.	.	PUNCT
ejpam-4753	300	1	n∗	n∗	PROPN
ejpam-4753	300	2	:	:	PUNCT
ejpam-4753	300	3	·	·	PUNCT
ejpam-4753	300	4	·	·	PUNCT
ejpam-4753	300	5	·	·	PUNCT
ejpam-4753	301	1	//	//	SYM
ejpam-4753	301	2	n(n+	n(n+	NUM
ejpam-4753	301	3	1	1	NUM
ejpam-4753	301	4	)	)	PUNCT
ejpam-4753	301	5	d′n+1+k	d′n+1+k	PROPN
ejpam-4753	301	6	//	//	NUM
ejpam-4753	301	7	n(n	n(n	NUM
ejpam-4753	301	8	)	)	PUNCT
ejpam-4753	302	1	d′n+k//	d′n+k//	X
ejpam-4753	302	2	n(n−	n(n−	PROPN
ejpam-4753	302	3	1	1	NUM
ejpam-4753	302	4	)	)	PUNCT
ejpam-4753	302	5	//	//	NOUN
ejpam-4753	302	6	·	·	PUNCT
ejpam-4753	302	7	·	·	PUNCT
ejpam-4753	302	8	·	·	PUNCT
ejpam-4753	302	9	.	.	PUNCT
ejpam-4753	303	1	proof	proof	NOUN
ejpam-4753	303	2	.	.	PUNCT
ejpam-4753	304	1	prove	prove	VERB
ejpam-4753	304	2	that	that	SCONJ
ejpam-4753	304	3	for	for	ADP
ejpam-4753	304	4	all	all	PRON
ejpam-4753	304	5	n	n	PRON
ejpam-4753	304	6	∈	∈	PROPN
ejpam-4753	304	7	z	z	NOUN
ejpam-4753	304	8	,	,	PUNCT
ejpam-4753	304	9	fk(n	fk(n	PUNCT
ejpam-4753	304	10	)	)	PUNCT
ejpam-4753	304	11	◦	◦	NOUN
ejpam-4753	304	12	dn+1	dn+1	NOUN
ejpam-4753	304	13	=	=	PUNCT
ejpam-4753	304	14	d	d	NOUN
ejpam-4753	304	15	′	′	NOUN
ejpam-4753	304	16	n+1+k	n+1+k	NOUN
ejpam-4753	304	17	◦	◦	NOUN
ejpam-4753	304	18	fk(n+	fk(n+	PROPN
ejpam-4753	304	19	1	1	NUM
ejpam-4753	304	20	)	)	PUNCT
ejpam-4753	304	21	.	.	PUNCT
ejpam-4753	305	1	let	let	VERB
ejpam-4753	305	2	x	x	X
ejpam-4753	305	3	∈	∈	PROPN
ejpam-4753	305	4	m(n+1	m(n+1	NUM
ejpam-4753	305	5	)	)	PUNCT
ejpam-4753	305	6	,	,	PUNCT
ejpam-4753	305	7	then	then	ADV
ejpam-4753	305	8	there	there	PRON
ejpam-4753	305	9	exist	exist	VERB
ejpam-4753	305	10	the	the	DET
ejpam-4753	305	11	unique	unique	ADJ
ejpam-4753	305	12	couple	couple	NOUN
ejpam-4753	305	13	(	(	PUNCT
ejpam-4753	305	14	y	y	NOUN
ejpam-4753	305	15	,	,	PUNCT
ejpam-4753	305	16	z	z	NOUN
ejpam-4753	305	17	)	)	PUNCT
ejpam-4753	305	18	∈	∈	PROPN
ejpam-4753	305	19	mn+1	mn+1	NOUN
ejpam-4753	305	20	×m(n+2	×m(n+2	NUM
ejpam-4753	305	21	)	)	PUNCT
ejpam-4753	305	22	such	such	ADJ
ejpam-4753	305	23	that	that	SCONJ
ejpam-4753	305	24	x	x	X
ejpam-4753	305	25	=	=	PUNCT
ejpam-4753	305	26	y	y	PROPN
ejpam-4753	306	1	+	+	PROPN
ejpam-4753	306	2	z	z	PROPN
ejpam-4753	306	3	,	,	PUNCT
ejpam-4753	306	4	so	so	CCONJ
ejpam-4753	306	5	(	(	PUNCT
ejpam-4753	306	6	fk(n	fk(n	NOUN
ejpam-4753	306	7	)	)	PUNCT
ejpam-4753	306	8	◦	◦	NOUN
ejpam-4753	306	9	dn+1)(x	dn+1)(x	NOUN
ejpam-4753	306	10	)	)	PUNCT
ejpam-4753	306	11	=	=	SYM
ejpam-4753	306	12	fk(n)[dn+1(x	fk(n)[dn+1(x	PROPN
ejpam-4753	306	13	)	)	PUNCT
ejpam-4753	306	14	]	]	PUNCT
ejpam-4753	307	1	=	=	PUNCT
ejpam-4753	307	2	f	f	X
ejpam-4753	308	1	[	[	X
ejpam-4753	308	2	dn+1(x	dn+1(x	X
ejpam-4753	308	3	)	)	PUNCT
ejpam-4753	308	4	]	]	PUNCT
ejpam-4753	309	1	=	=	PUNCT
ejpam-4753	309	2	f	f	X
ejpam-4753	310	1	[	[	X
ejpam-4753	310	2	y	y	X
ejpam-4753	310	3	]	]	X
ejpam-4753	310	4	=	=	SYM
ejpam-4753	310	5	f(y	f(y	NOUN
ejpam-4753	310	6	)	)	PUNCT
ejpam-4753	310	7	,	,	PUNCT
ejpam-4753	310	8	and	and	CCONJ
ejpam-4753	310	9	(	(	PUNCT
ejpam-4753	310	10	d	d	NOUN
ejpam-4753	310	11	′	′	NOUN
ejpam-4753	310	12	n+1+k	n+1+k	NOUN
ejpam-4753	310	13	◦	◦	NOUN
ejpam-4753	310	14	fk(n	fk(n	X
ejpam-4753	311	1	+	+	X
ejpam-4753	311	2	1))(x	1))(x	NUM
ejpam-4753	311	3	)	)	PUNCT
ejpam-4753	312	1	=	=	PUNCT
ejpam-4753	313	1	d	d	NOUN
ejpam-4753	313	2	′	′	NUM
ejpam-4753	313	3	n+1+k[f	n+1+k[f	NOUN
ejpam-4753	313	4	k(n	k(n	NOUN
ejpam-4753	313	5	+	+	CCONJ
ejpam-4753	313	6	1)(x	1)(x	NUM
ejpam-4753	313	7	)	)	PUNCT
ejpam-4753	313	8	]	]	PUNCT
ejpam-4753	314	1	=	=	PUNCT
ejpam-4753	314	2	d	d	NOUN
ejpam-4753	314	3	′	′	NUM
ejpam-4753	314	4	n+1+k[f(x	n+1+k[f(x	NOUN
ejpam-4753	314	5	)	)	PUNCT
ejpam-4753	314	6	]	]	PUNCT
ejpam-4753	315	1	=	=	PUNCT
ejpam-4753	315	2	d	d	NOUN
ejpam-4753	315	3	′	′	NUM
ejpam-4753	316	1	n+1+k[f(y	n+1+k[f(y	NOUN
ejpam-4753	316	2	+	+	X
ejpam-4753	316	3	z	z	X
ejpam-4753	316	4	)	)	PUNCT
ejpam-4753	316	5	]	]	PUNCT
ejpam-4753	317	1	=	=	PUNCT
ejpam-4753	318	1	d	d	NOUN
ejpam-4753	318	2	′	′	NUM
ejpam-4753	318	3	n+1+k[f(y	n+1+k[f(y	NOUN
ejpam-4753	318	4	)	)	PUNCT
ejpam-4753	319	1	+	+	CCONJ
ejpam-4753	319	2	f(z	f(z	NOUN
ejpam-4753	319	3	)	)	PUNCT
ejpam-4753	319	4	]	]	PUNCT
ejpam-4753	320	1	=	=	SYM
ejpam-4753	320	2	f(y	f(y	NOUN
ejpam-4753	320	3	)	)	PUNCT
ejpam-4753	320	4	,	,	PUNCT
ejpam-4753	320	5	because	because	SCONJ
ejpam-4753	320	6	f(y	f(y	NOUN
ejpam-4753	320	7	)	)	PUNCT
ejpam-4753	320	8	∈	∈	PROPN
ejpam-4753	320	9	nn+1+k	nn+1+k	PRON
ejpam-4753	320	10	and	and	CCONJ
ejpam-4753	320	11	f(z	f(z	PROPN
ejpam-4753	320	12	)	)	PUNCT
ejpam-4753	320	13	∈	∈	PROPN
ejpam-4753	320	14	n(n+	n(n+	NOUN
ejpam-4753	320	15	2	2	NUM
ejpam-4753	320	16	+	+	SYM
ejpam-4753	320	17	k	k	NOUN
ejpam-4753	320	18	)	)	PUNCT
ejpam-4753	320	19	,	,	PUNCT
ejpam-4753	321	1	=	=	NOUN
ejpam-4753	321	2	⇒	⇒	NOUN
ejpam-4753	321	3	(	(	PUNCT
ejpam-4753	321	4	fk(n	fk(n	NOUN
ejpam-4753	321	5	)	)	PUNCT
ejpam-4753	321	6	◦	◦	NOUN
ejpam-4753	321	7	dn+1)(x	dn+1)(x	NUM
ejpam-4753	321	8	)	)	PUNCT
ejpam-4753	321	9	=	=	PUNCT
ejpam-4753	322	1	(	(	PUNCT
ejpam-4753	322	2	d	d	NOUN
ejpam-4753	322	3	′	′	NOUN
ejpam-4753	322	4	n+1+k	n+1+k	NOUN
ejpam-4753	322	5	◦	◦	VERB
ejpam-4753	322	6	fk(n+	fk(n+	PROPN
ejpam-4753	322	7	1))(x	1))(x	PROPN
ejpam-4753	322	8	)	)	PUNCT
ejpam-4753	322	9	,	,	PUNCT
ejpam-4753	322	10	∀	∀	PUNCT
ejpam-4753	322	11	x	x	X
ejpam-4753	322	12	∈	∈	NOUN
ejpam-4753	322	13	m(n+	m(n+	NOUN
ejpam-4753	322	14	1	1	NUM
ejpam-4753	322	15	)	)	PUNCT
ejpam-4753	322	16	,	,	PUNCT
ejpam-4753	322	17	so	so	ADV
ejpam-4753	322	18	fk(n	fk(n	PUNCT
ejpam-4753	322	19	)	)	PUNCT
ejpam-4753	322	20	◦	◦	NOUN
ejpam-4753	322	21	dn+1	dn+1	NOUN
ejpam-4753	322	22	=	=	PUNCT
ejpam-4753	322	23	d	d	NOUN
ejpam-4753	322	24	′	′	NOUN
ejpam-4753	322	25	n+1+k	n+1+k	NOUN
ejpam-4753	322	26	◦	◦	NOUN
ejpam-4753	322	27	fk(n+	fk(n+	PROPN
ejpam-4753	322	28	1	1	NUM
ejpam-4753	322	29	)	)	PUNCT
ejpam-4753	322	30	,	,	PUNCT
ejpam-4753	322	31	thus	thus	ADV
ejpam-4753	322	32	fk	fk	INTJ
ejpam-4753	322	33	∗	∗	NOUN
ejpam-4753	322	34	is	be	AUX
ejpam-4753	322	35	a	a	DET
ejpam-4753	322	36	complex	complex	ADJ
ejpam-4753	322	37	chain	chain	NOUN
ejpam-4753	322	38	.	.	PUNCT
ejpam-4753	323	1	a.	a.	PROPN
ejpam-4753	323	2	o.	o.	PROPN
ejpam-4753	323	3	chbih	chbih	PROPN
ejpam-4753	323	4	,	,	PUNCT
ejpam-4753	323	5	m.	m.	PROPN
ejpam-4753	323	6	b.	b.	PROPN
ejpam-4753	323	7	maaouia	maaouia	PROPN
ejpam-4753	323	8	,	,	PUNCT
ejpam-4753	323	9	m.	m.	NOUN
ejpam-4753	323	10	sanghare	sanghare	PROPN
ejpam-4753	323	11	/	/	SYM
ejpam-4753	323	12	eur	eur	PROPN
ejpam-4753	323	13	.	.	PUNCT
ejpam-4753	324	1	j.	j.	PROPN
ejpam-4753	324	2	pure	pure	PROPN
ejpam-4753	324	3	appl	appl	PROPN
ejpam-4753	324	4	.	.	PROPN
ejpam-4753	324	5	math	math	PROPN
ejpam-4753	324	6	,	,	PUNCT
ejpam-4753	324	7	16	16	NUM
ejpam-4753	324	8	(	(	PUNCT
ejpam-4753	324	9	3	3	NUM
ejpam-4753	324	10	)	)	PUNCT
ejpam-4753	324	11	(	(	PUNCT
ejpam-4753	324	12	2023	2023	NUM
ejpam-4753	324	13	)	)	PUNCT
ejpam-4753	324	14	,	,	PUNCT
ejpam-4753	324	15	1913	1913	NUM
ejpam-4753	324	16	-	-	SYM
ejpam-4753	324	17	1939	1939	NUM
ejpam-4753	324	18	1928	1928	NUM
ejpam-4753	324	19	theorem	theorem	VERB
ejpam-4753	324	20	6	6	NUM
ejpam-4753	324	21	.	.	PUNCT
ejpam-4753	325	1	let	let	VERB
ejpam-4753	325	2	a	a	DET
ejpam-4753	325	3	=	=	SYM
ejpam-4753	325	4	⊕	⊕	PROPN
ejpam-4753	325	5	n∈z	n∈z	VERB
ejpam-4753	325	6	an	an	DET
ejpam-4753	325	7	be	be	AUX
ejpam-4753	325	8	a	a	DET
ejpam-4753	325	9	graded	grade	VERB
ejpam-4753	325	10	ring	ring	NOUN
ejpam-4753	325	11	,	,	PUNCT
ejpam-4753	325	12	then	then	ADV
ejpam-4753	325	13	the	the	DET
ejpam-4753	325	14	following	follow	VERB
ejpam-4753	325	15	information	information	NOUN
ejpam-4753	325	16	:	:	PUNCT
ejpam-4753	325	17	(	(	PUNCT
ejpam-4753	325	18	i	i	NOUN
ejpam-4753	325	19	)	)	PUNCT
ejpam-4753	325	20	the	the	DET
ejpam-4753	325	21	objets	objet	NOUN
ejpam-4753	325	22	are	be	AUX
ejpam-4753	325	23	the	the	DET
ejpam-4753	325	24	associate	associate	ADJ
ejpam-4753	325	25	complex	complex	ADJ
ejpam-4753	325	26	sequences	sequence	NOUN
ejpam-4753	325	27	of	of	ADP
ejpam-4753	325	28	a	a	DET
ejpam-4753	325	29	graded	grade	VERB
ejpam-4753	325	30	left	leave	VERB
ejpam-4753	325	31	a−modules	a−module	NOUN
ejpam-4753	325	32	;	;	PUNCT
ejpam-4753	325	33	(	(	PUNCT
ejpam-4753	325	34	ii	ii	NOUN
ejpam-4753	325	35	)	)	PUNCT
ejpam-4753	325	36	the	the	DET
ejpam-4753	325	37	morphisms	morphism	NOUN
ejpam-4753	325	38	are	be	AUX
ejpam-4753	325	39	the	the	DET
ejpam-4753	325	40	associate	associate	ADJ
ejpam-4753	325	41	complex	complex	ADJ
ejpam-4753	325	42	chains	chain	NOUN
ejpam-4753	325	43	of	of	ADP
ejpam-4753	325	44	a	a	DET
ejpam-4753	325	45	graded	grade	VERB
ejpam-4753	325	46	morphism	morphism	NOUN
ejpam-4753	325	47	of	of	ADP
ejpam-4753	325	48	a	a	DET
ejpam-4753	325	49	graded	grade	VERB
ejpam-4753	325	50	left	leave	VERB
ejpam-4753	325	51	a−modules	a−module	NOUN
ejpam-4753	325	52	.	.	PUNCT
ejpam-4753	326	1	formed	form	VERB
ejpam-4753	326	2	a	a	DET
ejpam-4753	326	3	category	category	NOUN
ejpam-4753	326	4	called	call	VERB
ejpam-4753	326	5	the	the	DET
ejpam-4753	326	6	category	category	NOUN
ejpam-4753	326	7	of	of	ADP
ejpam-4753	326	8	associate	associate	ADJ
ejpam-4753	326	9	complex	complex	NOUN
ejpam-4753	326	10	of	of	ADP
ejpam-4753	326	11	a	a	DET
ejpam-4753	326	12	graded	grade	VERB
ejpam-4753	326	13	left	leave	VERB
ejpam-4753	326	14	a−modules	a−module	NOUN
ejpam-4753	326	15	and	and	CCONJ
ejpam-4753	326	16	denoted	denote	VERB
ejpam-4753	326	17	by	by	ADP
ejpam-4753	326	18	comp	comp	NOUN
ejpam-4753	326	19	(	(	PUNCT
ejpam-4753	326	20	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	326	21	)	)	PUNCT
ejpam-4753	326	22	)	)	PUNCT
ejpam-4753	326	23	.	.	PUNCT
ejpam-4753	327	1	proof	proof	NOUN
ejpam-4753	327	2	.	.	PUNCT
ejpam-4753	328	1	let	let	VERB
ejpam-4753	328	2	m∗	m∗	NOUN
ejpam-4753	328	3	and	and	CCONJ
ejpam-4753	328	4	n∗	n∗	VERB
ejpam-4753	328	5	two	two	NUM
ejpam-4753	328	6	objets	objet	NOUN
ejpam-4753	328	7	of	of	ADP
ejpam-4753	328	8	comp	comp	NOUN
ejpam-4753	328	9	(	(	PUNCT
ejpam-4753	328	10	gr(a−mod)),then	gr(a−mod)),then	ADV
ejpam-4753	328	11	:	:	PUNCT
ejpam-4753	328	12	(	(	PUNCT
ejpam-4753	328	13	i	i	NOUN
ejpam-4753	328	14	)	)	PUNCT
ejpam-4753	328	15	homcomp	homcomp	NOUN
ejpam-4753	328	16	(	(	PUNCT
ejpam-4753	328	17	gr(a−mod))(m∗	gr(a−mod))(m∗	PROPN
ejpam-4753	328	18	,	,	PUNCT
ejpam-4753	328	19	n∗	n∗	PROPN
ejpam-4753	328	20	)	)	PUNCT
ejpam-4753	328	21	=	=	PRON
ejpam-4753	328	22	{	{	PUNCT
ejpam-4753	328	23	the	the	DET
ejpam-4753	328	24	set	set	NOUN
ejpam-4753	328	25	of	of	ADP
ejpam-4753	328	26	associate	associate	ADJ
ejpam-4753	328	27	complex	complex	ADJ
ejpam-4753	328	28	chains	chain	NOUN
ejpam-4753	328	29	fk	fk	INTJ
ejpam-4753	328	30	∗	∗	NOUN
ejpam-4753	328	31	,	,	PUNCT
ejpam-4753	328	32	of	of	ADP
ejpam-4753	328	33	m∗	m∗	NOUN
ejpam-4753	328	34	to	to	PART
ejpam-4753	328	35	n∗	n∗	PROPN
ejpam-4753	328	36	}	}	PUNCT
ejpam-4753	328	37	;	;	PUNCT
ejpam-4753	328	38	(	(	PUNCT
ejpam-4753	328	39	ii	ii	X
ejpam-4753	328	40	)	)	PUNCT
ejpam-4753	328	41	the	the	DET
ejpam-4753	328	42	morphisms	morphism	NOUN
ejpam-4753	328	43	are	be	AUX
ejpam-4753	328	44	the	the	DET
ejpam-4753	328	45	associate	associate	ADJ
ejpam-4753	328	46	complex	complex	ADJ
ejpam-4753	328	47	chains	chain	NOUN
ejpam-4753	328	48	of	of	ADP
ejpam-4753	328	49	a	a	DET
ejpam-4753	328	50	graded	grade	VERB
ejpam-4753	328	51	morphism	morphism	NOUN
ejpam-4753	328	52	of	of	ADP
ejpam-4753	328	53	degrees	degree	NOUN
ejpam-4753	328	54	k	k	PROPN
ejpam-4753	328	55	of	of	ADP
ejpam-4753	328	56	a	a	DET
ejpam-4753	328	57	graded	grade	VERB
ejpam-4753	328	58	left	leave	VERB
ejpam-4753	328	59	a−modules	a−module	NOUN
ejpam-4753	328	60	.	.	PUNCT
ejpam-4753	329	1	then	then	ADV
ejpam-4753	329	2	we	we	PRON
ejpam-4753	329	3	have	have	VERB
ejpam-4753	329	4	:	:	PUNCT
ejpam-4753	329	5	(	(	PUNCT
ejpam-4753	329	6	a	a	X
ejpam-4753	329	7	)	)	PUNCT
ejpam-4753	329	8	∀	∀	NOUN
ejpam-4753	329	9	fk	fk	INTJ
ejpam-4753	329	10	∗	∗	NOUN
ejpam-4753	329	11	∈	∈	PROPN
ejpam-4753	329	12	homcomp	homcomp	NOUN
ejpam-4753	329	13	(	(	PUNCT
ejpam-4753	329	14	gr(a−mod))(m∗	gr(a−mod))(m∗	PROPN
ejpam-4753	329	15	,	,	PUNCT
ejpam-4753	329	16	n∗	n∗	PROPN
ejpam-4753	329	17	)	)	PUNCT
ejpam-4753	329	18	;	;	PUNCT
ejpam-4753	329	19	∀	∀	NUM
ejpam-4753	329	20	gr∗	gr∗	PROPN
ejpam-4753	329	21	∈	∈	PROPN
ejpam-4753	329	22	homcomp	homcomp	PROPN
ejpam-4753	329	23	(	(	PUNCT
ejpam-4753	329	24	gr(a−mod))(n∗	gr(a−mod))(n∗	PROPN
ejpam-4753	329	25	,	,	PUNCT
ejpam-4753	329	26	p∗	p∗	PROPN
ejpam-4753	329	27	)	)	PUNCT
ejpam-4753	329	28	;	;	PUNCT
ejpam-4753	329	29	∀	∀	X
ejpam-4753	329	30	hs∗	hs∗	PROPN
ejpam-4753	329	31	∈	∈	PROPN
ejpam-4753	329	32	homcomp	homcomp	PROPN
ejpam-4753	329	33	(	(	PUNCT
ejpam-4753	329	34	gr(a−mod))(p∗	gr(a−mod))(p∗	NOUN
ejpam-4753	329	35	,	,	PUNCT
ejpam-4753	329	36	q∗	q∗	PROPN
ejpam-4753	329	37	)	)	PUNCT
ejpam-4753	329	38	on	on	ADP
ejpam-4753	329	39	a	a	DET
ejpam-4753	329	40	:	:	PUNCT
ejpam-4753	329	41	m∗	m∗	NOUN
ejpam-4753	329	42	:	:	PUNCT
ejpam-4753	329	43	·	·	PUNCT
ejpam-4753	329	44	·	·	PUNCT
ejpam-4753	329	45	·	·	PUNCT
ejpam-4753	330	1	//	//	PUNCT
ejpam-4753	330	2	fk	fk	INTJ
ejpam-4753	330	3	∗	∗	X
ejpam-4753	330	4	�	�	PROPN
ejpam-4753	330	5	�	�	PROPN
ejpam-4753	330	6	//m(n+	//m(n+	PUNCT
ejpam-4753	330	7	1	1	NUM
ejpam-4753	330	8	)	)	PUNCT
ejpam-4753	330	9	fk(n+1	fk(n+1	PROPN
ejpam-4753	330	10	)	)	PUNCT
ejpam-4753	330	11	�	�	PROPN
ejpam-4753	330	12	�	�	PROPN
ejpam-4753	330	13	dn+1	dn+1	PROPN
ejpam-4753	330	14	//m(n	//m(n	X
ejpam-4753	330	15	)	)	PUNCT
ejpam-4753	330	16	fk(n	fk(n	NOUN
ejpam-4753	330	17	)	)	PUNCT
ejpam-4753	330	18	�	�	PROPN
ejpam-4753	330	19	�	�	PROPN
ejpam-4753	330	20	dn	dn	PROPN
ejpam-4753	330	21	//	//	PROPN
ejpam-4753	330	22	.	.	PUNCT
ejpam-4753	330	23	.	.	PUNCT
ejpam-4753	330	24	.	.	PUNCT
ejpam-4753	331	1	n∗	n∗	PROPN
ejpam-4753	331	2	:	:	PUNCT
ejpam-4753	331	3	·	·	PUNCT
ejpam-4753	331	4	·	·	PUNCT
ejpam-4753	331	5	·	·	PUNCT
ejpam-4753	331	6	gr∗	gr∗	PROPN
ejpam-4753	331	7	�	�	PROPN
ejpam-4753	331	8	�	�	PROPN
ejpam-4753	331	9	//	//	NUM
ejpam-4753	331	10	n(n+	n(n+	NUM
ejpam-4753	331	11	1	1	NUM
ejpam-4753	331	12	)	)	PUNCT
ejpam-4753	331	13	gr(n+1	gr(n+1	NOUN
ejpam-4753	331	14	)	)	PUNCT
ejpam-4753	331	15	�	�	NOUN
ejpam-4753	331	16	�	�	PROPN
ejpam-4753	331	17	d	d	NOUN
ejpam-4753	331	18	′	′	NUM
ejpam-4753	331	19	n+1+k	n+1+k	NOUN
ejpam-4753	331	20	//	//	SYM
ejpam-4753	331	21	n(n	n(n	NOUN
ejpam-4753	331	22	)	)	PUNCT
ejpam-4753	331	23	gr(n	gr(n	NOUN
ejpam-4753	331	24	)	)	PUNCT
ejpam-4753	331	25	�	�	NOUN
ejpam-4753	331	26	�	�	PROPN
ejpam-4753	331	27	d	d	ADP
ejpam-4753	331	28	′	′	NUM
ejpam-4753	331	29	n+k	n+k	PROPN
ejpam-4753	331	30	//	//	NUM
ejpam-4753	331	31	.	.	PUNCT
ejpam-4753	331	32	.	.	PUNCT
ejpam-4753	331	33	.	.	PUNCT
ejpam-4753	332	1	p∗	p∗	ADJ
ejpam-4753	332	2	:	:	PUNCT
ejpam-4753	332	3	·	·	PUNCT
ejpam-4753	332	4	·	·	PUNCT
ejpam-4753	332	5	·	·	PUNCT
ejpam-4753	332	6	hs	hs	PROPN
ejpam-4753	332	7	∗	∗	PROPN
ejpam-4753	332	8	�	�	PROPN
ejpam-4753	332	9	�	�	PROPN
ejpam-4753	332	10	//	//	NUM
ejpam-4753	332	11	p	p	X
ejpam-4753	332	12	(	(	PUNCT
ejpam-4753	332	13	n+	n+	NOUN
ejpam-4753	332	14	1	1	NUM
ejpam-4753	332	15	)	)	PUNCT
ejpam-4753	332	16	hs(n+1	hs(n+1	NOUN
ejpam-4753	332	17	)	)	PUNCT
ejpam-4753	332	18	�	�	PROPN
ejpam-4753	332	19	�	�	PROPN
ejpam-4753	332	20	d	d	PROPN
ejpam-4753	332	21	′′	′′	PROPN
ejpam-4753	332	22	n+1+k+r	n+1+k+r	PROPN
ejpam-4753	332	23	//	//	SYM
ejpam-4753	332	24	p	p	X
ejpam-4753	332	25	(	(	PUNCT
ejpam-4753	332	26	n	n	CCONJ
ejpam-4753	332	27	)	)	PUNCT
ejpam-4753	332	28	hs(n	hs(n	ADJ
ejpam-4753	332	29	)	)	PUNCT
ejpam-4753	332	30	�	�	PROPN
ejpam-4753	332	31	�	�	PROPN
ejpam-4753	332	32	d	d	PROPN
ejpam-4753	332	33	′′	′′	PROPN
ejpam-4753	332	34	n+k+r	n+k+r	ADJ
ejpam-4753	332	35	//	//	NOUN
ejpam-4753	332	36	.	.	PUNCT
ejpam-4753	332	37	.	.	PUNCT
ejpam-4753	332	38	.	.	PUNCT
ejpam-4753	333	1	q∗	q∗	NOUN
ejpam-4753	333	2	:	:	PUNCT
ejpam-4753	333	3	·	·	PUNCT
ejpam-4753	333	4	·	·	PUNCT
ejpam-4753	333	5	·	·	PUNCT
ejpam-4753	334	1	//	//	NUM
ejpam-4753	334	2	q(n+	q(n+	PROPN
ejpam-4753	334	3	1	1	NUM
ejpam-4753	334	4	)	)	PUNCT
ejpam-4753	334	5	d	d	NOUN
ejpam-4753	334	6	′′′	′′′	ADV
ejpam-4753	334	7	n+1+k+r+s//	n+1+k+r+s//	INTJ
ejpam-4753	334	8	q(n	q(n	PROPN
ejpam-4753	334	9	)	)	PUNCT
ejpam-4753	334	10	d	d	X
ejpam-4753	334	11	′′′	′′′	PROPN
ejpam-4753	334	12	n+k+r+s//	n+k+r+s//	NOUN
ejpam-4753	334	13	.	.	PUNCT
ejpam-4753	334	14	.	.	PUNCT
ejpam-4753	334	15	.	.	PUNCT
ejpam-4753	335	1	so	so	ADV
ejpam-4753	335	2	(	(	PUNCT
ejpam-4753	335	3	hs∗	hs∗	X
ejpam-4753	335	4	◦	◦	PROPN
ejpam-4753	335	5	gr∗	gr∗	NOUN
ejpam-4753	335	6	)	)	PUNCT
ejpam-4753	335	7	◦	◦	NOUN
ejpam-4753	335	8	fk	fk	INTJ
ejpam-4753	335	9	∗	∗	NOUN
ejpam-4753	335	10	=	=	PUNCT
ejpam-4753	335	11	hs∗	hs∗	PROPN
ejpam-4753	335	12	◦	◦	NOUN
ejpam-4753	335	13	(	(	PUNCT
ejpam-4753	335	14	gr∗	gr∗	NOUN
ejpam-4753	335	15	◦	◦	NOUN
ejpam-4753	335	16	fk	fk	INTJ
ejpam-4753	335	17	∗	∗	NOUN
ejpam-4753	335	18	)	)	PUNCT
ejpam-4753	335	19	;	;	PUNCT
ejpam-4753	335	20	(	(	PUNCT
ejpam-4753	335	21	b	b	X
ejpam-4753	335	22	)	)	PUNCT
ejpam-4753	335	23	let	let	AUX
ejpam-4753	335	24	m∗	m∗	VERB
ejpam-4753	335	25	the	the	DET
ejpam-4753	335	26	object	object	NOUN
ejpam-4753	335	27	of	of	ADP
ejpam-4753	335	28	comp	comp	NOUN
ejpam-4753	335	29	(	(	PUNCT
ejpam-4753	335	30	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	335	31	)	)	PUNCT
ejpam-4753	335	32	)	)	PUNCT
ejpam-4753	335	33	,	,	PUNCT
ejpam-4753	335	34	we	we	PRON
ejpam-4753	335	35	have	have	VERB
ejpam-4753	335	36	:	:	PUNCT
ejpam-4753	335	37	1m∗	1m∗	NUM
ejpam-4753	335	38	:	:	PUNCT
ejpam-4753	335	39	m∗	m∗	VERB
ejpam-4753	335	40	−→	−→	ADJ
ejpam-4753	335	41	m∗	m∗	NOUN
ejpam-4753	335	42	m∗	m∗	VERB
ejpam-4753	335	43	:	:	PUNCT
ejpam-4753	335	44	·	·	PUNCT
ejpam-4753	335	45	·	·	PUNCT
ejpam-4753	335	46	·	·	PUNCT
ejpam-4753	336	1	//	//	SYM
ejpam-4753	336	2	1m∗	1m∗	NUM
ejpam-4753	336	3	�	�	PROPN
ejpam-4753	336	4	�	�	PROPN
ejpam-4753	336	5	//m(n+	//m(n+	PUNCT
ejpam-4753	336	6	1	1	NUM
ejpam-4753	336	7	)	)	PUNCT
ejpam-4753	336	8	1(n+1	1(n+1	NUM
ejpam-4753	336	9	)	)	PUNCT
ejpam-4753	336	10	�	�	PROPN
ejpam-4753	336	11	�	�	PROPN
ejpam-4753	336	12	dn+1	dn+1	PROPN
ejpam-4753	336	13	//m(n	//m(n	X
ejpam-4753	336	14	)	)	PUNCT
ejpam-4753	336	15	1(n	1(n	NUM
ejpam-4753	336	16	)	)	PUNCT
ejpam-4753	336	17	�	�	PROPN
ejpam-4753	336	18	�	�	PROPN
ejpam-4753	336	19	dn	dn	PROPN
ejpam-4753	336	20	//	//	PROPN
ejpam-4753	336	21	.	.	PUNCT
ejpam-4753	336	22	.	.	PUNCT
ejpam-4753	337	1	.	.	PUNCT
ejpam-4753	338	1	m∗	m∗	VERB
ejpam-4753	338	2	:	:	PUNCT
ejpam-4753	338	3	·	·	PUNCT
ejpam-4753	338	4	·	·	PUNCT
ejpam-4753	338	5	·	·	PUNCT
ejpam-4753	338	6	//m(n+	//m(n+	PUNCT
ejpam-4753	339	1	1	1	X
ejpam-4753	339	2	)	)	PUNCT
ejpam-4753	339	3	dn+1	dn+1	X
ejpam-4753	339	4	//m(n	//m(n	PUNCT
ejpam-4753	339	5	)	)	PUNCT
ejpam-4753	340	1	dn	dn	PROPN
ejpam-4753	340	2	//	//	PROPN
ejpam-4753	340	3	.	.	PUNCT
ejpam-4753	340	4	.	.	PUNCT
ejpam-4753	341	1	.	.	PUNCT
ejpam-4753	342	1	1m∗	1m∗	NUM
ejpam-4753	342	2	verified	verify	VERB
ejpam-4753	342	3	f∗	f∗	NOUN
ejpam-4753	342	4	◦	◦	NOUN
ejpam-4753	342	5	1m∗	1m∗	NUM
ejpam-4753	342	6	=	=	NOUN
ejpam-4753	342	7	f∗	f∗	NOUN
ejpam-4753	342	8	∀	∀	X
ejpam-4753	342	9	f∗	f∗	NOUN
ejpam-4753	342	10	∈	∈	PROPN
ejpam-4753	342	11	homcomp	homcomp	NOUN
ejpam-4753	342	12	(	(	PUNCT
ejpam-4753	342	13	gr(a−mod))(m∗	gr(a−mod))(m∗	PROPN
ejpam-4753	342	14	,	,	PUNCT
ejpam-4753	342	15	n∗	n∗	PROPN
ejpam-4753	342	16	)	)	PUNCT
ejpam-4753	342	17	.	.	PUNCT
ejpam-4753	343	1	furthermore	furthermore	ADV
ejpam-4753	343	2	1m∗	1m∗	NUM
ejpam-4753	343	3	◦	◦	NOUN
ejpam-4753	343	4	g∗	g∗	NOUN
ejpam-4753	343	5	=	=	SYM
ejpam-4753	343	6	g∗	g∗	PROPN
ejpam-4753	343	7	∀	∀	PUNCT
ejpam-4753	343	8	g∗	g∗	PROPN
ejpam-4753	343	9	∈	∈	PROPN
ejpam-4753	343	10	homcomp	homcomp	NOUN
ejpam-4753	343	11	(	(	PUNCT
ejpam-4753	343	12	gr(a−mod))(n∗,m∗	gr(a−mod))(n∗,m∗	PROPN
ejpam-4753	343	13	)	)	PUNCT
ejpam-4753	343	14	.	.	PUNCT
ejpam-4753	344	1	a.	a.	PROPN
ejpam-4753	344	2	o.	o.	PROPN
ejpam-4753	344	3	chbih	chbih	PROPN
ejpam-4753	344	4	,	,	PUNCT
ejpam-4753	344	5	m.	m.	PROPN
ejpam-4753	344	6	b.	b.	PROPN
ejpam-4753	344	7	maaouia	maaouia	PROPN
ejpam-4753	344	8	,	,	PUNCT
ejpam-4753	344	9	m.	m.	NOUN
ejpam-4753	344	10	sanghare	sanghare	PROPN
ejpam-4753	344	11	/	/	SYM
ejpam-4753	344	12	eur	eur	PROPN
ejpam-4753	344	13	.	.	PUNCT
ejpam-4753	345	1	j.	j.	PROPN
ejpam-4753	345	2	pure	pure	PROPN
ejpam-4753	345	3	appl	appl	PROPN
ejpam-4753	345	4	.	.	PROPN
ejpam-4753	345	5	math	math	PROPN
ejpam-4753	345	6	,	,	PUNCT
ejpam-4753	345	7	16	16	NUM
ejpam-4753	345	8	(	(	PUNCT
ejpam-4753	345	9	3	3	NUM
ejpam-4753	345	10	)	)	PUNCT
ejpam-4753	345	11	(	(	PUNCT
ejpam-4753	345	12	2023	2023	NUM
ejpam-4753	345	13	)	)	PUNCT
ejpam-4753	345	14	,	,	PUNCT
ejpam-4753	345	15	1913	1913	NUM
ejpam-4753	345	16	-	-	SYM
ejpam-4753	345	17	1939	1939	NUM
ejpam-4753	345	18	1929	1929	NUM
ejpam-4753	345	19	thus	thus	ADV
ejpam-4753	345	20	comp	comp	NOUN
ejpam-4753	345	21	(	(	PUNCT
ejpam-4753	345	22	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	345	23	)	)	PUNCT
ejpam-4753	345	24	)	)	PUNCT
ejpam-4753	345	25	is	be	AUX
ejpam-4753	345	26	a	a	DET
ejpam-4753	345	27	category	category	NOUN
ejpam-4753	345	28	.	.	PUNCT
ejpam-4753	346	1	proposition	proposition	NOUN
ejpam-4753	346	2	16	16	NUM
ejpam-4753	346	3	.	.	PUNCT
ejpam-4753	347	1	let	let	VERB
ejpam-4753	347	2	a	a	DET
ejpam-4753	347	3	=	=	SYM
ejpam-4753	347	4	⊕	⊕	PROPN
ejpam-4753	347	5	n∈z	n∈z	VERB
ejpam-4753	347	6	an	an	DET
ejpam-4753	347	7	be	be	AUX
ejpam-4753	347	8	a	a	DET
ejpam-4753	347	9	graded	grade	VERB
ejpam-4753	347	10	ring	ring	NOUN
ejpam-4753	347	11	,	,	PUNCT
ejpam-4753	347	12	m	m	VERB
ejpam-4753	347	13	=	=	ADJ
ejpam-4753	347	14	⊕	⊕	PROPN
ejpam-4753	347	15	n∈z	n∈z	VERB
ejpam-4753	347	16	mn	mn	PROPN
ejpam-4753	347	17	and	and	CCONJ
ejpam-4753	347	18	n	n	PROPN
ejpam-4753	347	19	=	=	PROPN
ejpam-4753	347	20	⊕	⊕	PROPN
ejpam-4753	347	21	n∈z	n∈z	ADJ
ejpam-4753	347	22	nn	nn	PROPN
ejpam-4753	347	23	are	be	AUX
ejpam-4753	347	24	two	two	NUM
ejpam-4753	347	25	graded	grade	VERB
ejpam-4753	347	26	left	leave	VERB
ejpam-4753	347	27	a−modules	a−module	NOUN
ejpam-4753	347	28	,	,	PUNCT
ejpam-4753	347	29	f	f	X
ejpam-4753	347	30	:	:	PUNCT
ejpam-4753	347	31	m	m	VERB
ejpam-4753	347	32	−→	−→	ADJ
ejpam-4753	348	1	n	n	NOUN
ejpam-4753	348	2	is	be	AUX
ejpam-4753	348	3	a	a	DET
ejpam-4753	348	4	graded	grade	VERB
ejpam-4753	348	5	morphism	morphism	NOUN
ejpam-4753	348	6	of	of	ADP
ejpam-4753	348	7	degree	degree	NOUN
ejpam-4753	348	8	k	k	PROPN
ejpam-4753	348	9	and	and	CCONJ
ejpam-4753	348	10	s	s	AUX
ejpam-4753	348	11	be	be	AUX
ejpam-4753	348	12	a	a	DET
ejpam-4753	348	13	multiplicatively	multiplicatively	ADV
ejpam-4753	348	14	closed	close	VERB
ejpam-4753	348	15	subset	subset	NOUN
ejpam-4753	348	16	satisfying	satisfy	VERB
ejpam-4753	348	17	the	the	DET
ejpam-4753	348	18	left	left	ADJ
ejpam-4753	348	19	conditions	condition	NOUN
ejpam-4753	348	20	of	of	ADP
ejpam-4753	348	21	ore	ore	NOUN
ejpam-4753	348	22	formed	form	VERB
ejpam-4753	348	23	of	of	ADP
ejpam-4753	348	24	homogeneous	homogeneous	ADJ
ejpam-4753	348	25	elements	element	NOUN
ejpam-4753	348	26	of	of	ADP
ejpam-4753	348	27	a	a	PRON
ejpam-4753	348	28	,	,	PUNCT
ejpam-4753	348	29	then	then	ADV
ejpam-4753	348	30	we	we	PRON
ejpam-4753	348	31	have	have	VERB
ejpam-4753	348	32	:	:	PUNCT
ejpam-4753	348	33	(	(	PUNCT
ejpam-4753	348	34	i	i	NOUN
ejpam-4753	348	35	)	)	PUNCT
ejpam-4753	348	36	the	the	DET
ejpam-4753	348	37	following	follow	VERB
ejpam-4753	348	38	complex	complex	ADJ
ejpam-4753	348	39	sequence	sequence	NOUN
ejpam-4753	348	40	:	:	PUNCT
ejpam-4753	348	41	s−1(m∗	s−1(m∗	NOUN
ejpam-4753	348	42	)	)	PUNCT
ejpam-4753	348	43	:	:	PUNCT
ejpam-4753	348	44	·	·	PUNCT
ejpam-4753	348	45	·	·	PUNCT
ejpam-4753	348	46	·	·	PUNCT
ejpam-4753	349	1	−→	−→	ADJ
ejpam-4753	349	2	s−1(m(n+	s−1(m(n+	NOUN
ejpam-4753	349	3	1	1	NUM
ejpam-4753	349	4	)	)	PUNCT
ejpam-4753	349	5	)	)	PUNCT
ejpam-4753	349	6	s−1(dn+1)−→	s−1(dn+1)−→	ADP
ejpam-4753	349	7	s−1(m(n	s−1(m(n	NOUN
ejpam-4753	349	8	)	)	PUNCT
ejpam-4753	349	9	)	)	PUNCT
ejpam-4753	350	1	s−1(dn)−→	s−1(dn)−→	PROPN
ejpam-4753	350	2	s−1(m(n−	s−1(m(n−	PROPN
ejpam-4753	350	3	1	1	NUM
ejpam-4753	350	4	)	)	PUNCT
ejpam-4753	350	5	)	)	PUNCT
ejpam-4753	351	1	−→	−→	NOUN
ejpam-4753	351	2	·	·	PUNCT
ejpam-4753	351	3	·	·	PUNCT
ejpam-4753	351	4	·	·	PUNCT
ejpam-4753	351	5	(	(	PUNCT
ejpam-4753	351	6	ii	ii	NOUN
ejpam-4753	351	7	)	)	PUNCT
ejpam-4753	351	8	the	the	DET
ejpam-4753	351	9	following	follow	VERB
ejpam-4753	351	10	complex	complex	ADJ
ejpam-4753	351	11	chain	chain	NOUN
ejpam-4753	351	12	:	:	PUNCT
ejpam-4753	351	13	s−1(m∗	s−1(m∗	PROPN
ejpam-4753	351	14	)	)	PUNCT
ejpam-4753	351	15	:	:	PUNCT
ejpam-4753	351	16	·	·	PUNCT
ejpam-4753	351	17	·	·	PUNCT
ejpam-4753	351	18	·	·	PUNCT
ejpam-4753	351	19	//	//	PUNCT
ejpam-4753	352	1	s−1(fk	s−1(fk	PROPN
ejpam-4753	352	2	∗	∗	NOUN
ejpam-4753	352	3	)	)	PUNCT
ejpam-4753	352	4	�	�	PROPN
ejpam-4753	352	5	�	�	PROPN
ejpam-4753	352	6	//	//	NUM
ejpam-4753	352	7	s−1(m(n+	s−1(m(n+	NOUN
ejpam-4753	352	8	1	1	NUM
ejpam-4753	352	9	)	)	PUNCT
ejpam-4753	352	10	)	)	PUNCT
ejpam-4753	352	11	s−1(dn+1)//	s−1(dn+1)//	VERB
ejpam-4753	352	12	s−1(fk(n+1	s−1(fk(n+1	NOUN
ejpam-4753	352	13	)	)	PUNCT
ejpam-4753	352	14	)	)	PUNCT
ejpam-4753	352	15	�	�	PROPN
ejpam-4753	352	16	�	�	PROPN
ejpam-4753	352	17	s−1(m(n	s−1(m(n	NOUN
ejpam-4753	352	18	)	)	PUNCT
ejpam-4753	352	19	)	)	PUNCT
ejpam-4753	353	1	s−1(dn)//	s−1(dn)//	PROPN
ejpam-4753	353	2	s−1(fk(n	s−1(fk(n	NOUN
ejpam-4753	353	3	)	)	PUNCT
ejpam-4753	353	4	)	)	PUNCT
ejpam-4753	353	5	�	�	PROPN
ejpam-4753	353	6	�	�	PROPN
ejpam-4753	353	7	s−1(m(n−	s−1(m(n−	VERB
ejpam-4753	353	8	1	1	NUM
ejpam-4753	353	9	)	)	PUNCT
ejpam-4753	353	10	)	)	PUNCT
ejpam-4753	354	1	//	//	X
ejpam-4753	354	2	s−1(fk(n−1	s−1(fk(n−1	NUM
ejpam-4753	354	3	)	)	PUNCT
ejpam-4753	354	4	)	)	PUNCT
ejpam-4753	354	5	�	�	PROPN
ejpam-4753	354	6	�	�	PROPN
ejpam-4753	354	7	·	·	PUNCT
ejpam-4753	354	8	·	·	PUNCT
ejpam-4753	354	9	·	·	PUNCT
ejpam-4753	354	10	s−1(n∗	s−1(n∗	PROPN
ejpam-4753	354	11	)	)	PUNCT
ejpam-4753	354	12	:	:	PUNCT
ejpam-4753	354	13	·	·	PUNCT
ejpam-4753	354	14	·	·	PUNCT
ejpam-4753	354	15	·	·	PUNCT
ejpam-4753	354	16	//	//	NUM
ejpam-4753	354	17	s−1(n(n+	s−1(n(n+	NOUN
ejpam-4753	354	18	1	1	NUM
ejpam-4753	354	19	)	)	PUNCT
ejpam-4753	354	20	)	)	PUNCT
ejpam-4753	354	21	s−1(d′n+1+k)//	s−1(d′n+1+k)//	ADP
ejpam-4753	354	22	s−1(n(n	s−1(n(n	NOUN
ejpam-4753	354	23	)	)	PUNCT
ejpam-4753	354	24	)	)	PUNCT
ejpam-4753	354	25	s−1(d′n+k)//	s−1(d′n+k)//	NOUN
ejpam-4753	354	26	s−1(n(n−	s−1(n(n−	NOUN
ejpam-4753	354	27	1	1	NUM
ejpam-4753	354	28	)	)	PUNCT
ejpam-4753	354	29	)	)	PUNCT
ejpam-4753	354	30	//	//	X
ejpam-4753	354	31	·	·	PUNCT
ejpam-4753	354	32	·	·	PUNCT
ejpam-4753	354	33	·	·	PUNCT
ejpam-4753	355	1	proof	proof	NOUN
ejpam-4753	355	2	.	.	PUNCT
ejpam-4753	356	1	as	as	ADP
ejpam-4753	356	2	for	for	ADP
ejpam-4753	356	3	all	all	DET
ejpam-4753	356	4	n	n	PRON
ejpam-4753	356	5	∈	∈	PROPN
ejpam-4753	356	6	z	z	PROPN
ejpam-4753	356	7	,	,	PUNCT
ejpam-4753	356	8	m∗	m∗	NOUN
ejpam-4753	356	9	and	and	CCONJ
ejpam-4753	356	10	n∗	n∗	NOUN
ejpam-4753	356	11	are	be	AUX
ejpam-4753	356	12	two	two	NUM
ejpam-4753	356	13	complex	complex	ADJ
ejpam-4753	356	14	sequences	sequence	NOUN
ejpam-4753	356	15	of	of	ADP
ejpam-4753	356	16	graded	grade	VERB
ejpam-4753	356	17	left	leave	VERB
ejpam-4753	356	18	a−module	a−module	ADP
ejpam-4753	356	19	,	,	PUNCT
ejpam-4753	356	20	then	then	ADV
ejpam-4753	356	21	s−1(m∗	s−1(m∗	PROPN
ejpam-4753	356	22	)	)	PUNCT
ejpam-4753	356	23	and	and	CCONJ
ejpam-4753	356	24	s−1(n∗	s−1(n∗	PROPN
ejpam-4753	356	25	)	)	PUNCT
ejpam-4753	356	26	are	be	AUX
ejpam-4753	356	27	two	two	NUM
ejpam-4753	356	28	complex	complex	ADJ
ejpam-4753	356	29	sequences	sequence	NOUN
ejpam-4753	356	30	of	of	ADP
ejpam-4753	356	31	a	a	DET
ejpam-4753	356	32	graded	grade	VERB
ejpam-4753	356	33	left	leave	VERB
ejpam-4753	356	34	s−1a−module	s−1a−module	PROPN
ejpam-4753	356	35	.	.	PUNCT
ejpam-4753	357	1	prove	prove	VERB
ejpam-4753	357	2	that	that	SCONJ
ejpam-4753	357	3	for	for	ADP
ejpam-4753	357	4	all	all	DET
ejpam-4753	357	5	n	n	PRON
ejpam-4753	357	6	∈	∈	PROPN
ejpam-4753	357	7	z	z	PROPN
ejpam-4753	357	8	,	,	PUNCT
ejpam-4753	357	9	s−1(fk(n	s−1(fk(n	NOUN
ejpam-4753	357	10	)	)	PUNCT
ejpam-4753	357	11	)	)	PUNCT
ejpam-4753	357	12	◦	◦	NOUN
ejpam-4753	357	13	s−1(dn+1	s−1(dn+1	NOUN
ejpam-4753	357	14	)	)	PUNCT
ejpam-4753	358	1	=	=	PUNCT
ejpam-4753	358	2	s−1(d	s−1(d	ADJ
ejpam-4753	358	3	′	′	NUM
ejpam-4753	358	4	n+1+k	n+1+k	NOUN
ejpam-4753	358	5	)	)	PUNCT
ejpam-4753	358	6	◦	◦	NOUN
ejpam-4753	358	7	s−1(fk(n+	s−1(fk(n+	NOUN
ejpam-4753	358	8	1	1	NUM
ejpam-4753	358	9	)	)	PUNCT
ejpam-4753	358	10	)	)	PUNCT
ejpam-4753	358	11	.	.	PUNCT
ejpam-4753	359	1	let	let	VERB
ejpam-4753	359	2	x	x	X
ejpam-4753	359	3	s	s	PROPN
ejpam-4753	359	4	∈	∈	PROPN
ejpam-4753	359	5	s−1(m(n+1	s−1(m(n+1	PROPN
ejpam-4753	359	6	)	)	PUNCT
ejpam-4753	359	7	)	)	PUNCT
ejpam-4753	359	8	,	,	PUNCT
ejpam-4753	359	9	then	then	ADV
ejpam-4753	359	10	it	it	PRON
ejpam-4753	359	11	is	be	AUX
ejpam-4753	359	12	exist	exist	VERB
ejpam-4753	359	13	a	a	DET
ejpam-4753	359	14	unique	unique	ADJ
ejpam-4753	359	15	couple	couple	NOUN
ejpam-4753	359	16	(	(	PUNCT
ejpam-4753	359	17	y	y	PROPN
ejpam-4753	359	18	t	t	PROPN
ejpam-4753	359	19	,	,	PUNCT
ejpam-4753	359	20	z	z	NOUN
ejpam-4753	359	21	r	r	NOUN
ejpam-4753	359	22	)	)	PUNCT
ejpam-4753	359	23	∈	∈	PROPN
ejpam-4753	359	24	s−1mn+1×s−1m(n+2	s−1mn+1×s−1m(n+2	PROPN
ejpam-4753	359	25	)	)	PUNCT
ejpam-4753	359	26	such	such	ADJ
ejpam-4753	359	27	that	that	SCONJ
ejpam-4753	359	28	x	x	SYM
ejpam-4753	359	29	s	s	X
ejpam-4753	359	30	=	=	X
ejpam-4753	359	31	y	y	PROPN
ejpam-4753	359	32	t	t	NOUN
ejpam-4753	360	1	+	+	CCONJ
ejpam-4753	360	2	z	z	NOUN
ejpam-4753	360	3	r	r	NOUN
ejpam-4753	360	4	,	,	PUNCT
ejpam-4753	360	5	so	so	CCONJ
ejpam-4753	360	6	(	(	PUNCT
ejpam-4753	360	7	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	360	8	)	)	PUNCT
ejpam-4753	360	9	◦	◦	NOUN
ejpam-4753	360	10	s−1dn+1	s−1dn+1	NOUN
ejpam-4753	360	11	)	)	PUNCT
ejpam-4753	360	12	(	(	PUNCT
ejpam-4753	360	13	x	x	SYM
ejpam-4753	360	14	s	s	X
ejpam-4753	360	15	)	)	PUNCT
ejpam-4753	360	16	=	=	SYM
ejpam-4753	360	17	s−1fk(n)[s−1dn+1	s−1fk(n)[s−1dn+1	NOUN
ejpam-4753	360	18	(	(	PUNCT
ejpam-4753	360	19	x	x	NOUN
ejpam-4753	360	20	s	s	X
ejpam-4753	360	21	)	)	PUNCT
ejpam-4753	360	22	]	]	PUNCT
ejpam-4753	360	23	=	=	SYM
ejpam-4753	360	24	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	360	25	)	)	PUNCT
ejpam-4753	361	1	[	[	PUNCT
ejpam-4753	361	2	y	y	NOUN
ejpam-4753	361	3	t	t	X
ejpam-4753	361	4	]	]	PUNCT
ejpam-4753	361	5	=	=	SYM
ejpam-4753	361	6	s−1f	s−1f	NOUN
ejpam-4753	361	7	[	[	PUNCT
ejpam-4753	361	8	y	y	PROPN
ejpam-4753	361	9	t	t	X
ejpam-4753	361	10	]	]	PUNCT
ejpam-4753	361	11	=	=	SYM
ejpam-4753	361	12	f(y	f(y	NOUN
ejpam-4753	361	13	)	)	PUNCT
ejpam-4753	361	14	t	t	NOUN
ejpam-4753	361	15	,	,	PUNCT
ejpam-4753	361	16	and	and	CCONJ
ejpam-4753	361	17	(	(	PUNCT
ejpam-4753	361	18	s−1d	s−1d	NOUN
ejpam-4753	361	19	′	′	NOUN
ejpam-4753	361	20	n+1+k	n+1+k	NOUN
ejpam-4753	361	21	◦	◦	VERB
ejpam-4753	361	22	s−1fk(n+	s−1fk(n+	NOUN
ejpam-4753	361	23	1	1	NUM
ejpam-4753	361	24	)	)	PUNCT
ejpam-4753	361	25	)	)	PUNCT
ejpam-4753	362	1	(	(	PUNCT
ejpam-4753	362	2	x	x	SYM
ejpam-4753	362	3	s	s	X
ejpam-4753	362	4	)	)	PUNCT
ejpam-4753	362	5	=	=	SYM
ejpam-4753	362	6	s−1d	s−1d	NOUN
ejpam-4753	362	7	′	′	VERB
ejpam-4753	362	8	n+1+k[s	n+1+k[s	ADP
ejpam-4753	362	9	−1fk(n+	−1fk(n+	PROPN
ejpam-4753	362	10	1	1	NUM
ejpam-4753	362	11	)	)	PUNCT
ejpam-4753	362	12	(	(	PUNCT
ejpam-4753	362	13	y	y	PROPN
ejpam-4753	362	14	t	t	PROPN
ejpam-4753	363	1	+	+	CCONJ
ejpam-4753	363	2	z	z	NOUN
ejpam-4753	363	3	r	r	NOUN
ejpam-4753	363	4	)	)	PUNCT
ejpam-4753	363	5	]	]	PUNCT
ejpam-4753	364	1	=	=	PUNCT
ejpam-4753	364	2	s−1d	s−1d	NOUN
ejpam-4753	364	3	′	′	VERB
ejpam-4753	364	4	n+1+k[s	n+1+k[s	ADP
ejpam-4753	364	5	−1f	−1f	PROPN
ejpam-4753	364	6	(	(	PUNCT
ejpam-4753	364	7	y	y	PROPN
ejpam-4753	364	8	t	t	PROPN
ejpam-4753	365	1	+	+	CCONJ
ejpam-4753	365	2	z	z	NOUN
ejpam-4753	365	3	r	r	NOUN
ejpam-4753	365	4	)	)	PUNCT
ejpam-4753	365	5	]	]	PUNCT
ejpam-4753	366	1	=	=	PUNCT
ejpam-4753	366	2	s−1d	s−1d	NOUN
ejpam-4753	366	3	′	′	VERB
ejpam-4753	366	4	n+1+k[s	n+1+k[s	ADP
ejpam-4753	366	5	−1f	−1f	PROPN
ejpam-4753	366	6	(	(	PUNCT
ejpam-4753	366	7	y	y	PROPN
ejpam-4753	366	8	t	t	PROPN
ejpam-4753	366	9	)	)	PUNCT
ejpam-4753	367	1	+	+	CCONJ
ejpam-4753	367	2	s−1f	s−1f	NOUN
ejpam-4753	367	3	(	(	PUNCT
ejpam-4753	367	4	z	z	NOUN
ejpam-4753	367	5	r	r	NOUN
ejpam-4753	367	6	)	)	PUNCT
ejpam-4753	367	7	]	]	PUNCT
ejpam-4753	368	1	=	=	SYM
ejpam-4753	368	2	s−1f	s−1f	NOUN
ejpam-4753	368	3	(	(	PUNCT
ejpam-4753	368	4	y	y	PROPN
ejpam-4753	368	5	t	t	PROPN
ejpam-4753	368	6	)	)	PUNCT
ejpam-4753	368	7	=	=	SYM
ejpam-4753	368	8	f(y	f(y	NOUN
ejpam-4753	368	9	)	)	PUNCT
ejpam-4753	368	10	t	t	PROPN
ejpam-4753	368	11	a.	a.	NOUN
ejpam-4753	368	12	o.	o.	PROPN
ejpam-4753	368	13	chbih	chbih	PROPN
ejpam-4753	368	14	,	,	PUNCT
ejpam-4753	368	15	m.	m.	PROPN
ejpam-4753	368	16	b.	b.	PROPN
ejpam-4753	368	17	maaouia	maaouia	PROPN
ejpam-4753	368	18	,	,	PUNCT
ejpam-4753	368	19	m.	m.	NOUN
ejpam-4753	368	20	sanghare	sanghare	PROPN
ejpam-4753	368	21	/	/	SYM
ejpam-4753	368	22	eur	eur	PROPN
ejpam-4753	368	23	.	.	PUNCT
ejpam-4753	369	1	j.	j.	PROPN
ejpam-4753	369	2	pure	pure	PROPN
ejpam-4753	369	3	appl	appl	PROPN
ejpam-4753	369	4	.	.	PROPN
ejpam-4753	369	5	math	math	PROPN
ejpam-4753	369	6	,	,	PUNCT
ejpam-4753	369	7	16	16	NUM
ejpam-4753	369	8	(	(	PUNCT
ejpam-4753	369	9	3	3	NUM
ejpam-4753	369	10	)	)	PUNCT
ejpam-4753	369	11	(	(	PUNCT
ejpam-4753	369	12	2023	2023	NUM
ejpam-4753	369	13	)	)	PUNCT
ejpam-4753	369	14	,	,	PUNCT
ejpam-4753	369	15	1913	1913	NUM
ejpam-4753	369	16	-	-	SYM
ejpam-4753	369	17	1939	1939	NUM
ejpam-4753	369	18	1930	1930	NUM
ejpam-4753	369	19	because	because	SCONJ
ejpam-4753	369	20	s−1f(yt	s−1f(yt	NOUN
ejpam-4753	369	21	)	)	PUNCT
ejpam-4753	370	1	∈	∈	PROPN
ejpam-4753	370	2	s−1nn+1+k	s−1nn+1+k	NOUN
ejpam-4753	370	3	and	and	CCONJ
ejpam-4753	370	4	s−1f	s−1f	NOUN
ejpam-4753	370	5	(	(	PUNCT
ejpam-4753	370	6	zr	zr	NOUN
ejpam-4753	370	7	)	)	PUNCT
ejpam-4753	370	8	∈	∈	PROPN
ejpam-4753	370	9	s−1n(n+	s−1n(n+	NOUN
ejpam-4753	370	10	2	2	NUM
ejpam-4753	370	11	+	+	CCONJ
ejpam-4753	370	12	k	k	X
ejpam-4753	370	13	)	)	PUNCT
ejpam-4753	371	1	=	=	NOUN
ejpam-4753	371	2	⇒	⇒	NOUN
ejpam-4753	371	3	(	(	PUNCT
ejpam-4753	371	4	s−1d	s−1d	X
ejpam-4753	371	5	′	′	NUM
ejpam-4753	371	6	n+1+k	n+1+k	NOUN
ejpam-4753	371	7	◦	◦	NOUN
ejpam-4753	371	8	s−1fk(n+1	s−1fk(n+1	NOUN
ejpam-4753	371	9	)	)	PUNCT
ejpam-4753	371	10	)	)	PUNCT
ejpam-4753	371	11	(	(	PUNCT
ejpam-4753	371	12	x	x	X
ejpam-4753	371	13	s	s	X
ejpam-4753	371	14	)	)	PUNCT
ejpam-4753	371	15	=	=	SYM
ejpam-4753	371	16	(	(	PUNCT
ejpam-4753	371	17	s−1fk(n)	s−1fk(n)	NOUN
ejpam-4753	371	18	◦	◦	NOUN
ejpam-4753	371	19	s−1dn+1	s−1dn+1	NOUN
ejpam-4753	371	20	)	)	PUNCT
ejpam-4753	371	21	(	(	PUNCT
ejpam-4753	371	22	x	x	SYM
ejpam-4753	371	23	s	s	X
ejpam-4753	371	24	)	)	PUNCT
ejpam-4753	371	25	∀	∀	X
ejpam-4753	371	26	x	x	SYM
ejpam-4753	371	27	s	s	PROPN
ejpam-4753	371	28	∈	∈	PROPN
ejpam-4753	371	29	s−1m(n+1	s−1m(n+1	NOUN
ejpam-4753	371	30	)	)	PUNCT
ejpam-4753	372	1	so	so	ADV
ejpam-4753	372	2	(	(	PUNCT
ejpam-4753	372	3	s−1d	s−1d	X
ejpam-4753	372	4	′	′	NOUN
ejpam-4753	372	5	n+1+k	n+1+k	NOUN
ejpam-4753	372	6	◦	◦	VERB
ejpam-4753	372	7	s−1fk(n+	s−1fk(n+	NOUN
ejpam-4753	372	8	1	1	NUM
ejpam-4753	372	9	)	)	PUNCT
ejpam-4753	372	10	)	)	PUNCT
ejpam-4753	372	11	)	)	PUNCT
ejpam-4753	373	1	=	=	PRON
ejpam-4753	373	2	(	(	PUNCT
ejpam-4753	373	3	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	373	4	)	)	PUNCT
ejpam-4753	373	5	◦	◦	NOUN
ejpam-4753	373	6	s−1dn+1	s−1dn+1	NOUN
ejpam-4753	373	7	)	)	PUNCT
ejpam-4753	373	8	thus	thus	ADV
ejpam-4753	373	9	s−1(fk	s−1(fk	PROPN
ejpam-4753	373	10	∗	∗	NOUN
ejpam-4753	373	11	)	)	PUNCT
ejpam-4753	373	12	is	be	AUX
ejpam-4753	373	13	a	a	DET
ejpam-4753	373	14	complex	complex	ADJ
ejpam-4753	373	15	chain	chain	NOUN
ejpam-4753	373	16	.	.	PUNCT
ejpam-4753	374	1	corollary	corollary	ADJ
ejpam-4753	374	2	8	8	NUM
ejpam-4753	374	3	.	.	PUNCT
ejpam-4753	375	1	let	let	VERB
ejpam-4753	375	2	a	a	DET
ejpam-4753	375	3	=	=	SYM
ejpam-4753	375	4	⊕	⊕	PROPN
ejpam-4753	375	5	n∈z	n∈z	VERB
ejpam-4753	375	6	an	an	DET
ejpam-4753	375	7	be	be	AUX
ejpam-4753	375	8	a	a	DET
ejpam-4753	375	9	graded	grade	VERB
ejpam-4753	375	10	duo	duo	NOUN
ejpam-4753	375	11	-	-	PUNCT
ejpam-4753	375	12	ring	ring	NOUN
ejpam-4753	375	13	,	,	PUNCT
ejpam-4753	375	14	m	m	VERB
ejpam-4753	375	15	=	=	ADJ
ejpam-4753	375	16	⊕	⊕	PROPN
ejpam-4753	375	17	n∈z	n∈z	VERB
ejpam-4753	375	18	mn	mn	PROPN
ejpam-4753	375	19	and	and	CCONJ
ejpam-4753	375	20	n	n	PROPN
ejpam-4753	375	21	=	=	PROPN
ejpam-4753	375	22	⊕	⊕	PROPN
ejpam-4753	375	23	n∈z	n∈z	ADJ
ejpam-4753	375	24	nn	nn	PROPN
ejpam-4753	375	25	are	be	AUX
ejpam-4753	375	26	two	two	NUM
ejpam-4753	375	27	graded	grade	VERB
ejpam-4753	375	28	left	leave	VERB
ejpam-4753	375	29	a−modules	a−module	NOUN
ejpam-4753	375	30	,	,	PUNCT
ejpam-4753	375	31	f	f	X
ejpam-4753	375	32	:	:	PUNCT
ejpam-4753	375	33	m	m	VERB
ejpam-4753	375	34	−→	−→	ADJ
ejpam-4753	376	1	n	n	NOUN
ejpam-4753	376	2	is	be	AUX
ejpam-4753	376	3	a	a	DET
ejpam-4753	376	4	graded	grade	VERB
ejpam-4753	376	5	morphism	morphism	NOUN
ejpam-4753	376	6	of	of	ADP
ejpam-4753	376	7	degree	degree	NOUN
ejpam-4753	376	8	k	k	PROPN
ejpam-4753	377	1	and	and	CCONJ
ejpam-4753	377	2	sh	sh	PROPN
ejpam-4753	377	3	be	be	AUX
ejpam-4753	377	4	a	a	DET
ejpam-4753	377	5	part	part	NOUN
ejpam-4753	377	6	formed	form	VERB
ejpam-4753	377	7	of	of	ADP
ejpam-4753	377	8	regulars	regular	NOUN
ejpam-4753	377	9	homogeneous	homogeneous	ADJ
ejpam-4753	377	10	elements	element	NOUN
ejpam-4753	377	11	of	of	ADP
ejpam-4753	377	12	a	a	PRON
ejpam-4753	377	13	,	,	PUNCT
ejpam-4753	377	14	then	then	ADV
ejpam-4753	377	15	we	we	PRON
ejpam-4753	377	16	have	have	VERB
ejpam-4753	377	17	:	:	PUNCT
ejpam-4753	377	18	(	(	PUNCT
ejpam-4753	377	19	i	i	NOUN
ejpam-4753	377	20	)	)	PUNCT
ejpam-4753	377	21	the	the	DET
ejpam-4753	377	22	following	follow	VERB
ejpam-4753	377	23	complex	complex	ADJ
ejpam-4753	377	24	sequence	sequence	NOUN
ejpam-4753	377	25	:	:	PUNCT
ejpam-4753	377	26	s	s	VERB
ejpam-4753	377	27	−1	−1	NOUN
ejpam-4753	377	28	h	h	NOUN
ejpam-4753	377	29	(	(	PUNCT
ejpam-4753	377	30	m∗	m∗	PROPN
ejpam-4753	377	31	)	)	PUNCT
ejpam-4753	377	32	:	:	PUNCT
ejpam-4753	377	33	·	·	PUNCT
ejpam-4753	377	34	·	·	PUNCT
ejpam-4753	377	35	·	·	PUNCT
ejpam-4753	378	1	−→	−→	NOUN
ejpam-4753	378	2	s	s	PRON
ejpam-4753	378	3	−1	−1	NOUN
ejpam-4753	378	4	h	h	NOUN
ejpam-4753	378	5	(	(	PUNCT
ejpam-4753	378	6	m(n+1	m(n+1	NOUN
ejpam-4753	378	7	)	)	PUNCT
ejpam-4753	378	8	)	)	PUNCT
ejpam-4753	378	9	s	s	VERB
ejpam-4753	378	10	−1	−1	NOUN
ejpam-4753	378	11	h	h	NOUN
ejpam-4753	378	12	(	(	PUNCT
ejpam-4753	378	13	dn+1)−→	dn+1)−→	PROPN
ejpam-4753	378	14	s	s	PART
ejpam-4753	378	15	−1	−1	NOUN
ejpam-4753	378	16	h	h	NOUN
ejpam-4753	378	17	(	(	PUNCT
ejpam-4753	378	18	m(n	m(n	PROPN
ejpam-4753	378	19	)	)	PUNCT
ejpam-4753	378	20	)	)	PUNCT
ejpam-4753	379	1	s	s	VERB
ejpam-4753	379	2	−1	−1	NOUN
ejpam-4753	379	3	h	h	NOUN
ejpam-4753	379	4	(	(	PUNCT
ejpam-4753	379	5	dn)−→	dn)−→	PROPN
ejpam-4753	379	6	s	s	PART
ejpam-4753	379	7	−1	−1	NOUN
ejpam-4753	379	8	h	h	NOUN
ejpam-4753	379	9	(	(	PUNCT
ejpam-4753	379	10	m(n−1	m(n−1	PROPN
ejpam-4753	379	11	)	)	PUNCT
ejpam-4753	379	12	)	)	PUNCT
ejpam-4753	380	1	−→	−→	NOUN
ejpam-4753	380	2	·	·	PUNCT
ejpam-4753	380	3	·	·	PUNCT
ejpam-4753	380	4	·	·	PUNCT
ejpam-4753	380	5	(	(	PUNCT
ejpam-4753	380	6	ii	ii	NOUN
ejpam-4753	380	7	)	)	PUNCT
ejpam-4753	380	8	the	the	DET
ejpam-4753	380	9	following	follow	VERB
ejpam-4753	380	10	complex	complex	ADJ
ejpam-4753	380	11	chain	chain	NOUN
ejpam-4753	380	12	:	:	PUNCT
ejpam-4753	380	13	s	s	X
ejpam-4753	380	14	−1	−1	NOUN
ejpam-4753	380	15	h	h	NOUN
ejpam-4753	380	16	(	(	PUNCT
ejpam-4753	380	17	m∗	m∗	PROPN
ejpam-4753	380	18	)	)	PUNCT
ejpam-4753	380	19	:	:	PUNCT
ejpam-4753	380	20	·	·	PUNCT
ejpam-4753	380	21	·	·	PUNCT
ejpam-4753	380	22	·	·	PUNCT
ejpam-4753	381	1	//	//	PUNCT
ejpam-4753	381	2	s	s	PART
ejpam-4753	381	3	−1	−1	NOUN
ejpam-4753	381	4	h	h	NOUN
ejpam-4753	381	5	(	(	PUNCT
ejpam-4753	381	6	fk	fk	INTJ
ejpam-4753	381	7	∗	∗	NOUN
ejpam-4753	381	8	)	)	PUNCT
ejpam-4753	381	9	�	�	PROPN
ejpam-4753	381	10	�	�	PROPN
ejpam-4753	381	11	//	//	PROPN
ejpam-4753	381	12	s	s	PART
ejpam-4753	381	13	−1	−1	NOUN
ejpam-4753	381	14	h	h	NOUN
ejpam-4753	381	15	(	(	PUNCT
ejpam-4753	381	16	m(n+	m(n+	NOUN
ejpam-4753	381	17	1	1	NUM
ejpam-4753	381	18	)	)	PUNCT
ejpam-4753	381	19	)	)	PUNCT
ejpam-4753	381	20	s	s	VERB
ejpam-4753	381	21	−1	−1	NOUN
ejpam-4753	381	22	h	h	NOUN
ejpam-4753	381	23	(	(	PUNCT
ejpam-4753	381	24	dn+1)//	dn+1)//	PROPN
ejpam-4753	381	25	s	s	PART
ejpam-4753	381	26	−1	−1	NOUN
ejpam-4753	381	27	h	h	NOUN
ejpam-4753	381	28	(	(	PUNCT
ejpam-4753	381	29	fk(n+1	fk(n+1	PROPN
ejpam-4753	381	30	)	)	PUNCT
ejpam-4753	381	31	)	)	PUNCT
ejpam-4753	381	32	�	�	PROPN
ejpam-4753	381	33	�	�	PROPN
ejpam-4753	381	34	s	s	PART
ejpam-4753	381	35	−1	−1	NOUN
ejpam-4753	381	36	h	h	NOUN
ejpam-4753	381	37	(	(	PUNCT
ejpam-4753	381	38	m(n	m(n	PROPN
ejpam-4753	381	39	)	)	PUNCT
ejpam-4753	381	40	)	)	PUNCT
ejpam-4753	382	1	s	s	VERB
ejpam-4753	382	2	−1	−1	NOUN
ejpam-4753	382	3	h	h	NOUN
ejpam-4753	382	4	(	(	PUNCT
ejpam-4753	382	5	dn)//	dn)//	PROPN
ejpam-4753	382	6	s	s	PART
ejpam-4753	382	7	−1	−1	NOUN
ejpam-4753	382	8	h	h	NOUN
ejpam-4753	382	9	(	(	PUNCT
ejpam-4753	382	10	fk(n	fk(n	NOUN
ejpam-4753	382	11	)	)	PUNCT
ejpam-4753	382	12	)	)	PUNCT
ejpam-4753	382	13	�	�	PROPN
ejpam-4753	382	14	�	�	PROPN
ejpam-4753	382	15	s	s	PART
ejpam-4753	382	16	−1	−1	NOUN
ejpam-4753	382	17	h	h	NOUN
ejpam-4753	382	18	(	(	PUNCT
ejpam-4753	382	19	m(n−	m(n−	NOUN
ejpam-4753	382	20	1	1	NUM
ejpam-4753	382	21	)	)	PUNCT
ejpam-4753	382	22	)	)	PUNCT
ejpam-4753	383	1	//	//	PUNCT
ejpam-4753	383	2	s	s	PART
ejpam-4753	383	3	−1	−1	NOUN
ejpam-4753	383	4	h	h	NOUN
ejpam-4753	383	5	(	(	PUNCT
ejpam-4753	383	6	fk(n−1	fk(n−1	PROPN
ejpam-4753	383	7	)	)	PUNCT
ejpam-4753	383	8	)	)	PUNCT
ejpam-4753	383	9	�	�	PROPN
ejpam-4753	383	10	�	�	PROPN
ejpam-4753	383	11	·	·	PUNCT
ejpam-4753	383	12	·	·	PUNCT
ejpam-4753	383	13	·	·	PUNCT
ejpam-4753	383	14	s	s	X
ejpam-4753	383	15	−1	−1	NOUN
ejpam-4753	383	16	h	h	NOUN
ejpam-4753	383	17	(	(	PUNCT
ejpam-4753	383	18	n∗	n∗	PROPN
ejpam-4753	383	19	)	)	PUNCT
ejpam-4753	383	20	:	:	PUNCT
ejpam-4753	383	21	·	·	PUNCT
ejpam-4753	383	22	·	·	PUNCT
ejpam-4753	383	23	·	·	PUNCT
ejpam-4753	384	1	//	//	PUNCT
ejpam-4753	384	2	s	s	PART
ejpam-4753	384	3	−1	−1	NOUN
ejpam-4753	384	4	h	h	NOUN
ejpam-4753	384	5	(	(	PUNCT
ejpam-4753	384	6	n(n+	n(n+	NOUN
ejpam-4753	384	7	1	1	NUM
ejpam-4753	384	8	)	)	PUNCT
ejpam-4753	384	9	)	)	PUNCT
ejpam-4753	384	10	s	s	VERB
ejpam-4753	384	11	−1	−1	NOUN
ejpam-4753	384	12	h	h	NOUN
ejpam-4753	384	13	(	(	PUNCT
ejpam-4753	384	14	d′n+1+k)//	d′n+1+k)//	PROPN
ejpam-4753	384	15	s	s	PART
ejpam-4753	384	16	−1	−1	NOUN
ejpam-4753	384	17	h	h	NOUN
ejpam-4753	384	18	(	(	PUNCT
ejpam-4753	384	19	n(n	n(n	NOUN
ejpam-4753	384	20	)	)	PUNCT
ejpam-4753	384	21	)	)	PUNCT
ejpam-4753	385	1	s	s	VERB
ejpam-4753	385	2	−1	−1	NOUN
ejpam-4753	385	3	h	h	NOUN
ejpam-4753	385	4	(	(	PUNCT
ejpam-4753	385	5	d′n+k)//	d′n+k)//	NOUN
ejpam-4753	385	6	s	s	PART
ejpam-4753	385	7	−1	−1	NOUN
ejpam-4753	385	8	h	h	NOUN
ejpam-4753	385	9	(	(	PUNCT
ejpam-4753	385	10	n(n−	n(n−	PROPN
ejpam-4753	385	11	1	1	NUM
ejpam-4753	385	12	)	)	PUNCT
ejpam-4753	385	13	)	)	PUNCT
ejpam-4753	385	14	//	//	X
ejpam-4753	385	15	·	·	PUNCT
ejpam-4753	385	16	·	·	PUNCT
ejpam-4753	386	1	·	·	PUNCT
ejpam-4753	386	2	proof	proof	NOUN
ejpam-4753	386	3	.	.	PUNCT
ejpam-4753	387	1	since	since	SCONJ
ejpam-4753	387	2	the	the	DET
ejpam-4753	387	3	proposition	proposition	NOUN
ejpam-4753	387	4	6	6	NUM
ejpam-4753	387	5	sh	sh	NOUN
ejpam-4753	387	6	is	be	AUX
ejpam-4753	387	7	multiplicatively	multiplicatively	ADV
ejpam-4753	387	8	closed	close	VERB
ejpam-4753	387	9	subset	subset	NOUN
ejpam-4753	387	10	satisfying	satisfy	VERB
ejpam-4753	387	11	the	the	DET
ejpam-4753	387	12	left	left	ADJ
ejpam-4753	387	13	conditions	condition	NOUN
ejpam-4753	387	14	of	of	ADP
ejpam-4753	387	15	ore	ore	NOUN
ejpam-4753	387	16	formed	form	VERB
ejpam-4753	387	17	of	of	ADP
ejpam-4753	387	18	homogeneous	homogeneous	ADJ
ejpam-4753	387	19	elements	element	NOUN
ejpam-4753	387	20	of	of	ADP
ejpam-4753	387	21	a	a	PRON
ejpam-4753	387	22	,	,	PUNCT
ejpam-4753	387	23	and	and	CCONJ
ejpam-4753	387	24	the	the	DET
ejpam-4753	387	25	rest	rest	NOUN
ejpam-4753	387	26	is	be	AUX
ejpam-4753	387	27	similarly	similarly	ADV
ejpam-4753	387	28	to	to	ADP
ejpam-4753	387	29	the	the	DET
ejpam-4753	387	30	proof	proof	NOUN
ejpam-4753	387	31	of	of	ADP
ejpam-4753	387	32	the	the	DET
ejpam-4753	387	33	proposition	proposition	NOUN
ejpam-4753	387	34	16	16	NUM
ejpam-4753	387	35	.	.	PUNCT
ejpam-4753	388	1	proposition	proposition	NOUN
ejpam-4753	388	2	17	17	NUM
ejpam-4753	388	3	.	.	PUNCT
ejpam-4753	389	1	let	let	VERB
ejpam-4753	389	2	a	a	DET
ejpam-4753	389	3	=	=	SYM
ejpam-4753	389	4	⊕	⊕	PROPN
ejpam-4753	389	5	n∈z	n∈z	VERB
ejpam-4753	389	6	an	an	DET
ejpam-4753	389	7	be	be	AUX
ejpam-4753	389	8	a	a	DET
ejpam-4753	389	9	graded	grade	VERB
ejpam-4753	389	10	ring	ring	NOUN
ejpam-4753	389	11	,	,	PUNCT
ejpam-4753	389	12	m	m	VERB
ejpam-4753	389	13	=	=	ADJ
ejpam-4753	389	14	⊕	⊕	PROPN
ejpam-4753	389	15	n∈z	n∈z	VERB
ejpam-4753	389	16	mn	mn	PROPN
ejpam-4753	389	17	and	and	CCONJ
ejpam-4753	389	18	n	n	PROPN
ejpam-4753	389	19	=	=	PROPN
ejpam-4753	389	20	⊕	⊕	PROPN
ejpam-4753	389	21	n∈z	n∈z	VERB
ejpam-4753	389	22	nn	nn	PROPN
ejpam-4753	389	23	two	two	NUM
ejpam-4753	389	24	graded	grade	VERB
ejpam-4753	389	25	left	leave	VERB
ejpam-4753	389	26	a−modules	a−module	NOUN
ejpam-4753	389	27	,	,	PUNCT
ejpam-4753	389	28	f	f	X
ejpam-4753	389	29	:	:	PUNCT
ejpam-4753	389	30	m	m	VERB
ejpam-4753	389	31	−→	−→	ADJ
ejpam-4753	389	32	n	n	NOUN
ejpam-4753	389	33	is	be	AUX
ejpam-4753	389	34	graded	grade	VERB
ejpam-4753	389	35	morphism	morphism	NOUN
ejpam-4753	389	36	of	of	ADP
ejpam-4753	389	37	degree	degree	NOUN
ejpam-4753	389	38	k	k	PROPN
ejpam-4753	389	39	∈	∈	PROPN
ejpam-4753	389	40	z	z	PROPN
ejpam-4753	389	41	and	and	CCONJ
ejpam-4753	389	42	s	s	AUX
ejpam-4753	389	43	be	be	AUX
ejpam-4753	389	44	a	a	DET
ejpam-4753	389	45	multiplicatively	multiplicatively	ADV
ejpam-4753	389	46	closed	close	VERB
ejpam-4753	389	47	subset	subset	NOUN
ejpam-4753	389	48	satisfying	satisfy	VERB
ejpam-4753	389	49	the	the	DET
ejpam-4753	389	50	left	left	ADJ
ejpam-4753	389	51	conditions	condition	NOUN
ejpam-4753	389	52	of	of	ADP
ejpam-4753	389	53	ore	ore	NOUN
ejpam-4753	389	54	formed	form	VERB
ejpam-4753	389	55	of	of	ADP
ejpam-4753	389	56	homogeneous	homogeneous	ADJ
ejpam-4753	389	57	elements	element	NOUN
ejpam-4753	389	58	of	of	ADP
ejpam-4753	389	59	a	a	PRON
ejpam-4753	389	60	,	,	PUNCT
ejpam-4753	389	61	then	then	ADV
ejpam-4753	389	62	we	we	PRON
ejpam-4753	389	63	have	have	VERB
ejpam-4753	389	64	:	:	PUNCT
ejpam-4753	389	65	(	(	PUNCT
ejpam-4753	389	66	i	i	NOUN
ejpam-4753	389	67	)	)	PUNCT
ejpam-4753	389	68	the	the	DET
ejpam-4753	389	69	following	follow	VERB
ejpam-4753	389	70	complex	complex	ADJ
ejpam-4753	389	71	sequence	sequence	NOUN
ejpam-4753	389	72	:	:	PUNCT
ejpam-4753	389	73	b∗	b∗	ADJ
ejpam-4753	389	74	:	:	PUNCT
ejpam-4753	389	75	·	·	PUNCT
ejpam-4753	389	76	·	·	PUNCT
ejpam-4753	389	77	·	·	PUNCT
ejpam-4753	390	1	−→	−→	NOUN
ejpam-4753	390	2	s−1a⊗	s−1a⊗	PRON
ejpam-4753	390	3	(	(	PUNCT
ejpam-4753	390	4	m(n+	m(n+	NOUN
ejpam-4753	390	5	1	1	NUM
ejpam-4753	390	6	)	)	PUNCT
ejpam-4753	390	7	)	)	PUNCT
ejpam-4753	390	8	s−1a⊗(dn+1)−→	s−1a⊗(dn+1)−→	NOUN
ejpam-4753	390	9	s−1a⊗	s−1a⊗	PUNCT
ejpam-4753	390	10	(	(	PUNCT
ejpam-4753	390	11	m(n	m(n	PROPN
ejpam-4753	390	12	)	)	PUNCT
ejpam-4753	390	13	)	)	PUNCT
ejpam-4753	390	14	s−1a⊗(dn)−→	s−1a⊗(dn)−→	PUNCT
ejpam-4753	390	15	s−1a⊗	s−1a⊗	X
ejpam-4753	390	16	(	(	PUNCT
ejpam-4753	390	17	m(n−	m(n−	NOUN
ejpam-4753	390	18	1	1	NUM
ejpam-4753	390	19	)	)	PUNCT
ejpam-4753	390	20	)	)	PUNCT
ejpam-4753	391	1	−→	−→	NOUN
ejpam-4753	391	2	·	·	PUNCT
ejpam-4753	391	3	·	·	PUNCT
ejpam-4753	391	4	·	·	PUNCT
ejpam-4753	391	5	(	(	PUNCT
ejpam-4753	391	6	ii	ii	NOUN
ejpam-4753	391	7	)	)	PUNCT
ejpam-4753	391	8	the	the	DET
ejpam-4753	391	9	following	follow	VERB
ejpam-4753	391	10	complex	complex	ADJ
ejpam-4753	391	11	chain	chain	NOUN
ejpam-4753	391	12	:	:	PUNCT
ejpam-4753	391	13	b∗	b∗	ADJ
ejpam-4753	391	14	:	:	PUNCT
ejpam-4753	391	15	·	·	PUNCT
ejpam-4753	391	16	·	·	PUNCT
ejpam-4753	391	17	·	·	PUNCT
ejpam-4753	391	18	//	//	PUNCT
ejpam-4753	392	1	s−1a⊗fk	s−1a⊗fk	NOUN
ejpam-4753	392	2	∗	∗	NOUN
ejpam-4753	392	3	�	�	PROPN
ejpam-4753	392	4	�	�	PROPN
ejpam-4753	392	5	//	//	NUM
ejpam-4753	392	6	s−1a⊗	s−1a⊗	PROPN
ejpam-4753	392	7	(	(	PUNCT
ejpam-4753	392	8	m(n+	m(n+	NOUN
ejpam-4753	392	9	1	1	NUM
ejpam-4753	392	10	)	)	PUNCT
ejpam-4753	392	11	)	)	PUNCT
ejpam-4753	392	12	s−1a⊗(dn+1)//	s−1a⊗(dn+1)//	PROPN
ejpam-4753	392	13	s−1a⊗(fk(n+1	s−1a⊗(fk(n+1	PROPN
ejpam-4753	392	14	)	)	PUNCT
ejpam-4753	392	15	)	)	PUNCT
ejpam-4753	392	16	�	�	PROPN
ejpam-4753	392	17	�	�	PROPN
ejpam-4753	392	18	s−1a⊗	s−1a⊗	PROPN
ejpam-4753	392	19	(	(	PUNCT
ejpam-4753	392	20	m(n	m(n	PROPN
ejpam-4753	392	21	)	)	PUNCT
ejpam-4753	392	22	)	)	PUNCT
ejpam-4753	392	23	s−1a⊗(dn)//	s−1a⊗(dn)//	NOUN
ejpam-4753	392	24	s−1a⊗(fk(n	s−1a⊗(fk(n	PROPN
ejpam-4753	392	25	)	)	PUNCT
ejpam-4753	392	26	)	)	PUNCT
ejpam-4753	392	27	�	�	PROPN
ejpam-4753	392	28	�	�	PROPN
ejpam-4753	392	29	s−1a⊗	s−1a⊗	PROPN
ejpam-4753	392	30	(	(	PUNCT
ejpam-4753	392	31	m(n−	m(n−	NOUN
ejpam-4753	392	32	1	1	NUM
ejpam-4753	392	33	)	)	PUNCT
ejpam-4753	392	34	)	)	PUNCT
ejpam-4753	392	35	//	//	PUNCT
ejpam-4753	393	1	s−1a⊗(fk(n−1	s−1a⊗(fk(n−1	PROPN
ejpam-4753	393	2	)	)	PUNCT
ejpam-4753	393	3	)	)	PUNCT
ejpam-4753	393	4	�	�	PROPN
ejpam-4753	393	5	�	�	PROPN
ejpam-4753	393	6	·	·	PUNCT
ejpam-4753	393	7	·	·	PUNCT
ejpam-4753	393	8	·	·	PUNCT
ejpam-4753	393	9	d∗	d∗	INTJ
ejpam-4753	393	10	:	:	PUNCT
ejpam-4753	393	11	·	·	PUNCT
ejpam-4753	393	12	·	·	PUNCT
ejpam-4753	393	13	·	·	PUNCT
ejpam-4753	394	1	//	//	SYM
ejpam-4753	394	2	s−1a⊗	s−1a⊗	PROPN
ejpam-4753	394	3	(	(	PUNCT
ejpam-4753	394	4	n(n+	n(n+	NOUN
ejpam-4753	394	5	1	1	NUM
ejpam-4753	394	6	)	)	PUNCT
ejpam-4753	394	7	)	)	PUNCT
ejpam-4753	394	8	s−1a⊗(d′	s−1a⊗(d′	NUM
ejpam-4753	395	1	n+1+k)//	n+1+k)//	NOUN
ejpam-4753	395	2	s−1a⊗n(n	s−1a⊗n(n	NOUN
ejpam-4753	395	3	)	)	PUNCT
ejpam-4753	395	4	)	)	PUNCT
ejpam-4753	396	1	s−1a⊗(d′	s−1a⊗(d′	X
ejpam-4753	396	2	n+k)//	n+k)//	ADV
ejpam-4753	396	3	s−1a⊗	s−1a⊗	PROPN
ejpam-4753	396	4	(	(	PUNCT
ejpam-4753	396	5	n(n−	n(n−	PROPN
ejpam-4753	396	6	1	1	NUM
ejpam-4753	396	7	)	)	PUNCT
ejpam-4753	396	8	)	)	PUNCT
ejpam-4753	396	9	//	//	X
ejpam-4753	396	10	·	·	PUNCT
ejpam-4753	396	11	·	·	PUNCT
ejpam-4753	396	12	·	·	PUNCT
ejpam-4753	396	13	a.	a.	PROPN
ejpam-4753	396	14	o.	o.	PROPN
ejpam-4753	396	15	chbih	chbih	PROPN
ejpam-4753	396	16	,	,	PUNCT
ejpam-4753	396	17	m.	m.	PROPN
ejpam-4753	396	18	b.	b.	PROPN
ejpam-4753	396	19	maaouia	maaouia	PROPN
ejpam-4753	396	20	,	,	PUNCT
ejpam-4753	396	21	m.	m.	NOUN
ejpam-4753	396	22	sanghare	sanghare	PROPN
ejpam-4753	396	23	/	/	SYM
ejpam-4753	396	24	eur	eur	PROPN
ejpam-4753	396	25	.	.	PUNCT
ejpam-4753	397	1	j.	j.	PROPN
ejpam-4753	397	2	pure	pure	PROPN
ejpam-4753	397	3	appl	appl	PROPN
ejpam-4753	397	4	.	.	PROPN
ejpam-4753	397	5	math	math	PROPN
ejpam-4753	397	6	,	,	PUNCT
ejpam-4753	397	7	16	16	NUM
ejpam-4753	397	8	(	(	PUNCT
ejpam-4753	397	9	3	3	NUM
ejpam-4753	397	10	)	)	PUNCT
ejpam-4753	397	11	(	(	PUNCT
ejpam-4753	397	12	2023	2023	NUM
ejpam-4753	397	13	)	)	PUNCT
ejpam-4753	397	14	,	,	PUNCT
ejpam-4753	397	15	1913	1913	NUM
ejpam-4753	397	16	-	-	SYM
ejpam-4753	397	17	1939	1939	NUM
ejpam-4753	397	18	1931	1931	NUM
ejpam-4753	397	19	with	with	ADP
ejpam-4753	397	20	b∗	b∗	ADJ
ejpam-4753	397	21	=	=	PUNCT
ejpam-4753	397	22	s−1a⊗	s−1a⊗	X
ejpam-4753	397	23	(	(	PUNCT
ejpam-4753	397	24	m∗	m∗	PROPN
ejpam-4753	397	25	)	)	PUNCT
ejpam-4753	397	26	and	and	CCONJ
ejpam-4753	397	27	d∗	d∗	PROPN
ejpam-4753	397	28	=	=	SYM
ejpam-4753	397	29	s−1a	s−1a	PROPN
ejpam-4753	397	30	⊗	⊗	PROPN
ejpam-4753	397	31	a(n∗	a(n∗	PROPN
ejpam-4753	397	32	)	)	PUNCT
ejpam-4753	397	33	.	.	PUNCT
ejpam-4753	398	1	proof	proof	NOUN
ejpam-4753	398	2	.	.	PUNCT
ejpam-4753	399	1	we	we	PRON
ejpam-4753	399	2	have	have	VERB
ejpam-4753	399	3	the	the	DET
ejpam-4753	399	4	functor	functor	PROPN
ejpam-4753	399	5	s−1	s−1	PROPN
ejpam-4753	399	6	(	(	PUNCT
ejpam-4753	399	7	)	)	PUNCT
ejpam-4753	399	8	and	and	CCONJ
ejpam-4753	400	1	the	the	DET
ejpam-4753	400	2	functor	functor	PROPN
ejpam-4753	400	3	s−1a	s−1a	PROPN
ejpam-4753	400	4	⊗	⊗	PROPN
ejpam-4753	400	5	a	a	X
ejpam-4753	400	6	(	(	PUNCT
ejpam-4753	400	7	)	)	PUNCT
ejpam-4753	400	8	are	be	AUX
ejpam-4753	400	9	isomorphs	isomorph	NOUN
ejpam-4753	400	10	.	.	PUNCT
ejpam-4753	401	1	on	on	ADP
ejpam-4753	401	2	the	the	DET
ejpam-4753	401	3	other	other	ADJ
ejpam-4753	401	4	hand	hand	NOUN
ejpam-4753	401	5	it	it	PRON
ejpam-4753	401	6	suffices	suffice	VERB
ejpam-4753	401	7	to	to	PART
ejpam-4753	401	8	prove	prove	VERB
ejpam-4753	401	9	that	that	SCONJ
ejpam-4753	401	10	the	the	DET
ejpam-4753	401	11	following	follow	VERB
ejpam-4753	401	12	diagram	diagram	NOUN
ejpam-4753	401	13	is	be	AUX
ejpam-4753	401	14	commutative	commutative	ADJ
ejpam-4753	401	15	b∗	b∗	ADV
ejpam-4753	401	16	:	:	PUNCT
ejpam-4753	401	17	·	·	PUNCT
ejpam-4753	401	18	·	·	PUNCT
ejpam-4753	401	19	·	·	PUNCT
ejpam-4753	402	1	//	//	NUM
ejpam-4753	402	2	γ	γ	PROPN
ejpam-4753	402	3	�	�	PROPN
ejpam-4753	402	4	�	�	PROPN
ejpam-4753	402	5	//	//	SYM
ejpam-4753	402	6	s−1a	s−1a	PROPN
ejpam-4753	402	7	⊗	⊗	PROPN
ejpam-4753	402	8	m(n+	m(n+	PROPN
ejpam-4753	402	9	1	1	NUM
ejpam-4753	402	10	)	)	PUNCT
ejpam-4753	402	11	γn+1	γn+1	NUM
ejpam-4753	402	12	�	�	PROPN
ejpam-4753	402	13	�	�	PROPN
ejpam-4753	402	14	s−1a	s−1a	PROPN
ejpam-4753	402	15	⊗	⊗	PROPN
ejpam-4753	402	16	dn+1//	dn+1//	PROPN
ejpam-4753	402	17	s−1a	s−1a	PROPN
ejpam-4753	402	18	⊗	⊗	PROPN
ejpam-4753	402	19	m(n	m(n	PROPN
ejpam-4753	402	20	)	)	PUNCT
ejpam-4753	402	21	γn	γn	ADP
ejpam-4753	402	22	�	�	PROPN
ejpam-4753	402	23	�	�	PROPN
ejpam-4753	402	24	s−1a	s−1a	PROPN
ejpam-4753	402	25	⊗	⊗	PROPN
ejpam-4753	402	26	dn	dn	PROPN
ejpam-4753	402	27	//	//	PROPN
ejpam-4753	402	28	.	.	PUNCT
ejpam-4753	402	29	.	.	PUNCT
ejpam-4753	402	30	.	.	PUNCT
ejpam-4753	403	1	s−1(m∗	s−1(m∗	NOUN
ejpam-4753	403	2	)	)	PUNCT
ejpam-4753	403	3	:	:	PUNCT
ejpam-4753	403	4	·	·	PUNCT
ejpam-4753	403	5	·	·	PUNCT
ejpam-4753	403	6	·	·	PUNCT
ejpam-4753	403	7	s−1fk	s−1fk	AUX
ejpam-4753	403	8	∗	∗	PROPN
ejpam-4753	403	9	�	�	PROPN
ejpam-4753	403	10	�	�	PROPN
ejpam-4753	403	11	//	//	NUM
ejpam-4753	403	12	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	403	13	1	1	NUM
ejpam-4753	403	14	)	)	PUNCT
ejpam-4753	403	15	s−1fk(n+1	s−1fk(n+1	PROPN
ejpam-4753	403	16	)	)	PUNCT
ejpam-4753	403	17	�	�	PROPN
ejpam-4753	403	18	�	�	PROPN
ejpam-4753	403	19	s−1dn+1	s−1dn+1	PROPN
ejpam-4753	403	20	//	//	SYM
ejpam-4753	403	21	s−1m(n	s−1m(n	NOUN
ejpam-4753	403	22	)	)	PUNCT
ejpam-4753	403	23	s−1fk(n	s−1fk(n	PROPN
ejpam-4753	403	24	)	)	PUNCT
ejpam-4753	403	25	�	�	PROPN
ejpam-4753	403	26	�	�	PROPN
ejpam-4753	403	27	s−1dn	s−1dn	PROPN
ejpam-4753	403	28	//	//	PROPN
ejpam-4753	403	29	.	.	PUNCT
ejpam-4753	403	30	.	.	PUNCT
ejpam-4753	403	31	.	.	PUNCT
ejpam-4753	404	1	(	(	PUNCT
ejpam-4753	404	2	s−1(n∗	s−1(n∗	PROPN
ejpam-4753	404	3	)	)	PUNCT
ejpam-4753	404	4	:	:	PUNCT
ejpam-4753	404	5	·	·	PUNCT
ejpam-4753	404	6	·	·	PUNCT
ejpam-4753	404	7	·	·	PUNCT
ejpam-4753	404	8	λ	λ	SYM
ejpam-4753	404	9	�	�	PROPN
ejpam-4753	404	10	�	�	PROPN
ejpam-4753	404	11	//	//	NUM
ejpam-4753	404	12	s−1n(n+	s−1n(n+	PROPN
ejpam-4753	404	13	1	1	NUM
ejpam-4753	404	14	)	)	PUNCT
ejpam-4753	404	15	λn+1	λn+1	ADP
ejpam-4753	404	16	�	�	PROPN
ejpam-4753	404	17	�	�	PROPN
ejpam-4753	404	18	s−1d′n+1+k	s−1d′n+1+k	VERB
ejpam-4753	404	19	//	//	X
ejpam-4753	404	20	s−1n(n	s−1n(n	NOUN
ejpam-4753	404	21	)	)	PUNCT
ejpam-4753	404	22	λn	λn	PROPN
ejpam-4753	404	23	�	�	PROPN
ejpam-4753	404	24	�	�	PROPN
ejpam-4753	404	25	s−1d′n+k	s−1d′n+k	PUNCT
ejpam-4753	405	1	//	//	PROPN
ejpam-4753	405	2	.	.	PUNCT
ejpam-4753	405	3	.	.	PUNCT
ejpam-4753	405	4	.	.	PUNCT
ejpam-4753	406	1	d∗	d∗	PROPN
ejpam-4753	406	2	:	:	PUNCT
ejpam-4753	406	3	·	·	PUNCT
ejpam-4753	406	4	·	·	PUNCT
ejpam-4753	406	5	·	·	PUNCT
ejpam-4753	407	1	//	//	PUNCT
ejpam-4753	408	1	s−1a	s−1a	PROPN
ejpam-4753	408	2	⊗	⊗	PROPN
ejpam-4753	408	3	n(n+	n(n+	NUM
ejpam-4753	408	4	1	1	NUM
ejpam-4753	408	5	)	)	PUNCT
ejpam-4753	408	6	s−1a	s−1a	VERB
ejpam-4753	408	7	⊗	⊗	PROPN
ejpam-4753	408	8	d′n+1+k//	d′n+1+k//	PROPN
ejpam-4753	408	9	s−1a	s−1a	VERB
ejpam-4753	408	10	⊗	⊗	PROPN
ejpam-4753	408	11	n(n	n(n	NUM
ejpam-4753	408	12	)	)	PUNCT
ejpam-4753	409	1	s−1a	s−1a	VERB
ejpam-4753	409	2	⊗	⊗	PROPN
ejpam-4753	409	3	d′n+k//	d′n+k//	PROPN
ejpam-4753	409	4	.	.	PUNCT
ejpam-4753	409	5	.	.	PUNCT
ejpam-4753	409	6	.	.	PUNCT
ejpam-4753	410	1	i.e.	i.e.	X
ejpam-4753	410	2	prove	prove	VERB
ejpam-4753	410	3	that	that	SCONJ
ejpam-4753	410	4	for	for	ADP
ejpam-4753	410	5	all	all	DET
ejpam-4753	410	6	n	n	PRON
ejpam-4753	410	7	∈	∈	NOUN
ejpam-4753	410	8	z	z	NOUN
ejpam-4753	410	9	we	we	PRON
ejpam-4753	410	10	have	have	VERB
ejpam-4753	410	11	λn	λn	NOUN
ejpam-4753	410	12	◦	◦	NOUN
ejpam-4753	410	13	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	410	14	)	)	PUNCT
ejpam-4753	411	1	◦	◦	NOUN
ejpam-4753	411	2	γn	γn	ADP
ejpam-4753	411	3	◦	◦	NOUN
ejpam-4753	411	4	s−1a	s−1a	PROPN
ejpam-4753	411	5	⊗	⊗	NUM
ejpam-4753	411	6	dn+1	dn+1	PROPN
ejpam-4753	412	1	=	=	PUNCT
ejpam-4753	412	2	s−1a	s−1a	PROPN
ejpam-4753	412	3	⊗	⊗	INTJ
ejpam-4753	412	4	d′n+1+k	d′n+1+k	PROPN
ejpam-4753	412	5	◦	◦	NOUN
ejpam-4753	412	6	λn+1	λn+1	ADP
ejpam-4753	412	7	◦	◦	VERB
ejpam-4753	412	8	s−1fk(n+	s−1fk(n+	NOUN
ejpam-4753	412	9	1	1	X
ejpam-4753	412	10	)	)	PUNCT
ejpam-4753	412	11	◦	◦	NOUN
ejpam-4753	412	12	γn+1	γn+1	NUM
ejpam-4753	412	13	or	or	CCONJ
ejpam-4753	412	14	for	for	ADP
ejpam-4753	412	15	all	all	DET
ejpam-4753	412	16	n	n	PRON
ejpam-4753	412	17	∈	∈	PROPN
ejpam-4753	412	18	z	z	NOUN
ejpam-4753	412	19	,	,	PUNCT
ejpam-4753	412	20	we	we	PRON
ejpam-4753	412	21	have	have	VERB
ejpam-4753	412	22	λn	λn	NOUN
ejpam-4753	412	23	◦	◦	NOUN
ejpam-4753	412	24	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	412	25	)	)	PUNCT
ejpam-4753	413	1	◦	◦	NOUN
ejpam-4753	413	2	γn	γn	NOUN
ejpam-4753	413	3	=	=	SYM
ejpam-4753	413	4	1s−1a	1s−1a	PROPN
ejpam-4753	413	5	⊗	⊗	NUM
ejpam-4753	413	6	fk(n	fk(n	NUM
ejpam-4753	413	7	)	)	PUNCT
ejpam-4753	413	8	.	.	PUNCT
ejpam-4753	414	1	let	let	VERB
ejpam-4753	414	2	1	1	NUM
ejpam-4753	414	3	s	s	PART
ejpam-4753	414	4	⊗m	⊗m	PROPN
ejpam-4753	414	5	∈	∈	PROPN
ejpam-4753	415	1	s−1a	s−1a	VERB
ejpam-4753	416	1	⊗	⊗	PROPN
ejpam-4753	416	2	a	a	PROPN
ejpam-4753	416	3	m(n+1	m(n+1	PROPN
ejpam-4753	416	4	)	)	PUNCT
ejpam-4753	416	5	,	,	PUNCT
ejpam-4753	416	6	then	then	ADV
ejpam-4753	416	7	it	it	PRON
ejpam-4753	416	8	is	be	AUX
ejpam-4753	416	9	exist	exist	VERB
ejpam-4753	416	10	an	an	DET
ejpam-4753	416	11	unique	unique	ADJ
ejpam-4753	416	12	couple	couple	NOUN
ejpam-4753	416	13	(	(	PUNCT
ejpam-4753	416	14	x	x	NOUN
ejpam-4753	416	15	,	,	PUNCT
ejpam-4753	416	16	y	y	NOUN
ejpam-4753	416	17	)	)	PUNCT
ejpam-4753	416	18	∈	∈	PROPN
ejpam-4753	416	19	mn+1×m(n+2	mn+1×m(n+2	NOUN
ejpam-4753	416	20	)	)	PUNCT
ejpam-4753	416	21	such	such	ADJ
ejpam-4753	416	22	that	that	SCONJ
ejpam-4753	416	23	m	m	VERB
ejpam-4753	416	24	=	=	ADJ
ejpam-4753	416	25	x+	x+	ADJ
ejpam-4753	417	1	y	y	PROPN
ejpam-4753	417	2	so	so	ADV
ejpam-4753	417	3	λn	λn	NOUN
ejpam-4753	417	4	◦	◦	NOUN
ejpam-4753	417	5	s−1fk(n)	s−1fk(n)	NOUN
ejpam-4753	417	6	◦	◦	NOUN
ejpam-4753	417	7	γn	γn	NOUN
ejpam-4753	417	8	◦	◦	NOUN
ejpam-4753	417	9	s−1a	s−1a	NOUN
ejpam-4753	417	10	⊗	⊗	PROPN
ejpam-4753	417	11	dn+1	dn+1	PROPN
ejpam-4753	417	12	[	[	PUNCT
ejpam-4753	417	13	1	1	NUM
ejpam-4753	417	14	s	s	NOUN
ejpam-4753	417	15	⊗m	⊗m	NOUN
ejpam-4753	417	16	]	]	X
ejpam-4753	418	1	=	=	PUNCT
ejpam-4753	418	2	λn	λn	NOUN
ejpam-4753	418	3	◦	◦	NOUN
ejpam-4753	418	4	s−1fk(n)	s−1fk(n)	NOUN
ejpam-4753	418	5	◦	◦	NOUN
ejpam-4753	418	6	γn	γn	NOUN
ejpam-4753	418	7	◦	◦	NOUN
ejpam-4753	418	8	s−1a	s−1a	NOUN
ejpam-4753	418	9	⊗	⊗	PROPN
ejpam-4753	418	10	dn+1	dn+1	PROPN
ejpam-4753	418	11	[	[	PUNCT
ejpam-4753	418	12	1	1	NUM
ejpam-4753	418	13	s	s	NOUN
ejpam-4753	418	14	⊗(x+y	⊗(x+y	NOUN
ejpam-4753	418	15	)	)	PUNCT
ejpam-4753	418	16	]	]	PUNCT
ejpam-4753	419	1	=	=	SYM
ejpam-4753	419	2	λn	λn	PROPN
ejpam-4753	419	3	◦	◦	NOUN
ejpam-4753	419	4	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	419	5	)	)	PUNCT
ejpam-4753	420	1	◦	◦	VERB
ejpam-4753	420	2	γn	γn	NUM
ejpam-4753	420	3	[	[	PUNCT
ejpam-4753	420	4	1	1	NUM
ejpam-4753	420	5	s	s	NOUN
ejpam-4753	420	6	⊗	⊗	NUM
ejpam-4753	420	7	x	x	X
ejpam-4753	420	8	]	]	X
ejpam-4753	420	9	=	=	SYM
ejpam-4753	420	10	1s−1a	1s−1a	PROPN
ejpam-4753	420	11	⊗	⊗	NUM
ejpam-4753	420	12	fk(n	fk(n	X
ejpam-4753	420	13	)	)	PUNCT
ejpam-4753	420	14	[	[	PUNCT
ejpam-4753	420	15	1	1	NUM
ejpam-4753	420	16	s	s	NOUN
ejpam-4753	420	17	⊗	⊗	NUM
ejpam-4753	420	18	x	x	X
ejpam-4753	420	19	]	]	X
ejpam-4753	420	20	=	=	SYM
ejpam-4753	420	21	1	1	NUM
ejpam-4753	420	22	s	s	X
ejpam-4753	420	23	⊗	⊗	PROPN
ejpam-4753	420	24	f(x	f(x	PROPN
ejpam-4753	420	25	)	)	PUNCT
ejpam-4753	420	26	.	.	PUNCT
ejpam-4753	421	1	on	on	ADP
ejpam-4753	421	2	the	the	DET
ejpam-4753	421	3	other	other	ADJ
ejpam-4753	421	4	hand	hand	NOUN
ejpam-4753	421	5	we	we	PRON
ejpam-4753	421	6	have	have	AUX
ejpam-4753	421	7	s−1a	s−1a	VERB
ejpam-4753	422	1	⊗	⊗	ADJ
ejpam-4753	422	2	d′n+1+k	d′n+1+k	PROPN
ejpam-4753	422	3	◦	◦	NOUN
ejpam-4753	422	4	λn+1	λn+1	ADP
ejpam-4753	422	5	◦	◦	VERB
ejpam-4753	422	6	s−1fk(n+	s−1fk(n+	NOUN
ejpam-4753	422	7	1	1	X
ejpam-4753	422	8	)	)	PUNCT
ejpam-4753	422	9	◦	◦	NOUN
ejpam-4753	422	10	γn+1	γn+1	NUM
ejpam-4753	422	11	[	[	PUNCT
ejpam-4753	422	12	1	1	NUM
ejpam-4753	422	13	s	s	NOUN
ejpam-4753	422	14	⊗m	⊗m	NOUN
ejpam-4753	422	15	]	]	X
ejpam-4753	423	1	=	=	PUNCT
ejpam-4753	423	2	s−1a	s−1a	PROPN
ejpam-4753	423	3	⊗	⊗	INTJ
ejpam-4753	423	4	d′n+1+k	d′n+1+k	PROPN
ejpam-4753	423	5	◦	◦	NOUN
ejpam-4753	423	6	λn+1	λn+1	ADP
ejpam-4753	423	7	◦	◦	NOUN
ejpam-4753	423	8	s−1fk(n	s−1fk(n	NOUN
ejpam-4753	423	9	)	)	PUNCT
ejpam-4753	423	10	◦	◦	NOUN
ejpam-4753	423	11	γn+1	γn+1	NUM
ejpam-4753	423	12	[	[	PUNCT
ejpam-4753	423	13	1	1	NUM
ejpam-4753	423	14	s	s	NOUN
ejpam-4753	423	15	⊗	⊗	NOUN
ejpam-4753	423	16	(	(	PUNCT
ejpam-4753	423	17	x+	x+	PROPN
ejpam-4753	423	18	y	y	NOUN
ejpam-4753	423	19	)	)	PUNCT
ejpam-4753	423	20	]	]	PUNCT
ejpam-4753	424	1	=	=	PUNCT
ejpam-4753	424	2	s−1a	s−1a	PROPN
ejpam-4753	424	3	⊗	⊗	PROPN
ejpam-4753	424	4	d′n+1+k	d′n+1+k	PROPN
ejpam-4753	425	1	[	[	PUNCT
ejpam-4753	425	2	1	1	NUM
ejpam-4753	425	3	s	s	NOUN
ejpam-4753	425	4	⊗	⊗	PROPN
ejpam-4753	425	5	f(x+	f(x+	PROPN
ejpam-4753	425	6	y	y	PROPN
ejpam-4753	425	7	)	)	PUNCT
ejpam-4753	425	8	]	]	PUNCT
ejpam-4753	426	1	=	=	PUNCT
ejpam-4753	426	2	1	1	NUM
ejpam-4753	426	3	s	s	X
ejpam-4753	426	4	⊗	⊗	PROPN
ejpam-4753	426	5	f(x	f(x	PROPN
ejpam-4753	426	6	)	)	PUNCT
ejpam-4753	426	7	thus	thus	ADV
ejpam-4753	426	8	s−1a⊗	s−1a⊗	X
ejpam-4753	426	9	fk	fk	INTJ
ejpam-4753	426	10	∗	∗	NOUN
ejpam-4753	426	11	is	be	AUX
ejpam-4753	426	12	a	a	DET
ejpam-4753	426	13	complex	complex	NOUN
ejpam-4753	426	14	the	the	DET
ejpam-4753	426	15	chain	chain	NOUN
ejpam-4753	426	16	.	.	PUNCT
ejpam-4753	427	1	corollary	corollary	ADJ
ejpam-4753	427	2	9	9	NUM
ejpam-4753	427	3	.	.	PUNCT
ejpam-4753	428	1	let	let	VERB
ejpam-4753	428	2	a	a	DET
ejpam-4753	428	3	=	=	SYM
ejpam-4753	428	4	⊕	⊕	PROPN
ejpam-4753	428	5	n∈z	n∈z	VERB
ejpam-4753	428	6	an	an	DET
ejpam-4753	428	7	be	be	AUX
ejpam-4753	428	8	a	a	DET
ejpam-4753	428	9	graded	grade	VERB
ejpam-4753	428	10	duo	duo	NOUN
ejpam-4753	428	11	-	-	PUNCT
ejpam-4753	428	12	ring	ring	NOUN
ejpam-4753	428	13	,	,	PUNCT
ejpam-4753	428	14	m	m	VERB
ejpam-4753	428	15	=	=	ADJ
ejpam-4753	428	16	⊕	⊕	PROPN
ejpam-4753	428	17	n∈z	n∈z	VERB
ejpam-4753	428	18	mn	mn	PROPN
ejpam-4753	428	19	and	and	CCONJ
ejpam-4753	428	20	n	n	PROPN
ejpam-4753	428	21	=	=	PROPN
ejpam-4753	428	22	⊕	⊕	PROPN
ejpam-4753	428	23	n∈z	n∈z	VERB
ejpam-4753	428	24	nn	nn	PROPN
ejpam-4753	428	25	two	two	NUM
ejpam-4753	428	26	graded	grade	VERB
ejpam-4753	428	27	left	leave	VERB
ejpam-4753	428	28	a−modules	a−module	NOUN
ejpam-4753	428	29	,	,	PUNCT
ejpam-4753	428	30	f	f	X
ejpam-4753	428	31	:	:	PUNCT
ejpam-4753	428	32	m	m	VERB
ejpam-4753	428	33	−→	−→	ADJ
ejpam-4753	428	34	n	n	NOUN
ejpam-4753	428	35	is	be	AUX
ejpam-4753	428	36	graded	grade	VERB
ejpam-4753	428	37	morphism	morphism	NOUN
ejpam-4753	428	38	of	of	ADP
ejpam-4753	428	39	degree	degree	NOUN
ejpam-4753	429	1	k	k	PROPN
ejpam-4753	429	2	∈	∈	PROPN
ejpam-4753	429	3	z	z	PROPN
ejpam-4753	429	4	and	and	CCONJ
ejpam-4753	429	5	sh	sh	PROPN
ejpam-4753	429	6	be	be	AUX
ejpam-4753	429	7	a	a	DET
ejpam-4753	429	8	part	part	NOUN
ejpam-4753	429	9	formed	form	VERB
ejpam-4753	429	10	of	of	ADP
ejpam-4753	429	11	regulars	regular	NOUN
ejpam-4753	429	12	homogeneous	homogeneous	ADJ
ejpam-4753	429	13	elements	element	NOUN
ejpam-4753	429	14	of	of	ADP
ejpam-4753	429	15	a	a	PRON
ejpam-4753	429	16	,	,	PUNCT
ejpam-4753	429	17	then	then	ADV
ejpam-4753	429	18	we	we	PRON
ejpam-4753	429	19	have	have	VERB
ejpam-4753	429	20	:	:	PUNCT
ejpam-4753	429	21	a.	a.	PROPN
ejpam-4753	429	22	o.	o.	PROPN
ejpam-4753	429	23	chbih	chbih	PROPN
ejpam-4753	429	24	,	,	PUNCT
ejpam-4753	429	25	m.	m.	PROPN
ejpam-4753	429	26	b.	b.	PROPN
ejpam-4753	429	27	maaouia	maaouia	PROPN
ejpam-4753	429	28	,	,	PUNCT
ejpam-4753	429	29	m.	m.	NOUN
ejpam-4753	429	30	sanghare	sanghare	PROPN
ejpam-4753	429	31	/	/	SYM
ejpam-4753	429	32	eur	eur	PROPN
ejpam-4753	429	33	.	.	PUNCT
ejpam-4753	430	1	j.	j.	PROPN
ejpam-4753	430	2	pure	pure	PROPN
ejpam-4753	430	3	appl	appl	PROPN
ejpam-4753	430	4	.	.	PROPN
ejpam-4753	430	5	math	math	PROPN
ejpam-4753	430	6	,	,	PUNCT
ejpam-4753	430	7	16	16	NUM
ejpam-4753	430	8	(	(	PUNCT
ejpam-4753	430	9	3	3	NUM
ejpam-4753	430	10	)	)	PUNCT
ejpam-4753	430	11	(	(	PUNCT
ejpam-4753	430	12	2023	2023	NUM
ejpam-4753	430	13	)	)	PUNCT
ejpam-4753	430	14	,	,	PUNCT
ejpam-4753	430	15	1913	1913	NUM
ejpam-4753	430	16	-	-	SYM
ejpam-4753	430	17	1939	1939	NUM
ejpam-4753	430	18	1932	1932	NUM
ejpam-4753	430	19	(	(	PUNCT
ejpam-4753	430	20	i	i	NOUN
ejpam-4753	430	21	)	)	PUNCT
ejpam-4753	430	22	the	the	DET
ejpam-4753	430	23	following	follow	VERB
ejpam-4753	430	24	complex	complex	ADJ
ejpam-4753	430	25	sequence	sequence	NOUN
ejpam-4753	430	26	:	:	PUNCT
ejpam-4753	430	27	b∗	b∗	ADJ
ejpam-4753	430	28	:	:	PUNCT
ejpam-4753	430	29	·	·	PUNCT
ejpam-4753	430	30	·	·	PUNCT
ejpam-4753	430	31	·	·	PUNCT
ejpam-4753	431	1	−→	−→	NOUN
ejpam-4753	431	2	s	s	PRON
ejpam-4753	431	3	−1	−1	NOUN
ejpam-4753	431	4	h	h	NOUN
ejpam-4753	431	5	a⊗	a⊗	NOUN
ejpam-4753	431	6	(	(	PUNCT
ejpam-4753	431	7	m(n+	m(n+	NOUN
ejpam-4753	431	8	1	1	NUM
ejpam-4753	431	9	)	)	PUNCT
ejpam-4753	431	10	)	)	PUNCT
ejpam-4753	432	1	s	s	VERB
ejpam-4753	432	2	−1	−1	NOUN
ejpam-4753	432	3	h	h	NOUN
ejpam-4753	432	4	a⊗(dn+1)−→	a⊗(dn+1)−→	PROPN
ejpam-4753	432	5	s	s	PART
ejpam-4753	432	6	−1	−1	NOUN
ejpam-4753	432	7	h	h	NOUN
ejpam-4753	432	8	a⊗	a⊗	NOUN
ejpam-4753	432	9	(	(	PUNCT
ejpam-4753	432	10	m(n	m(n	PROPN
ejpam-4753	432	11	)	)	PUNCT
ejpam-4753	432	12	)	)	PUNCT
ejpam-4753	433	1	s	s	VERB
ejpam-4753	433	2	−1	−1	NOUN
ejpam-4753	433	3	h	h	NOUN
ejpam-4753	433	4	a⊗(dn)−→	a⊗(dn)−→	PROPN
ejpam-4753	433	5	s	s	PART
ejpam-4753	433	6	−1	−1	NOUN
ejpam-4753	433	7	h	h	NOUN
ejpam-4753	433	8	a⊗	a⊗	NOUN
ejpam-4753	433	9	(	(	PUNCT
ejpam-4753	433	10	m(n−	m(n−	NOUN
ejpam-4753	433	11	1	1	NUM
ejpam-4753	433	12	)	)	PUNCT
ejpam-4753	433	13	)	)	PUNCT
ejpam-4753	434	1	−→	−→	NOUN
ejpam-4753	434	2	·	·	PUNCT
ejpam-4753	434	3	·	·	PUNCT
ejpam-4753	434	4	·	·	PUNCT
ejpam-4753	434	5	(	(	PUNCT
ejpam-4753	434	6	ii	ii	NOUN
ejpam-4753	434	7	)	)	PUNCT
ejpam-4753	434	8	the	the	DET
ejpam-4753	434	9	following	follow	VERB
ejpam-4753	434	10	complex	complex	ADJ
ejpam-4753	434	11	chain	chain	NOUN
ejpam-4753	434	12	:	:	PUNCT
ejpam-4753	434	13	b∗	b∗	ADJ
ejpam-4753	434	14	:	:	PUNCT
ejpam-4753	434	15	·	·	PUNCT
ejpam-4753	434	16	·	·	PUNCT
ejpam-4753	434	17	·	·	PUNCT
ejpam-4753	435	1	//	//	PUNCT
ejpam-4753	435	2	s	s	AUX
ejpam-4753	435	3	−1	−1	NOUN
ejpam-4753	435	4	h	h	NOUN
ejpam-4753	435	5	a⊗fk	a⊗fk	NOUN
ejpam-4753	435	6	∗	∗	NUM
ejpam-4753	435	7	�	�	PROPN
ejpam-4753	435	8	�	�	PROPN
ejpam-4753	435	9	//	//	PROPN
ejpam-4753	435	10	s	s	PART
ejpam-4753	435	11	−1	−1	NOUN
ejpam-4753	435	12	h	h	NOUN
ejpam-4753	435	13	a⊗	a⊗	NOUN
ejpam-4753	435	14	(	(	PUNCT
ejpam-4753	435	15	m(n+	m(n+	NOUN
ejpam-4753	435	16	1	1	NUM
ejpam-4753	435	17	)	)	PUNCT
ejpam-4753	435	18	)	)	PUNCT
ejpam-4753	436	1	s	s	VERB
ejpam-4753	436	2	−1	−1	NOUN
ejpam-4753	436	3	h	h	NOUN
ejpam-4753	436	4	a⊗(dn+1)//	a⊗(dn+1)//	ADP
ejpam-4753	436	5	s	s	PART
ejpam-4753	436	6	−1	−1	NOUN
ejpam-4753	436	7	h	h	NOUN
ejpam-4753	436	8	a⊗(fk(n+1	a⊗(fk(n+1	ADJ
ejpam-4753	436	9	)	)	PUNCT
ejpam-4753	436	10	)	)	PUNCT
ejpam-4753	437	1	�	�	PROPN
ejpam-4753	437	2	�	�	PROPN
ejpam-4753	437	3	s	s	PART
ejpam-4753	437	4	−1	−1	NOUN
ejpam-4753	437	5	h	h	NOUN
ejpam-4753	437	6	a⊗	a⊗	NOUN
ejpam-4753	437	7	(	(	PUNCT
ejpam-4753	437	8	m(n	m(n	PROPN
ejpam-4753	437	9	)	)	PUNCT
ejpam-4753	437	10	)	)	PUNCT
ejpam-4753	438	1	s	s	VERB
ejpam-4753	438	2	−1	−1	NOUN
ejpam-4753	438	3	h	h	NOUN
ejpam-4753	438	4	a⊗(dn)//	a⊗(dn)//	NOUN
ejpam-4753	438	5	s	s	PRON
ejpam-4753	438	6	−1	−1	NOUN
ejpam-4753	438	7	h	h	NOUN
ejpam-4753	438	8	a⊗(fk(n	a⊗(fk(n	NOUN
ejpam-4753	438	9	)	)	PUNCT
ejpam-4753	438	10	)	)	PUNCT
ejpam-4753	438	11	�	�	PROPN
ejpam-4753	438	12	�	�	PROPN
ejpam-4753	438	13	s	s	PART
ejpam-4753	438	14	−1	−1	NOUN
ejpam-4753	438	15	h	h	NOUN
ejpam-4753	438	16	a⊗	a⊗	NOUN
ejpam-4753	438	17	(	(	PUNCT
ejpam-4753	438	18	m(n−	m(n−	NOUN
ejpam-4753	438	19	1	1	NUM
ejpam-4753	438	20	)	)	PUNCT
ejpam-4753	438	21	)	)	PUNCT
ejpam-4753	439	1	//	//	PUNCT
ejpam-4753	439	2	s	s	PART
ejpam-4753	439	3	−1	−1	NOUN
ejpam-4753	439	4	h	h	NOUN
ejpam-4753	439	5	a⊗(fk(n−1	a⊗(fk(n−1	NOUN
ejpam-4753	439	6	)	)	PUNCT
ejpam-4753	439	7	)	)	PUNCT
ejpam-4753	439	8	�	�	PROPN
ejpam-4753	439	9	�	�	PROPN
ejpam-4753	439	10	·	·	PUNCT
ejpam-4753	439	11	·	·	PUNCT
ejpam-4753	439	12	·	·	PUNCT
ejpam-4753	440	1	d∗	d∗	INTJ
ejpam-4753	440	2	:	:	PUNCT
ejpam-4753	440	3	·	·	PUNCT
ejpam-4753	440	4	·	·	PUNCT
ejpam-4753	440	5	·	·	PUNCT
ejpam-4753	441	1	//	//	PUNCT
ejpam-4753	441	2	s	s	PART
ejpam-4753	441	3	−1	−1	NOUN
ejpam-4753	441	4	h	h	NOUN
ejpam-4753	441	5	a⊗	a⊗	NOUN
ejpam-4753	441	6	(	(	PUNCT
ejpam-4753	441	7	n(n+	n(n+	NOUN
ejpam-4753	441	8	1	1	NUM
ejpam-4753	441	9	)	)	PUNCT
ejpam-4753	441	10	)	)	PUNCT
ejpam-4753	442	1	s	s	VERB
ejpam-4753	442	2	−1	−1	NOUN
ejpam-4753	442	3	h	h	NOUN
ejpam-4753	442	4	a⊗(d′	a⊗(d′	NOUN
ejpam-4753	443	1	n+1+k)//	n+1+k)//	NOUN
ejpam-4753	443	2	s	s	PART
ejpam-4753	443	3	−1	−1	NOUN
ejpam-4753	443	4	h	h	NOUN
ejpam-4753	443	5	a⊗n(n	a⊗n(n	PROPN
ejpam-4753	443	6	)	)	PUNCT
ejpam-4753	443	7	)	)	PUNCT
ejpam-4753	444	1	s	s	VERB
ejpam-4753	444	2	−1	−1	NOUN
ejpam-4753	444	3	h	h	NOUN
ejpam-4753	444	4	a⊗(d′	a⊗(d′	PROPN
ejpam-4753	444	5	n+k)//	n+k)//	PROPN
ejpam-4753	444	6	s	s	PART
ejpam-4753	444	7	−1	−1	NOUN
ejpam-4753	444	8	h	h	NOUN
ejpam-4753	444	9	a⊗	a⊗	NOUN
ejpam-4753	444	10	(	(	PUNCT
ejpam-4753	444	11	n(n−	n(n−	NOUN
ejpam-4753	444	12	1	1	NUM
ejpam-4753	444	13	)	)	PUNCT
ejpam-4753	444	14	)	)	PUNCT
ejpam-4753	444	15	//	//	X
ejpam-4753	444	16	·	·	PUNCT
ejpam-4753	444	17	·	·	PUNCT
ejpam-4753	444	18	·	·	PUNCT
ejpam-4753	445	1	with	with	ADP
ejpam-4753	445	2	b∗	b∗	ADJ
ejpam-4753	445	3	=	=	SYM
ejpam-4753	445	4	s	s	PART
ejpam-4753	445	5	−1	−1	NOUN
ejpam-4753	445	6	h	h	NOUN
ejpam-4753	445	7	a⊗	a⊗	NOUN
ejpam-4753	445	8	(	(	PUNCT
ejpam-4753	445	9	m∗	m∗	PROPN
ejpam-4753	445	10	)	)	PUNCT
ejpam-4753	445	11	and	and	CCONJ
ejpam-4753	445	12	d∗	d∗	PROPN
ejpam-4753	445	13	=	=	SYM
ejpam-4753	445	14	s	s	VERB
ejpam-4753	445	15	−1	−1	NOUN
ejpam-4753	445	16	h	h	NOUN
ejpam-4753	445	17	a	a	DET
ejpam-4753	445	18	⊗	⊗	PROPN
ejpam-4753	445	19	a(n∗	a(n∗	PROPN
ejpam-4753	445	20	)	)	PUNCT
ejpam-4753	445	21	.	.	PUNCT
ejpam-4753	446	1	proof	proof	NOUN
ejpam-4753	446	2	.	.	PUNCT
ejpam-4753	447	1	it	it	PRON
ejpam-4753	447	2	is	be	AUX
ejpam-4753	447	3	sufficient	sufficient	ADJ
ejpam-4753	447	4	to	to	PART
ejpam-4753	447	5	note	note	VERB
ejpam-4753	447	6	that	that	SCONJ
ejpam-4753	447	7	sh	sh	PROPN
ejpam-4753	447	8	is	be	AUX
ejpam-4753	447	9	a	a	DET
ejpam-4753	447	10	multiplicatively	multiplicatively	ADV
ejpam-4753	447	11	closed	close	VERB
ejpam-4753	447	12	subset	subset	NOUN
ejpam-4753	447	13	satisfying	satisfy	VERB
ejpam-4753	447	14	the	the	DET
ejpam-4753	447	15	left	left	ADJ
ejpam-4753	447	16	conditions	condition	NOUN
ejpam-4753	447	17	of	of	ADP
ejpam-4753	447	18	ore	ore	NOUN
ejpam-4753	447	19	formed	form	VERB
ejpam-4753	447	20	of	of	ADP
ejpam-4753	447	21	homogeneous	homogeneous	ADJ
ejpam-4753	447	22	elements	element	NOUN
ejpam-4753	447	23	of	of	ADP
ejpam-4753	447	24	a.	a.	NOUN
ejpam-4753	447	25	corollary	corollary	PROPN
ejpam-4753	447	26	10	10	NUM
ejpam-4753	447	27	.	.	PUNCT
ejpam-4753	448	1	let	let	VERB
ejpam-4753	448	2	a	a	DET
ejpam-4753	448	3	=	=	SYM
ejpam-4753	448	4	⊕	⊕	PROPN
ejpam-4753	448	5	n∈z	n∈z	VERB
ejpam-4753	448	6	an	an	DET
ejpam-4753	448	7	be	be	AUX
ejpam-4753	448	8	a	a	DET
ejpam-4753	448	9	graded	grade	VERB
ejpam-4753	448	10	duo	duo	NOUN
ejpam-4753	448	11	-	-	PUNCT
ejpam-4753	448	12	ring	ring	NOUN
ejpam-4753	448	13	,	,	PUNCT
ejpam-4753	448	14	m	m	VERB
ejpam-4753	448	15	=	=	ADJ
ejpam-4753	448	16	⊕	⊕	PROPN
ejpam-4753	448	17	n∈z	n∈z	VERB
ejpam-4753	448	18	mn	mn	PROPN
ejpam-4753	448	19	and	and	CCONJ
ejpam-4753	448	20	n	n	PROPN
ejpam-4753	448	21	=	=	PROPN
ejpam-4753	448	22	⊕	⊕	PROPN
ejpam-4753	448	23	n∈z	n∈z	VERB
ejpam-4753	448	24	nn	nn	PROPN
ejpam-4753	448	25	two	two	NUM
ejpam-4753	448	26	graded	grade	VERB
ejpam-4753	448	27	left	leave	VERB
ejpam-4753	448	28	a−modules	a−module	NOUN
ejpam-4753	448	29	,	,	PUNCT
ejpam-4753	448	30	f	f	X
ejpam-4753	448	31	:	:	PUNCT
ejpam-4753	448	32	m	m	VERB
ejpam-4753	448	33	−→	−→	ADJ
ejpam-4753	448	34	n	n	NOUN
ejpam-4753	448	35	is	be	AUX
ejpam-4753	448	36	graded	grade	VERB
ejpam-4753	448	37	morphism	morphism	NOUN
ejpam-4753	448	38	of	of	ADP
ejpam-4753	448	39	degree	degree	NOUN
ejpam-4753	449	1	k	k	PROPN
ejpam-4753	450	1	and	and	CCONJ
ejpam-4753	450	2	sh	sh	PROPN
ejpam-4753	450	3	be	be	AUX
ejpam-4753	450	4	the	the	DET
ejpam-4753	450	5	set	set	NOUN
ejpam-4753	450	6	of	of	ADP
ejpam-4753	450	7	all	all	DET
ejpam-4753	450	8	regulars	regular	NOUN
ejpam-4753	450	9	homogeneous	homogeneous	ADJ
ejpam-4753	450	10	elements	element	NOUN
ejpam-4753	450	11	of	of	ADP
ejpam-4753	450	12	a	a	PRON
ejpam-4753	450	13	,	,	PUNCT
ejpam-4753	450	14	then	then	ADV
ejpam-4753	450	15	we	we	PRON
ejpam-4753	450	16	have	have	VERB
ejpam-4753	450	17	:	:	PUNCT
ejpam-4753	450	18	(	(	PUNCT
ejpam-4753	450	19	i	i	NOUN
ejpam-4753	450	20	)	)	PUNCT
ejpam-4753	450	21	the	the	DET
ejpam-4753	450	22	following	follow	VERB
ejpam-4753	450	23	complex	complex	ADJ
ejpam-4753	450	24	sequence	sequence	NOUN
ejpam-4753	450	25	:	:	PUNCT
ejpam-4753	450	26	b∗	b∗	ADJ
ejpam-4753	450	27	:	:	PUNCT
ejpam-4753	450	28	·	·	PUNCT
ejpam-4753	450	29	·	·	PUNCT
ejpam-4753	450	30	·	·	PUNCT
ejpam-4753	451	1	−→	−→	ADJ
ejpam-4753	451	2	s−1a⊗(m(n+1	s−1a⊗(m(n+1	NOUN
ejpam-4753	451	3	)	)	PUNCT
ejpam-4753	451	4	)	)	PUNCT
ejpam-4753	451	5	s−1a⊗(dn+1)−→	s−1a⊗(dn+1)−→	NUM
ejpam-4753	451	6	s−1a⊗(m(n	s−1a⊗(m(n	NOUN
ejpam-4753	451	7	)	)	PUNCT
ejpam-4753	451	8	)	)	PUNCT
ejpam-4753	451	9	s−1a⊗(dn)−→	s−1a⊗(dn)−→	PUNCT
ejpam-4753	451	10	s−1a⊗(m(n−1	s−1a⊗(m(n−1	NOUN
ejpam-4753	451	11	)	)	PUNCT
ejpam-4753	451	12	)	)	PUNCT
ejpam-4753	452	1	−→	−→	NOUN
ejpam-4753	452	2	·	·	PUNCT
ejpam-4753	452	3	·	·	PUNCT
ejpam-4753	452	4	·	·	PUNCT
ejpam-4753	452	5	(	(	PUNCT
ejpam-4753	452	6	ii	ii	NOUN
ejpam-4753	452	7	)	)	PUNCT
ejpam-4753	452	8	the	the	DET
ejpam-4753	452	9	following	follow	VERB
ejpam-4753	452	10	complex	complex	ADJ
ejpam-4753	452	11	chain	chain	NOUN
ejpam-4753	452	12	:	:	PUNCT
ejpam-4753	452	13	b∗	b∗	ADJ
ejpam-4753	452	14	:	:	PUNCT
ejpam-4753	452	15	·	·	PUNCT
ejpam-4753	452	16	·	·	PUNCT
ejpam-4753	452	17	·	·	PUNCT
ejpam-4753	453	1	//	//	PUNCT
ejpam-4753	453	2	s−1	s−1	PROPN
ejpam-4753	453	3	h	h	NOUN
ejpam-4753	453	4	a⊗fk	a⊗fk	VERB
ejpam-4753	453	5	∗	∗	NUM
ejpam-4753	453	6	�	�	PROPN
ejpam-4753	453	7	�	�	PROPN
ejpam-4753	453	8	//	//	NUM
ejpam-4753	453	9	s−1	s−1	PROPN
ejpam-4753	453	10	h	h	NOUN
ejpam-4753	453	11	a⊗	a⊗	NOUN
ejpam-4753	453	12	(	(	PUNCT
ejpam-4753	453	13	m(n+	m(n+	NOUN
ejpam-4753	453	14	1	1	NUM
ejpam-4753	453	15	)	)	PUNCT
ejpam-4753	453	16	)	)	PUNCT
ejpam-4753	454	1	s−1	s−1	PROPN
ejpam-4753	454	2	h	h	NOUN
ejpam-4753	454	3	a⊗(dn+1)//	a⊗(dn+1)//	ADV
ejpam-4753	454	4	s−1	s−1	PROPN
ejpam-4753	454	5	h	h	NOUN
ejpam-4753	454	6	a⊗(fk(n+1	a⊗(fk(n+1	ADJ
ejpam-4753	454	7	)	)	PUNCT
ejpam-4753	454	8	)	)	PUNCT
ejpam-4753	454	9	�	�	PROPN
ejpam-4753	454	10	�	�	PROPN
ejpam-4753	454	11	s−1	s−1	PROPN
ejpam-4753	454	12	h	h	NOUN
ejpam-4753	454	13	a⊗	a⊗	NOUN
ejpam-4753	454	14	(	(	PUNCT
ejpam-4753	454	15	m(n	m(n	PROPN
ejpam-4753	454	16	)	)	PUNCT
ejpam-4753	454	17	)	)	PUNCT
ejpam-4753	455	1	s−1	s−1	PROPN
ejpam-4753	455	2	h	h	NOUN
ejpam-4753	455	3	a⊗(dn)//	a⊗(dn)//	NOUN
ejpam-4753	455	4	s−1	s−1	PROPN
ejpam-4753	455	5	h	h	NOUN
ejpam-4753	455	6	a⊗(fk(n	a⊗(fk(n	NOUN
ejpam-4753	455	7	)	)	PUNCT
ejpam-4753	455	8	)	)	PUNCT
ejpam-4753	455	9	�	�	PROPN
ejpam-4753	455	10	�	�	PROPN
ejpam-4753	455	11	s−1	s−1	PROPN
ejpam-4753	455	12	h	h	NOUN
ejpam-4753	455	13	a⊗	a⊗	NOUN
ejpam-4753	455	14	(	(	PUNCT
ejpam-4753	455	15	m(n−	m(n−	NOUN
ejpam-4753	455	16	1	1	NUM
ejpam-4753	455	17	)	)	PUNCT
ejpam-4753	455	18	)	)	PUNCT
ejpam-4753	456	1	//	//	PUNCT
ejpam-4753	457	1	s−1	s−1	PROPN
ejpam-4753	457	2	h	h	NOUN
ejpam-4753	457	3	a⊗(fk(n−1	a⊗(fk(n−1	NOUN
ejpam-4753	457	4	)	)	PUNCT
ejpam-4753	457	5	)	)	PUNCT
ejpam-4753	457	6	�	�	PROPN
ejpam-4753	457	7	�	�	PROPN
ejpam-4753	457	8	·	·	PUNCT
ejpam-4753	457	9	·	·	PUNCT
ejpam-4753	457	10	·	·	PUNCT
ejpam-4753	457	11	d∗	d∗	INTJ
ejpam-4753	457	12	:	:	PUNCT
ejpam-4753	457	13	·	·	PUNCT
ejpam-4753	457	14	·	·	PUNCT
ejpam-4753	457	15	·	·	PUNCT
ejpam-4753	457	16	//	//	PUNCT
ejpam-4753	458	1	s−1	s−1	ADJ
ejpam-4753	458	2	h	h	NOUN
ejpam-4753	458	3	a⊗	a⊗	NOUN
ejpam-4753	458	4	(	(	PUNCT
ejpam-4753	458	5	n(n+	n(n+	NOUN
ejpam-4753	458	6	1	1	NUM
ejpam-4753	458	7	)	)	PUNCT
ejpam-4753	458	8	)	)	PUNCT
ejpam-4753	459	1	s−1	s−1	PROPN
ejpam-4753	459	2	h	h	NOUN
ejpam-4753	459	3	a⊗(d′n+1+k)//	a⊗(d′n+1+k)//	NOUN
ejpam-4753	459	4	s−1	s−1	PROPN
ejpam-4753	459	5	h	h	NOUN
ejpam-4753	459	6	a⊗n(n	a⊗n(n	NOUN
ejpam-4753	459	7	)	)	PUNCT
ejpam-4753	459	8	)	)	PUNCT
ejpam-4753	460	1	s−1	s−1	NOUN
ejpam-4753	460	2	h	h	NOUN
ejpam-4753	460	3	a⊗(d′n+k)//	a⊗(d′n+k)//	VERB
ejpam-4753	460	4	s−1	s−1	PROPN
ejpam-4753	460	5	h	h	NOUN
ejpam-4753	460	6	a⊗	a⊗	NOUN
ejpam-4753	460	7	(	(	PUNCT
ejpam-4753	460	8	n(n−	n(n−	NOUN
ejpam-4753	460	9	1	1	NUM
ejpam-4753	460	10	)	)	PUNCT
ejpam-4753	460	11	)	)	PUNCT
ejpam-4753	461	1	//	//	X
ejpam-4753	461	2	·	·	PUNCT
ejpam-4753	461	3	·	·	PUNCT
ejpam-4753	461	4	·	·	PUNCT
ejpam-4753	461	5	with	with	ADP
ejpam-4753	461	6	b∗	b∗	ADJ
ejpam-4753	461	7	=	=	SYM
ejpam-4753	461	8	s−1	s−1	ADJ
ejpam-4753	461	9	h	h	NOUN
ejpam-4753	461	10	a⊗	a⊗	NOUN
ejpam-4753	461	11	(	(	PUNCT
ejpam-4753	461	12	m∗	m∗	PROPN
ejpam-4753	461	13	)	)	PUNCT
ejpam-4753	461	14	and	and	CCONJ
ejpam-4753	461	15	d∗	d∗	NOUN
ejpam-4753	462	1	=	=	SYM
ejpam-4753	462	2	s−1	s−1	PROPN
ejpam-4753	462	3	h	h	NOUN
ejpam-4753	462	4	a	a	DET
ejpam-4753	462	5	⊗	⊗	PROPN
ejpam-4753	462	6	a(n∗	a(n∗	PROPN
ejpam-4753	462	7	)	)	PUNCT
ejpam-4753	462	8	.	.	PUNCT
ejpam-4753	463	1	proof	proof	NOUN
ejpam-4753	463	2	.	.	PUNCT
ejpam-4753	464	1	it	it	PRON
ejpam-4753	464	2	is	be	AUX
ejpam-4753	464	3	sufficient	sufficient	ADJ
ejpam-4753	464	4	to	to	PART
ejpam-4753	464	5	note	note	VERB
ejpam-4753	464	6	that	that	SCONJ
ejpam-4753	464	7	sh	sh	INTJ
ejpam-4753	464	8	=	=	SYM
ejpam-4753	464	9	sh	sh	INTJ
ejpam-4753	464	10	.	.	PUNCT
ejpam-4753	465	1	theorem	theorem	ADJ
ejpam-4753	465	2	7	7	NUM
ejpam-4753	465	3	.	.	PUNCT
ejpam-4753	466	1	let	let	VERB
ejpam-4753	466	2	a	a	DET
ejpam-4753	466	3	=	=	SYM
ejpam-4753	466	4	⊕	⊕	PROPN
ejpam-4753	466	5	n∈z	n∈z	VERB
ejpam-4753	466	6	an	an	DET
ejpam-4753	466	7	be	be	AUX
ejpam-4753	466	8	a	a	DET
ejpam-4753	466	9	graded	grade	VERB
ejpam-4753	466	10	ring	ring	NOUN
ejpam-4753	466	11	and	and	CCONJ
ejpam-4753	466	12	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	466	13	)	)	PUNCT
ejpam-4753	466	14	the	the	DET
ejpam-4753	466	15	category	category	NOUN
ejpam-4753	466	16	of	of	ADP
ejpam-4753	466	17	graded	grade	VERB
ejpam-4753	466	18	left	leave	VERB
ejpam-4753	466	19	a−modules	a−module	NOUN
ejpam-4753	466	20	,	,	PUNCT
ejpam-4753	466	21	then	then	ADV
ejpam-4753	466	22	the	the	DET
ejpam-4753	466	23	relation	relation	NOUN
ejpam-4753	466	24	c(−	c(−	NOUN
ejpam-4753	466	25	)	)	PUNCT
ejpam-4753	466	26	:	:	PUNCT
ejpam-4753	466	27	gr(a−mod	gr(a−mod	X
ejpam-4753	466	28	)	)	PUNCT
ejpam-4753	466	29	−→	−→	NOUN
ejpam-4753	466	30	comp	comp	NOUN
ejpam-4753	466	31	(	(	PUNCT
ejpam-4753	466	32	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	466	33	)	)	PUNCT
ejpam-4753	466	34	)	)	PUNCT
ejpam-4753	466	35	which	which	PRON
ejpam-4753	466	36	that	that	SCONJ
ejpam-4753	466	37	for	for	ADP
ejpam-4753	466	38	all	all	PRON
ejpam-4753	466	39	graded	grade	VERB
ejpam-4753	466	40	left	leave	VERB
ejpam-4753	466	41	a−module	a−module	ADP
ejpam-4753	466	42	m	m	PROPN
ejpam-4753	466	43	=	=	PROPN
ejpam-4753	466	44	⊕	⊕	PROPN
ejpam-4753	466	45	n∈z	n∈z	PROPN
ejpam-4753	466	46	mn	mn	PROPN
ejpam-4753	466	47	of	of	ADP
ejpam-4753	466	48	gr(a	gr(a	NOUN
ejpam-4753	466	49	−	−	PROPN
ejpam-4753	466	50	mod	mod	PROPN
ejpam-4753	466	51	)	)	PUNCT
ejpam-4753	466	52	we	we	PRON
ejpam-4753	466	53	correspond	correspond	VERB
ejpam-4753	466	54	the	the	DET
ejpam-4753	466	55	associate	associate	NOUN
ejpam-4753	466	56	a.	a.	NOUN
ejpam-4753	466	57	o.	o.	PROPN
ejpam-4753	466	58	chbih	chbih	PROPN
ejpam-4753	466	59	,	,	PUNCT
ejpam-4753	466	60	m.	m.	PROPN
ejpam-4753	466	61	b.	b.	PROPN
ejpam-4753	466	62	maaouia	maaouia	PROPN
ejpam-4753	466	63	,	,	PUNCT
ejpam-4753	466	64	m.	m.	NOUN
ejpam-4753	466	65	sanghare	sanghare	PROPN
ejpam-4753	466	66	/	/	SYM
ejpam-4753	466	67	eur	eur	PROPN
ejpam-4753	466	68	.	.	PUNCT
ejpam-4753	467	1	j.	j.	PROPN
ejpam-4753	467	2	pure	pure	PROPN
ejpam-4753	467	3	appl	appl	PROPN
ejpam-4753	467	4	.	.	PROPN
ejpam-4753	467	5	math	math	PROPN
ejpam-4753	467	6	,	,	PUNCT
ejpam-4753	467	7	16	16	NUM
ejpam-4753	467	8	(	(	PUNCT
ejpam-4753	467	9	3	3	NUM
ejpam-4753	467	10	)	)	PUNCT
ejpam-4753	467	11	(	(	PUNCT
ejpam-4753	467	12	2023	2023	NUM
ejpam-4753	467	13	)	)	PUNCT
ejpam-4753	467	14	,	,	PUNCT
ejpam-4753	467	15	1913	1913	NUM
ejpam-4753	467	16	-	-	SYM
ejpam-4753	467	17	1939	1939	NUM
ejpam-4753	467	18	1933	1933	NUM
ejpam-4753	467	19	complex	complex	ADJ
ejpam-4753	467	20	sequence	sequence	NOUN
ejpam-4753	467	21	m∗	m∗	VERB
ejpam-4753	467	22	to	to	ADP
ejpam-4753	467	23	a	a	DET
ejpam-4753	467	24	graded	grade	VERB
ejpam-4753	467	25	a−module	a−module	ADP
ejpam-4753	467	26	m	m	PROPN
ejpam-4753	467	27	=	=	PROPN
ejpam-4753	467	28	⊕	⊕	PROPN
ejpam-4753	467	29	n∈z	n∈z	VERB
ejpam-4753	467	30	mn	mn	PROPN
ejpam-4753	467	31	and	and	CCONJ
ejpam-4753	467	32	for	for	ADP
ejpam-4753	467	33	all	all	DET
ejpam-4753	467	34	graded	grade	VERB
ejpam-4753	467	35	morphism	morphism	NOUN
ejpam-4753	467	36	of	of	ADP
ejpam-4753	467	37	graded	grade	VERB
ejpam-4753	467	38	left	leave	VERB
ejpam-4753	467	39	a−modules	a−module	NOUN
ejpam-4753	467	40	f	f	X
ejpam-4753	467	41	:	:	PUNCT
ejpam-4753	467	42	m	m	VERB
ejpam-4753	467	43	=	=	SYM
ejpam-4753	467	44	⊕	⊕	PROPN
ejpam-4753	467	45	n∈z	n∈z	VERB
ejpam-4753	467	46	mn	mn	PROPN
ejpam-4753	468	1	−→	−→	NOUN
ejpam-4753	469	1	n	n	PROPN
ejpam-4753	469	2	=	=	SYM
ejpam-4753	469	3	⊕	⊕	PROPN
ejpam-4753	469	4	n∈z	n∈z	NOUN
ejpam-4753	469	5	nn	nn	PROPN
ejpam-4753	469	6	of	of	ADP
ejpam-4753	469	7	degree	degree	NOUN
ejpam-4753	470	1	k	k	NOUN
ejpam-4753	470	2	we	we	PRON
ejpam-4753	470	3	correspond	correspond	VERB
ejpam-4753	470	4	the	the	DET
ejpam-4753	470	5	associate	associate	ADJ
ejpam-4753	470	6	complex	complex	NOUN
ejpam-4753	470	7	chain	chain	NOUN
ejpam-4753	470	8	fk	fk	INTJ
ejpam-4753	470	9	∗	∗	NOUN
ejpam-4753	470	10	to	to	ADP
ejpam-4753	470	11	a	a	DET
ejpam-4753	470	12	morphism	morphism	NOUN
ejpam-4753	470	13	of	of	ADP
ejpam-4753	470	14	graded	grade	VERB
ejpam-4753	470	15	left	leave	VERB
ejpam-4753	470	16	a−module	a−module	ADP
ejpam-4753	470	17	f	f	X
ejpam-4753	470	18	:	:	PUNCT
ejpam-4753	470	19	m	m	VERB
ejpam-4753	470	20	=	=	SYM
ejpam-4753	470	21	⊕	⊕	PROPN
ejpam-4753	470	22	n∈z	n∈z	VERB
ejpam-4753	470	23	mn	mn	PROPN
ejpam-4753	471	1	−→	−→	NOUN
ejpam-4753	472	1	n	n	PROPN
ejpam-4753	472	2	=	=	SYM
ejpam-4753	472	3	⊕	⊕	PROPN
ejpam-4753	472	4	n∈z	n∈z	VERB
ejpam-4753	472	5	nn	nn	PROPN
ejpam-4753	472	6	is	be	AUX
ejpam-4753	472	7	exact	exact	ADJ
ejpam-4753	473	1	additively	additively	ADV
ejpam-4753	473	2	covariant	covariant	ADJ
ejpam-4753	473	3	functor	functor	PROPN
ejpam-4753	473	4	.	.	PUNCT
ejpam-4753	473	5	proof	proof	NOUN
ejpam-4753	473	6	.	.	PUNCT
ejpam-4753	474	1	let	let	VERB
ejpam-4753	474	2	m	m	PRON
ejpam-4753	474	3	,	,	PUNCT
ejpam-4753	474	4	n	n	PRON
ejpam-4753	474	5	two	two	NUM
ejpam-4753	474	6	graded	grade	VERB
ejpam-4753	474	7	left	leave	VERB
ejpam-4753	474	8	a−modules	a−module	NOUN
ejpam-4753	474	9	and	and	CCONJ
ejpam-4753	474	10	f	f	X
ejpam-4753	474	11	:	:	PUNCT
ejpam-4753	474	12	m	m	VERB
ejpam-4753	474	13	−→	−→	ADJ
ejpam-4753	474	14	n	n	ADV
ejpam-4753	474	15	graded	grade	VERB
ejpam-4753	474	16	morphism	morphism	NOUN
ejpam-4753	474	17	of	of	ADP
ejpam-4753	474	18	graded	grade	VERB
ejpam-4753	474	19	a−modules	a−module	NOUN
ejpam-4753	474	20	,	,	PUNCT
ejpam-4753	474	21	we	we	PRON
ejpam-4753	474	22	note	note	VERB
ejpam-4753	474	23	that	that	SCONJ
ejpam-4753	474	24	c(m	c(m	NOUN
ejpam-4753	474	25	)	)	PUNCT
ejpam-4753	475	1	=	=	PRON
ejpam-4753	475	2	m∗	m∗	NOUN
ejpam-4753	475	3	(	(	PUNCT
ejpam-4753	475	4	respectively	respectively	ADV
ejpam-4753	475	5	c(n	c(n	ADJ
ejpam-4753	475	6	)	)	PUNCT
ejpam-4753	475	7	=	=	PUNCT
ejpam-4753	475	8	n∗	n∗	PROPN
ejpam-4753	475	9	)	)	PUNCT
ejpam-4753	475	10	the	the	DET
ejpam-4753	475	11	associate	associate	ADJ
ejpam-4753	475	12	complex	complex	ADJ
ejpam-4753	475	13	sequence	sequence	NOUN
ejpam-4753	475	14	m∗	m∗	NOUN
ejpam-4753	475	15	(	(	PUNCT
ejpam-4753	475	16	respectively	respectively	ADV
ejpam-4753	475	17	n∗	n∗	PROPN
ejpam-4753	475	18	)	)	PUNCT
ejpam-4753	475	19	to	to	ADP
ejpam-4753	475	20	a	a	DET
ejpam-4753	475	21	graded	grade	VERB
ejpam-4753	475	22	a−module	a−module	ADP
ejpam-4753	475	23	m	m	PROPN
ejpam-4753	475	24	=	=	PROPN
ejpam-4753	475	25	⊕	⊕	PROPN
ejpam-4753	475	26	n∈z	n∈z	ADJ
ejpam-4753	475	27	mn	mn	PROPN
ejpam-4753	475	28	(	(	PUNCT
ejpam-4753	475	29	respectively	respectively	ADV
ejpam-4753	475	30	to	to	ADP
ejpam-4753	475	31	a	a	DET
ejpam-4753	475	32	graded	grade	VERB
ejpam-4753	475	33	a−module	a−module	ADP
ejpam-4753	475	34	n	n	NOUN
ejpam-4753	475	35	=	=	SYM
ejpam-4753	475	36	⊕	⊕	PROPN
ejpam-4753	475	37	n∈z	n∈z	VERB
ejpam-4753	475	38	nn	nn	NOUN
ejpam-4753	475	39	)	)	PUNCT
ejpam-4753	475	40	so	so	ADV
ejpam-4753	475	41	m∗	m∗	NOUN
ejpam-4753	475	42	,	,	PUNCT
ejpam-4753	475	43	n∗	n∗	PROPN
ejpam-4753	475	44	∈	∈	PROPN
ejpam-4753	475	45	comp	comp	NOUN
ejpam-4753	475	46	(	(	PUNCT
ejpam-4753	475	47	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	475	48	)	)	PUNCT
ejpam-4753	475	49	)	)	PUNCT
ejpam-4753	475	50	.	.	PUNCT
ejpam-4753	476	1	so	so	ADV
ejpam-4753	476	2	c(f	c(f	PROPN
ejpam-4753	476	3	)	)	PUNCT
ejpam-4753	476	4	:	:	PUNCT
ejpam-4753	476	5	m∗	m∗	VERB
ejpam-4753	476	6	−→	−→	ADJ
ejpam-4753	476	7	n∗	n∗	NOUN
ejpam-4753	476	8	has	have	VERB
ejpam-4753	476	9	a	a	DET
ejpam-4753	476	10	sense	sense	NOUN
ejpam-4753	476	11	.	.	PUNCT
ejpam-4753	477	1	(	(	PUNCT
ejpam-4753	477	2	i	i	NOUN
ejpam-4753	477	3	)	)	PUNCT
ejpam-4753	477	4	let	let	VERB
ejpam-4753	477	5	m	m	PRON
ejpam-4753	477	6	∈	∈	NOUN
ejpam-4753	477	7	gr(a	gr(a	PUNCT
ejpam-4753	477	8	−	−	PROPN
ejpam-4753	477	9	mod	mod	PROPN
ejpam-4753	477	10	)	)	PUNCT
ejpam-4753	477	11	then	then	ADV
ejpam-4753	477	12	c(m	c(m	NUM
ejpam-4753	477	13	)	)	PUNCT
ejpam-4753	478	1	=	=	PUNCT
ejpam-4753	478	2	m∗	m∗	PROPN
ejpam-4753	478	3	is	be	AUX
ejpam-4753	478	4	the	the	DET
ejpam-4753	478	5	associate	associate	ADJ
ejpam-4753	478	6	complex	complex	ADJ
ejpam-4753	478	7	sequence	sequence	NOUN
ejpam-4753	478	8	to	to	ADP
ejpam-4753	478	9	a	a	DET
ejpam-4753	478	10	graded	grade	VERB
ejpam-4753	478	11	a−module	a−module	ADP
ejpam-4753	478	12	m	m	PROPN
ejpam-4753	478	13	=	=	PROPN
ejpam-4753	478	14	⊕	⊕	PROPN
ejpam-4753	479	1	n∈z	n∈z	ADJ
ejpam-4753	479	2	mn	mn	PROPN
ejpam-4753	479	3	then	then	ADV
ejpam-4753	479	4	m∗	m∗	VERB
ejpam-4753	479	5	∈	∈	PROPN
ejpam-4753	479	6	comp	comp	NOUN
ejpam-4753	479	7	(	(	PUNCT
ejpam-4753	479	8	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	479	9	)	)	PUNCT
ejpam-4753	479	10	)	)	PUNCT
ejpam-4753	479	11	.	.	PUNCT
ejpam-4753	480	1	(	(	PUNCT
ejpam-4753	480	2	ii	ii	X
ejpam-4753	480	3	)	)	PUNCT
ejpam-4753	480	4	let	let	VERB
ejpam-4753	480	5	f	f	PRON
ejpam-4753	480	6	:	:	PUNCT
ejpam-4753	480	7	m	m	VERB
ejpam-4753	480	8	−→	−→	ADJ
ejpam-4753	480	9	n	n	ADV
ejpam-4753	480	10	graded	grade	VERB
ejpam-4753	480	11	morphism	morphism	NOUN
ejpam-4753	480	12	of	of	ADP
ejpam-4753	480	13	degree	degree	NOUN
ejpam-4753	480	14	k	k	PROPN
ejpam-4753	480	15	of	of	ADP
ejpam-4753	480	16	graded	grade	VERB
ejpam-4753	480	17	a−modules	a−module	NOUN
ejpam-4753	480	18	then	then	ADV
ejpam-4753	480	19	:	:	PUNCT
ejpam-4753	480	20	c(f	c(f	X
ejpam-4753	480	21	)	)	PUNCT
ejpam-4753	481	1	=	=	SYM
ejpam-4753	481	2	fk	fk	INTJ
ejpam-4753	481	3	∗	∗	NOUN
ejpam-4753	481	4	:	:	PUNCT
ejpam-4753	481	5	m∗	m∗	VERB
ejpam-4753	481	6	−→	−→	NOUN
ejpam-4753	481	7	n∗	n∗	VERB
ejpam-4753	481	8	the	the	DET
ejpam-4753	481	9	associate	associate	ADJ
ejpam-4753	481	10	complex	complex	ADJ
ejpam-4753	481	11	chain	chain	NOUN
ejpam-4753	481	12	to	to	ADP
ejpam-4753	481	13	a	a	DET
ejpam-4753	481	14	graded	grade	VERB
ejpam-4753	481	15	morphism	morphism	NOUN
ejpam-4753	481	16	of	of	ADP
ejpam-4753	481	17	degree	degree	NOUN
ejpam-4753	481	18	k	k	PROPN
ejpam-4753	481	19	of	of	ADP
ejpam-4753	481	20	graded	grade	VERB
ejpam-4753	481	21	left	leave	VERB
ejpam-4753	481	22	a−module	a−module	ADP
ejpam-4753	481	23	.	.	PUNCT
ejpam-4753	482	1	furthermore	furthermore	ADV
ejpam-4753	482	2	c(g	c(g	PROPN
ejpam-4753	482	3	◦	◦	NOUN
ejpam-4753	482	4	f	f	X
ejpam-4753	482	5	)	)	PUNCT
ejpam-4753	483	1	=	=	SYM
ejpam-4753	483	2	(	(	PUNCT
ejpam-4753	483	3	g	g	NOUN
ejpam-4753	483	4	◦	◦	NOUN
ejpam-4753	483	5	f)k∗	f)k∗	ADJ
ejpam-4753	484	1	=	=	SYM
ejpam-4753	484	2	g[f	g[f	X
ejpam-4753	484	3	]	]	PUNCT
ejpam-4753	484	4	k∗	k∗	NOUN
ejpam-4753	484	5	=	=	PUNCT
ejpam-4753	484	6	g[fk	g[fk	NOUN
ejpam-4753	484	7	∗	∗	NOUN
ejpam-4753	484	8	]	]	PUNCT
ejpam-4753	485	1	k	k	X
ejpam-4753	485	2	∗	∗	NOUN
ejpam-4753	485	3	=	=	PUNCT
ejpam-4753	485	4	gk∗	gk∗	NOUN
ejpam-4753	485	5	◦	◦	NOUN
ejpam-4753	485	6	fk	fk	INTJ
ejpam-4753	485	7	∗	∗	NOUN
ejpam-4753	485	8	=	=	SYM
ejpam-4753	485	9	c(g	c(g	PROPN
ejpam-4753	485	10	)	)	PUNCT
ejpam-4753	485	11	◦	◦	NOUN
ejpam-4753	485	12	c(f	c(f	PROPN
ejpam-4753	485	13	)	)	PUNCT
ejpam-4753	485	14	.	.	PUNCT
ejpam-4753	486	1	on	on	ADP
ejpam-4753	486	2	other	other	ADJ
ejpam-4753	486	3	hand	hand	NOUN
ejpam-4753	486	4	c(1m(n	c(1m(n	NOUN
ejpam-4753	486	5	)	)	PUNCT
ejpam-4753	486	6	)	)	PUNCT
ejpam-4753	486	7	:	:	PUNCT
ejpam-4753	486	8	m(n)∗	m(n)∗	INTJ
ejpam-4753	486	9	−→	−→	NOUN
ejpam-4753	486	10	m(n)∗	m(n)∗	PROPN
ejpam-4753	486	11	1m(n)∗	1m(n)∗	NUM
ejpam-4753	486	12	=	=	SYM
ejpam-4753	486	13	1c(m	1c(m	NUM
ejpam-4753	486	14	)	)	PUNCT
ejpam-4753	486	15	thus	thus	ADV
ejpam-4753	486	16	c	c	X
ejpam-4753	486	17	(	(	PUNCT
ejpam-4753	486	18	)	)	PUNCT
ejpam-4753	486	19	is	be	AUX
ejpam-4753	486	20	a	a	DET
ejpam-4753	486	21	covariant	covariant	ADJ
ejpam-4753	486	22	functor	functor	NOUN
ejpam-4753	486	23	of	of	ADP
ejpam-4753	486	24	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	486	25	)	)	PUNCT
ejpam-4753	486	26	to	to	ADP
ejpam-4753	486	27	comp	comp	NOUN
ejpam-4753	486	28	(	(	PUNCT
ejpam-4753	486	29	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	486	30	)	)	PUNCT
ejpam-4753	486	31	)	)	PUNCT
ejpam-4753	486	32	.	.	PUNCT
ejpam-4753	487	1	let	let	VERB
ejpam-4753	487	2	0	0	NUM
ejpam-4753	487	3	−→	−→	NOUN
ejpam-4753	487	4	m	m	ADP
ejpam-4753	487	5	f−→	f−→	NOUN
ejpam-4753	487	6	n	n	NOUN
ejpam-4753	487	7	g−→	g−→	NOUN
ejpam-4753	487	8	l	l	NOUN
ejpam-4753	487	9	−→	−→	NOUN
ejpam-4753	487	10	0	0	NUM
ejpam-4753	487	11	be	be	AUX
ejpam-4753	487	12	the	the	DET
ejpam-4753	487	13	short	short	ADJ
ejpam-4753	487	14	exact	exact	ADJ
ejpam-4753	487	15	sequence	sequence	NOUN
ejpam-4753	487	16	of	of	ADP
ejpam-4753	487	17	graded	grade	VERB
ejpam-4753	487	18	left	leave	VERB
ejpam-4753	487	19	a−modules	a−module	NOUN
ejpam-4753	487	20	then	then	ADV
ejpam-4753	487	21	we	we	PRON
ejpam-4753	487	22	make	make	VERB
ejpam-4753	487	23	the	the	DET
ejpam-4753	487	24	functor	functor	PROPN
ejpam-4753	487	25	c	c	PROPN
ejpam-4753	487	26	(	(	PUNCT
ejpam-4753	487	27	)	)	PUNCT
ejpam-4753	487	28	then	then	ADV
ejpam-4753	487	29	we	we	PRON
ejpam-4753	487	30	have	have	VERB
ejpam-4753	487	31	a.	a.	NOUN
ejpam-4753	487	32	o.	o.	PROPN
ejpam-4753	487	33	chbih	chbih	PROPN
ejpam-4753	487	34	,	,	PUNCT
ejpam-4753	487	35	m.	m.	PROPN
ejpam-4753	487	36	b.	b.	PROPN
ejpam-4753	487	37	maaouia	maaouia	PROPN
ejpam-4753	487	38	,	,	PUNCT
ejpam-4753	487	39	m.	m.	NOUN
ejpam-4753	487	40	sanghare	sanghare	PROPN
ejpam-4753	487	41	/	/	SYM
ejpam-4753	487	42	eur	eur	PROPN
ejpam-4753	487	43	.	.	PUNCT
ejpam-4753	488	1	j.	j.	PROPN
ejpam-4753	488	2	pure	pure	PROPN
ejpam-4753	488	3	appl	appl	PROPN
ejpam-4753	488	4	.	.	PROPN
ejpam-4753	488	5	math	math	PROPN
ejpam-4753	488	6	,	,	PUNCT
ejpam-4753	488	7	16	16	NUM
ejpam-4753	488	8	(	(	PUNCT
ejpam-4753	488	9	3	3	NUM
ejpam-4753	488	10	)	)	PUNCT
ejpam-4753	488	11	(	(	PUNCT
ejpam-4753	488	12	2023	2023	NUM
ejpam-4753	488	13	)	)	PUNCT
ejpam-4753	488	14	,	,	PUNCT
ejpam-4753	488	15	1913	1913	NUM
ejpam-4753	488	16	-	-	SYM
ejpam-4753	488	17	1939	1939	NUM
ejpam-4753	488	18	1934	1934	NUM
ejpam-4753	488	19	0	0	NUM
ejpam-4753	488	20	:	:	PUNCT
ejpam-4753	488	21	·	·	PUNCT
ejpam-4753	488	22	·	·	PUNCT
ejpam-4753	488	23	·	·	PUNCT
ejpam-4753	488	24	//	//	SYM
ejpam-4753	488	25	�	�	PROPN
ejpam-4753	488	26	�	�	PROPN
ejpam-4753	488	27	//	//	SYM
ejpam-4753	488	28	0	0	NUM
ejpam-4753	488	29	�	�	PROPN
ejpam-4753	488	30	�	�	PROPN
ejpam-4753	488	31	//	//	SYM
ejpam-4753	488	32	0	0	NUM
ejpam-4753	488	33	�	�	PROPN
ejpam-4753	488	34	�	�	PROPN
ejpam-4753	488	35	//	//	PROPN
ejpam-4753	488	36	.	.	PUNCT
ejpam-4753	488	37	.	.	PUNCT
ejpam-4753	488	38	.	.	PUNCT
ejpam-4753	489	1	m∗	m∗	VERB
ejpam-4753	489	2	:	:	PUNCT
ejpam-4753	489	3	·	·	PUNCT
ejpam-4753	489	4	·	·	PUNCT
ejpam-4753	489	5	·	·	PUNCT
ejpam-4753	490	1	//	//	PUNCT
ejpam-4753	490	2	fk	fk	INTJ
ejpam-4753	490	3	∗	∗	X
ejpam-4753	490	4	�	�	PROPN
ejpam-4753	490	5	�	�	PROPN
ejpam-4753	490	6	//m(n+	//m(n+	PUNCT
ejpam-4753	490	7	1	1	NUM
ejpam-4753	490	8	)	)	PUNCT
ejpam-4753	490	9	fk(n+1	fk(n+1	PROPN
ejpam-4753	490	10	)	)	PUNCT
ejpam-4753	490	11	�	�	PROPN
ejpam-4753	490	12	�	�	PROPN
ejpam-4753	490	13	dn+1	dn+1	PROPN
ejpam-4753	490	14	//m(n	//m(n	X
ejpam-4753	490	15	)	)	PUNCT
ejpam-4753	490	16	fk(n	fk(n	NOUN
ejpam-4753	490	17	)	)	PUNCT
ejpam-4753	490	18	�	�	PROPN
ejpam-4753	490	19	�	�	PROPN
ejpam-4753	490	20	dn	dn	PROPN
ejpam-4753	490	21	//	//	PROPN
ejpam-4753	490	22	.	.	PUNCT
ejpam-4753	490	23	.	.	PUNCT
ejpam-4753	490	24	.	.	PUNCT
ejpam-4753	491	1	n∗	n∗	PROPN
ejpam-4753	491	2	:	:	PUNCT
ejpam-4753	491	3	·	·	PUNCT
ejpam-4753	491	4	·	·	PUNCT
ejpam-4753	491	5	·	·	PUNCT
ejpam-4753	491	6	gr∗	gr∗	PROPN
ejpam-4753	491	7	�	�	PROPN
ejpam-4753	491	8	�	�	PROPN
ejpam-4753	491	9	//	//	NUM
ejpam-4753	491	10	n(n+	n(n+	NUM
ejpam-4753	491	11	1	1	NUM
ejpam-4753	491	12	)	)	PUNCT
ejpam-4753	491	13	gr(n+1	gr(n+1	NOUN
ejpam-4753	491	14	)	)	PUNCT
ejpam-4753	491	15	�	�	NOUN
ejpam-4753	491	16	�	�	PROPN
ejpam-4753	491	17	d	d	NOUN
ejpam-4753	491	18	′	′	NUM
ejpam-4753	491	19	n+1+k	n+1+k	NOUN
ejpam-4753	491	20	//	//	SYM
ejpam-4753	491	21	n(n	n(n	NOUN
ejpam-4753	491	22	)	)	PUNCT
ejpam-4753	491	23	gr(n	gr(n	NOUN
ejpam-4753	491	24	)	)	PUNCT
ejpam-4753	491	25	�	�	NOUN
ejpam-4753	491	26	�	�	PROPN
ejpam-4753	491	27	d	d	ADP
ejpam-4753	491	28	′	′	NUM
ejpam-4753	491	29	n+k	n+k	PROPN
ejpam-4753	491	30	//	//	NUM
ejpam-4753	491	31	.	.	PUNCT
ejpam-4753	491	32	.	.	PUNCT
ejpam-4753	491	33	.	.	PUNCT
ejpam-4753	492	1	l∗	l∗	PROPN
ejpam-4753	492	2	:	:	PUNCT
ejpam-4753	492	3	·	·	PUNCT
ejpam-4753	492	4	·	·	PUNCT
ejpam-4753	492	5	·	·	PUNCT
ejpam-4753	492	6	�	�	PROPN
ejpam-4753	492	7	�	�	PROPN
ejpam-4753	492	8	//	//	NUM
ejpam-4753	492	9	l(n+	l(n+	ADJ
ejpam-4753	492	10	1	1	NUM
ejpam-4753	492	11	)	)	PUNCT
ejpam-4753	492	12	�	�	PROPN
ejpam-4753	492	13	�	�	PROPN
ejpam-4753	492	14	d	d	PROPN
ejpam-4753	492	15	′′	′′	PROPN
ejpam-4753	492	16	n+1+k+r	n+1+k+r	PROPN
ejpam-4753	492	17	//	//	SYM
ejpam-4753	492	18	l(n	l(n	PROPN
ejpam-4753	492	19	)	)	PUNCT
ejpam-4753	492	20	�	�	PROPN
ejpam-4753	492	21	�	�	PROPN
ejpam-4753	492	22	d	d	PROPN
ejpam-4753	492	23	′′	′′	PROPN
ejpam-4753	492	24	n+k+r	n+k+r	ADJ
ejpam-4753	492	25	//	//	NOUN
ejpam-4753	492	26	.	.	PUNCT
ejpam-4753	492	27	.	.	PUNCT
ejpam-4753	492	28	.	.	PUNCT
ejpam-4753	493	1	0	0	NUM
ejpam-4753	494	1	:	:	PUNCT
ejpam-4753	495	1	·	·	PUNCT
ejpam-4753	495	2	·	·	PUNCT
ejpam-4753	495	3	·	·	PUNCT
ejpam-4753	495	4	//	//	PUNCT
ejpam-4753	495	5	0	0	NUM
ejpam-4753	495	6	//	//	SYM
ejpam-4753	495	7	0	0	NUM
ejpam-4753	495	8	//	//	PROPN
ejpam-4753	495	9	.	.	PUNCT
ejpam-4753	495	10	.	.	PUNCT
ejpam-4753	496	1	.	.	PUNCT
ejpam-4753	497	1	is	be	AUX
ejpam-4753	497	2	a	a	DET
ejpam-4753	497	3	short	short	ADJ
ejpam-4753	497	4	exact	exact	ADJ
ejpam-4753	497	5	complex	complex	ADJ
ejpam-4753	497	6	chain	chain	NOUN
ejpam-4753	497	7	associate	associate	NOUN
ejpam-4753	497	8	to	to	ADP
ejpam-4753	497	9	short	short	ADJ
ejpam-4753	497	10	exact	exact	ADJ
ejpam-4753	497	11	sequence	sequence	NOUN
ejpam-4753	497	12	of	of	ADP
ejpam-4753	497	13	graded	grade	VERB
ejpam-4753	497	14	left	leave	VERB
ejpam-4753	497	15	a−modules	a−module	NOUN
ejpam-4753	497	16	then	then	ADV
ejpam-4753	497	17	0	0	NUM
ejpam-4753	497	18	−→	−→	NOUN
ejpam-4753	497	19	m∗	m∗	NOUN
ejpam-4753	497	20	fk	fk	INTJ
ejpam-4753	497	21	∗−→	∗−→	NUM
ejpam-4753	497	22	n∗	n∗	PROPN
ejpam-4753	497	23	gk∗−→	gk∗−→	VERB
ejpam-4753	497	24	l∗	l∗	VERB
ejpam-4753	497	25	−→	−→	NOUN
ejpam-4753	497	26	0	0	NUM
ejpam-4753	497	27	is	be	AUX
ejpam-4753	497	28	exact	exact	ADJ
ejpam-4753	497	29	complex	complex	ADJ
ejpam-4753	497	30	chain	chain	NOUN
ejpam-4753	497	31	.	.	PUNCT
ejpam-4753	498	1	thus	thus	ADV
ejpam-4753	498	2	c	c	X
ejpam-4753	498	3	(	(	PUNCT
ejpam-4753	498	4	)	)	PUNCT
ejpam-4753	498	5	is	be	AUX
ejpam-4753	498	6	exact	exact	ADJ
ejpam-4753	498	7	additively	additively	ADV
ejpam-4753	498	8	covariant	covariant	ADJ
ejpam-4753	498	9	functor	functor	NOUN
ejpam-4753	498	10	of	of	ADP
ejpam-4753	498	11	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	498	12	)	)	PUNCT
ejpam-4753	498	13	to	to	ADP
ejpam-4753	498	14	comp	comp	NOUN
ejpam-4753	498	15	(	(	PUNCT
ejpam-4753	498	16	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	498	17	)	)	PUNCT
ejpam-4753	498	18	)	)	PUNCT
ejpam-4753	498	19	.	.	PUNCT
ejpam-4753	499	1	theorem	theorem	ADJ
ejpam-4753	499	2	8	8	NUM
ejpam-4753	499	3	.	.	PUNCT
ejpam-4753	500	1	let	let	VERB
ejpam-4753	500	2	a	a	DET
ejpam-4753	500	3	=	=	SYM
ejpam-4753	500	4	⊕	⊕	PROPN
ejpam-4753	500	5	n∈z	n∈z	VERB
ejpam-4753	500	6	an	an	DET
ejpam-4753	500	7	be	be	AUX
ejpam-4753	500	8	a	a	DET
ejpam-4753	500	9	graded	grade	VERB
ejpam-4753	500	10	ring	ring	NOUN
ejpam-4753	500	11	,	,	PUNCT
ejpam-4753	500	12	s	s	PART
ejpam-4753	500	13	is	be	AUX
ejpam-4753	500	14	a	a	DET
ejpam-4753	500	15	multiplicatively	multiplicatively	ADV
ejpam-4753	500	16	closed	close	VERB
ejpam-4753	500	17	subset	subset	NOUN
ejpam-4753	500	18	satisfying	satisfy	VERB
ejpam-4753	500	19	the	the	DET
ejpam-4753	500	20	left	left	ADJ
ejpam-4753	500	21	conditions	condition	NOUN
ejpam-4753	500	22	of	of	ADP
ejpam-4753	500	23	ore	ore	NOUN
ejpam-4753	500	24	formed	form	VERB
ejpam-4753	500	25	of	of	ADP
ejpam-4753	500	26	homogeneous	homogeneous	ADJ
ejpam-4753	500	27	elements	element	NOUN
ejpam-4753	500	28	of	of	ADP
ejpam-4753	500	29	a	a	PRON
ejpam-4753	500	30	and	and	CCONJ
ejpam-4753	500	31	gr(s	gr(s	ADJ
ejpam-4753	501	1	−1a	−1a	NUM
ejpam-4753	501	2	−	−	PROPN
ejpam-4753	501	3	mod	mod	PROPN
ejpam-4753	501	4	)	)	PUNCT
ejpam-4753	501	5	the	the	DET
ejpam-4753	501	6	category	category	NOUN
ejpam-4753	501	7	of	of	ADP
ejpam-4753	501	8	graded	grade	VERB
ejpam-4753	501	9	left	left	ADJ
ejpam-4753	501	10	s−1a−modules	s−1a−module	NOUN
ejpam-4753	501	11	,	,	PUNCT
ejpam-4753	501	12	then	then	ADV
ejpam-4753	501	13	the	the	DET
ejpam-4753	501	14	relation	relation	NOUN
ejpam-4753	501	15	ch(−	ch(−	PUNCT
ejpam-4753	501	16	)	)	PUNCT
ejpam-4753	501	17	:	:	PUNCT
ejpam-4753	501	18	gr(s	gr(s	X
ejpam-4753	501	19	−1a	−1a	NUM
ejpam-4753	501	20	−	−	NOUN
ejpam-4753	501	21	mod	mod	ADJ
ejpam-4753	501	22	)	)	PUNCT
ejpam-4753	501	23	−→	−→	NOUN
ejpam-4753	501	24	comp	comp	NOUN
ejpam-4753	501	25	(	(	PUNCT
ejpam-4753	501	26	gr(s	gr(s	X
ejpam-4753	501	27	−1a−mod	−1a−mod	PROPN
ejpam-4753	501	28	)	)	PUNCT
ejpam-4753	501	29	)	)	PUNCT
ejpam-4753	501	30	which	which	PRON
ejpam-4753	501	31	that	that	SCONJ
ejpam-4753	501	32	for	for	ADP
ejpam-4753	501	33	all	all	PRON
ejpam-4753	501	34	graded	grade	VERB
ejpam-4753	501	35	left	leave	VERB
ejpam-4753	501	36	s−1a−module	s−1a−module	NOUN
ejpam-4753	501	37	s−1	s−1	PROPN
ejpam-4753	501	38	m	m	PART
ejpam-4753	501	39	of	of	ADP
ejpam-4753	501	40	gr(s	gr(s	X
ejpam-4753	501	41	−1a	−1a	NUM
ejpam-4753	501	42	−mod	−mod	ADV
ejpam-4753	501	43	)	)	PUNCT
ejpam-4753	501	44	we	we	PRON
ejpam-4753	501	45	correspond	correspond	VERB
ejpam-4753	501	46	the	the	DET
ejpam-4753	501	47	associate	associate	ADJ
ejpam-4753	501	48	complex	complex	ADJ
ejpam-4753	501	49	sequence	sequence	NOUN
ejpam-4753	501	50	(	(	PUNCT
ejpam-4753	501	51	s−1m)∗	s−1m)∗	NOUN
ejpam-4753	501	52	to	to	ADP
ejpam-4753	501	53	a	a	DET
ejpam-4753	501	54	graded	grade	VERB
ejpam-4753	501	55	s−1a−module	s−1a−module	NOUN
ejpam-4753	501	56	s−1	s−1	PROPN
ejpam-4753	501	57	m	m	PROPN
ejpam-4753	501	58	and	and	CCONJ
ejpam-4753	501	59	for	for	ADP
ejpam-4753	501	60	all	all	DET
ejpam-4753	501	61	graded	grade	VERB
ejpam-4753	501	62	morphism	morphism	NOUN
ejpam-4753	501	63	of	of	ADP
ejpam-4753	501	64	graded	grade	VERB
ejpam-4753	501	65	left	leave	VERB
ejpam-4753	501	66	s−1a−modules	s−1a−module	NOUN
ejpam-4753	501	67	s−1f	s−1f	NOUN
ejpam-4753	501	68	:	:	PUNCT
ejpam-4753	501	69	s−1	s−1	PROPN
ejpam-4753	501	70	m	m	VERB
ejpam-4753	501	71	−→	−→	NOUN
ejpam-4753	501	72	s−1n	s−1n	NOUN
ejpam-4753	501	73	of	of	ADP
ejpam-4753	501	74	degree	degree	NOUN
ejpam-4753	502	1	k	k	NOUN
ejpam-4753	503	1	we	we	PRON
ejpam-4753	503	2	correspond	correspond	VERB
ejpam-4753	503	3	the	the	DET
ejpam-4753	503	4	associate	associate	ADJ
ejpam-4753	503	5	complex	complex	NOUN
ejpam-4753	503	6	chain	chain	NOUN
ejpam-4753	503	7	(	(	PUNCT
ejpam-4753	503	8	s−1f)k∗	s−1f)k∗	NOUN
ejpam-4753	503	9	to	to	ADP
ejpam-4753	503	10	a	a	DET
ejpam-4753	503	11	morphism	morphism	NOUN
ejpam-4753	503	12	of	of	ADP
ejpam-4753	503	13	graded	grade	VERB
ejpam-4753	503	14	left	leave	VERB
ejpam-4753	503	15	s−1a−module	s−1a−module	NOUN
ejpam-4753	503	16	s−1f	s−1f	NOUN
ejpam-4753	503	17	:	:	PUNCT
ejpam-4753	504	1	s−1	s−1	PROPN
ejpam-4753	504	2	m	m	VERB
ejpam-4753	504	3	−→	−→	ADJ
ejpam-4753	504	4	s−1n	s−1n	NOUN
ejpam-4753	504	5	is	be	AUX
ejpam-4753	504	6	additively	additively	ADV
ejpam-4753	504	7	exact	exact	ADJ
ejpam-4753	504	8	covariant	covariant	ADJ
ejpam-4753	504	9	functor	functor	PROPN
ejpam-4753	504	10	.	.	PUNCT
ejpam-4753	504	11	proof	proof	NOUN
ejpam-4753	504	12	.	.	PUNCT
ejpam-4753	505	1	similarly	similarly	ADV
ejpam-4753	505	2	to	to	ADP
ejpam-4753	505	3	the	the	DET
ejpam-4753	505	4	proof	proof	NOUN
ejpam-4753	505	5	of	of	ADP
ejpam-4753	505	6	theorem	theorem	ADJ
ejpam-4753	505	7	precedent	precedent	NOUN
ejpam-4753	505	8	7	7	NUM
ejpam-4753	505	9	theorem	theorem	NOUN
ejpam-4753	505	10	9	9	NUM
ejpam-4753	505	11	.	.	PUNCT
ejpam-4753	506	1	let	let	VERB
ejpam-4753	506	2	a	a	DET
ejpam-4753	506	3	=	=	SYM
ejpam-4753	506	4	⊕	⊕	PROPN
ejpam-4753	506	5	n∈z	n∈z	VERB
ejpam-4753	506	6	an	an	DET
ejpam-4753	506	7	be	be	AUX
ejpam-4753	506	8	a	a	DET
ejpam-4753	506	9	graded	grade	VERB
ejpam-4753	506	10	duo	duo	NOUN
ejpam-4753	506	11	-	-	PUNCT
ejpam-4753	506	12	ring	ring	NOUN
ejpam-4753	506	13	,	,	PUNCT
ejpam-4753	506	14	sh	sh	PROPN
ejpam-4753	506	15	is	be	AUX
ejpam-4753	506	16	a	a	DET
ejpam-4753	506	17	part	part	NOUN
ejpam-4753	506	18	formed	form	VERB
ejpam-4753	506	19	of	of	ADP
ejpam-4753	506	20	regulars	regular	NOUN
ejpam-4753	506	21	homogeneous	homogeneous	ADJ
ejpam-4753	506	22	elements	element	NOUN
ejpam-4753	506	23	of	of	ADP
ejpam-4753	506	24	a	a	DET
ejpam-4753	506	25	and	and	CCONJ
ejpam-4753	506	26	gr(s	gr(s	NOUN
ejpam-4753	506	27	−1	−1	NOUN
ejpam-4753	506	28	h	h	NOUN
ejpam-4753	506	29	a−mod	a−mod	NOUN
ejpam-4753	506	30	)	)	PUNCT
ejpam-4753	506	31	the	the	DET
ejpam-4753	506	32	category	category	NOUN
ejpam-4753	506	33	of	of	ADP
ejpam-4753	506	34	graded	grade	VERB
ejpam-4753	506	35	left	leave	VERB
ejpam-4753	506	36	s	s	PRON
ejpam-4753	506	37	−1	−1	NOUN
ejpam-4753	506	38	h	h	NOUN
ejpam-4753	506	39	a−modules	a−module	NOUN
ejpam-4753	506	40	,	,	PUNCT
ejpam-4753	506	41	then	then	ADV
ejpam-4753	506	42	the	the	DET
ejpam-4753	506	43	relation	relation	NOUN
ejpam-4753	506	44	ch(−	ch(−	PUNCT
ejpam-4753	506	45	)	)	PUNCT
ejpam-4753	506	46	:	:	PUNCT
ejpam-4753	506	47	gr(s	gr(s	X
ejpam-4753	506	48	−1	−1	NOUN
ejpam-4753	506	49	h	h	PROPN
ejpam-4753	506	50	a−mod	a−mod	NOUN
ejpam-4753	506	51	)	)	PUNCT
ejpam-4753	506	52	−→	−→	NOUN
ejpam-4753	506	53	comp	comp	NOUN
ejpam-4753	506	54	(	(	PUNCT
ejpam-4753	506	55	gr(s	gr(s	NOUN
ejpam-4753	506	56	−1	−1	NOUN
ejpam-4753	506	57	h	h	NOUN
ejpam-4753	506	58	a−mod	a−mod	NOUN
ejpam-4753	506	59	)	)	PUNCT
ejpam-4753	506	60	)	)	PUNCT
ejpam-4753	506	61	which	which	PRON
ejpam-4753	506	62	that	that	SCONJ
ejpam-4753	506	63	for	for	ADP
ejpam-4753	506	64	all	all	PRON
ejpam-4753	506	65	graded	grade	VERB
ejpam-4753	506	66	left	leave	VERB
ejpam-4753	506	67	s	s	PRON
ejpam-4753	506	68	−1	−1	NOUN
ejpam-4753	506	69	h	h	NOUN
ejpam-4753	506	70	a−module	a−module	ADP
ejpam-4753	506	71	s	s	NUM
ejpam-4753	506	72	−1	−1	NOUN
ejpam-4753	506	73	h	h	NOUN
ejpam-4753	506	74	m	m	VERB
ejpam-4753	506	75	of	of	ADP
ejpam-4753	506	76	gr(s	gr(s	PUNCT
ejpam-4753	506	77	−1	−1	NOUN
ejpam-4753	506	78	h	h	PROPN
ejpam-4753	506	79	a−mod	a−mod	NOUN
ejpam-4753	506	80	)	)	PUNCT
ejpam-4753	507	1	we	we	PRON
ejpam-4753	507	2	correspond	correspond	VERB
ejpam-4753	507	3	the	the	DET
ejpam-4753	507	4	associate	associate	ADJ
ejpam-4753	507	5	complex	complex	ADJ
ejpam-4753	507	6	sequence	sequence	NOUN
ejpam-4753	507	7	(	(	PUNCT
ejpam-4753	507	8	s	s	NOUN
ejpam-4753	507	9	−1	−1	NOUN
ejpam-4753	507	10	h	h	NOUN
ejpam-4753	507	11	m)∗	m)∗	VERB
ejpam-4753	507	12	to	to	ADP
ejpam-4753	507	13	a	a	DET
ejpam-4753	507	14	graded	grade	VERB
ejpam-4753	507	15	s	s	PRON
ejpam-4753	507	16	−1	−1	NOUN
ejpam-4753	507	17	h	h	NOUN
ejpam-4753	507	18	a−module	a−module	ADP
ejpam-4753	507	19	s	s	X
ejpam-4753	507	20	−1	−1	NOUN
ejpam-4753	507	21	h	h	NOUN
ejpam-4753	507	22	m	m	PROPN
ejpam-4753	507	23	and	and	CCONJ
ejpam-4753	507	24	for	for	ADP
ejpam-4753	507	25	all	all	DET
ejpam-4753	507	26	graded	grade	VERB
ejpam-4753	507	27	morphism	morphism	NOUN
ejpam-4753	507	28	of	of	ADP
ejpam-4753	507	29	graded	grade	VERB
ejpam-4753	507	30	left	leave	VERB
ejpam-4753	507	31	s	s	PRON
ejpam-4753	507	32	−1	−1	NOUN
ejpam-4753	507	33	h	h	NOUN
ejpam-4753	508	1	a−modules	a−module	NOUN
ejpam-4753	508	2	s	s	PART
ejpam-4753	508	3	−1	−1	NOUN
ejpam-4753	508	4	h	h	NOUN
ejpam-4753	508	5	f	f	NOUN
ejpam-4753	508	6	:	:	PUNCT
ejpam-4753	508	7	s	s	VERB
ejpam-4753	508	8	−1	−1	NOUN
ejpam-4753	508	9	h	h	NOUN
ejpam-4753	508	10	m	m	VERB
ejpam-4753	508	11	−→	−→	NOUN
ejpam-4753	508	12	s	s	PART
ejpam-4753	508	13	−1	−1	NOUN
ejpam-4753	508	14	h	h	NOUN
ejpam-4753	508	15	n	n	PROPN
ejpam-4753	508	16	of	of	ADP
ejpam-4753	508	17	degree	degree	NOUN
ejpam-4753	509	1	k	k	NOUN
ejpam-4753	509	2	we	we	PRON
ejpam-4753	509	3	correspond	correspond	VERB
ejpam-4753	509	4	the	the	DET
ejpam-4753	509	5	associate	associate	ADJ
ejpam-4753	509	6	complex	complex	NOUN
ejpam-4753	509	7	chain	chain	NOUN
ejpam-4753	509	8	(	(	PUNCT
ejpam-4753	509	9	s	s	NOUN
ejpam-4753	509	10	−1	−1	NOUN
ejpam-4753	509	11	h	h	NOUN
ejpam-4753	509	12	f)k∗	f)k∗	ADJ
ejpam-4753	509	13	to	to	ADP
ejpam-4753	509	14	a	a	DET
ejpam-4753	509	15	morphism	morphism	NOUN
ejpam-4753	509	16	of	of	ADP
ejpam-4753	509	17	graded	grade	VERB
ejpam-4753	509	18	left	leave	VERB
ejpam-4753	509	19	s	s	PRON
ejpam-4753	509	20	−1	−1	NOUN
ejpam-4753	509	21	h	h	NOUN
ejpam-4753	509	22	a−module	a−module	ADP
ejpam-4753	509	23	s	s	NUM
ejpam-4753	509	24	−1	−1	NOUN
ejpam-4753	509	25	h	h	NOUN
ejpam-4753	509	26	f	f	NOUN
ejpam-4753	509	27	:	:	PUNCT
ejpam-4753	509	28	s	s	VERB
ejpam-4753	509	29	−1	−1	NOUN
ejpam-4753	509	30	h	h	NOUN
ejpam-4753	509	31	m	m	VERB
ejpam-4753	509	32	−→	−→	NOUN
ejpam-4753	509	33	s	s	PART
ejpam-4753	509	34	−1	−1	NOUN
ejpam-4753	509	35	h	h	NOUN
ejpam-4753	509	36	n	n	ADV
ejpam-4753	509	37	is	be	AUX
ejpam-4753	509	38	additively	additively	ADV
ejpam-4753	509	39	exact	exact	ADJ
ejpam-4753	509	40	covariant	covariant	PROPN
ejpam-4753	509	41	functor	functor	PROPN
ejpam-4753	509	42	.	.	PUNCT
ejpam-4753	509	43	a.	a.	PROPN
ejpam-4753	509	44	o.	o.	PROPN
ejpam-4753	509	45	chbih	chbih	PROPN
ejpam-4753	509	46	,	,	PUNCT
ejpam-4753	509	47	m.	m.	PROPN
ejpam-4753	509	48	b.	b.	PROPN
ejpam-4753	509	49	maaouia	maaouia	PROPN
ejpam-4753	509	50	,	,	PUNCT
ejpam-4753	509	51	m.	m.	NOUN
ejpam-4753	509	52	sanghare	sanghare	PROPN
ejpam-4753	509	53	/	/	SYM
ejpam-4753	509	54	eur	eur	PROPN
ejpam-4753	509	55	.	.	PUNCT
ejpam-4753	510	1	j.	j.	PROPN
ejpam-4753	510	2	pure	pure	PROPN
ejpam-4753	510	3	appl	appl	PROPN
ejpam-4753	510	4	.	.	PROPN
ejpam-4753	510	5	math	math	PROPN
ejpam-4753	510	6	,	,	PUNCT
ejpam-4753	510	7	16	16	NUM
ejpam-4753	510	8	(	(	PUNCT
ejpam-4753	510	9	3	3	NUM
ejpam-4753	510	10	)	)	PUNCT
ejpam-4753	510	11	(	(	PUNCT
ejpam-4753	510	12	2023	2023	NUM
ejpam-4753	510	13	)	)	PUNCT
ejpam-4753	510	14	,	,	PUNCT
ejpam-4753	510	15	1913	1913	NUM
ejpam-4753	510	16	-	-	SYM
ejpam-4753	510	17	1939	1939	NUM
ejpam-4753	510	18	1935	1935	NUM
ejpam-4753	510	19	proof	proof	NOUN
ejpam-4753	510	20	.	.	PUNCT
ejpam-4753	511	1	similarly	similarly	ADV
ejpam-4753	511	2	to	to	ADP
ejpam-4753	511	3	the	the	DET
ejpam-4753	511	4	proof	proof	NOUN
ejpam-4753	511	5	of	of	ADP
ejpam-4753	511	6	theorem	theorem	ADJ
ejpam-4753	511	7	precedent	precedent	NOUN
ejpam-4753	511	8	7	7	NUM
ejpam-4753	511	9	theorem	theorem	NOUN
ejpam-4753	511	10	10	10	NUM
ejpam-4753	511	11	.	.	PUNCT
ejpam-4753	512	1	let	let	VERB
ejpam-4753	512	2	a	a	DET
ejpam-4753	512	3	=	=	SYM
ejpam-4753	512	4	⊕	⊕	PROPN
ejpam-4753	512	5	n∈z	n∈z	VERB
ejpam-4753	512	6	an	an	DET
ejpam-4753	512	7	be	be	AUX
ejpam-4753	512	8	a	a	DET
ejpam-4753	512	9	graded	grade	VERB
ejpam-4753	512	10	ring	ring	NOUN
ejpam-4753	512	11	,	,	PUNCT
ejpam-4753	512	12	s	s	PART
ejpam-4753	512	13	is	be	AUX
ejpam-4753	512	14	a	a	DET
ejpam-4753	512	15	multiplicatively	multiplicatively	ADV
ejpam-4753	512	16	closed	close	VERB
ejpam-4753	512	17	subset	subset	NOUN
ejpam-4753	512	18	satisfying	satisfy	VERB
ejpam-4753	512	19	the	the	DET
ejpam-4753	512	20	left	left	ADJ
ejpam-4753	512	21	conditions	condition	NOUN
ejpam-4753	512	22	of	of	ADP
ejpam-4753	512	23	ore	ore	NOUN
ejpam-4753	512	24	formed	form	VERB
ejpam-4753	512	25	of	of	ADP
ejpam-4753	512	26	homogeneous	homogeneous	ADJ
ejpam-4753	512	27	elements	element	NOUN
ejpam-4753	512	28	of	of	ADP
ejpam-4753	512	29	a	a	DET
ejpam-4753	512	30	and	and	CCONJ
ejpam-4753	512	31	gr(s	gr(s	PUNCT
ejpam-4753	512	32	−1a−	−1a−	NOUN
ejpam-4753	512	33	mod	mod	PROPN
ejpam-4753	512	34	)	)	PUNCT
ejpam-4753	512	35	the	the	DET
ejpam-4753	512	36	category	category	NOUN
ejpam-4753	512	37	of	of	ADP
ejpam-4753	512	38	graded	grade	VERB
ejpam-4753	512	39	left	left	ADJ
ejpam-4753	512	40	s−1a−modules	s−1a−module	NOUN
ejpam-4753	512	41	,	,	PUNCT
ejpam-4753	512	42	then	then	ADV
ejpam-4753	512	43	the	the	DET
ejpam-4753	512	44	relation	relation	NOUN
ejpam-4753	512	45	(	(	PUNCT
ejpam-4753	512	46	ch	ch	NOUN
ejpam-4753	512	47	◦	◦	NOUN
ejpam-4753	512	48	s−1)(−	s−1)(−	NUM
ejpam-4753	512	49	)	)	PUNCT
ejpam-4753	512	50	:	:	PUNCT
ejpam-4753	512	51	gr(a−	gr(a−	X
ejpam-4753	512	52	mod	mod	ADJ
ejpam-4753	512	53	)	)	PUNCT
ejpam-4753	512	54	−→	−→	NOUN
ejpam-4753	512	55	comp	comp	NOUN
ejpam-4753	512	56	(	(	PUNCT
ejpam-4753	512	57	gr(s	gr(s	NOUN
ejpam-4753	512	58	−1a	−1a	NUM
ejpam-4753	512	59	−	−	NOUN
ejpam-4753	512	60	mod	mod	PROPN
ejpam-4753	512	61	)	)	PUNCT
ejpam-4753	512	62	)	)	PUNCT
ejpam-4753	512	63	which	which	PRON
ejpam-4753	512	64	that	that	SCONJ
ejpam-4753	512	65	for	for	ADP
ejpam-4753	512	66	all	all	PRON
ejpam-4753	512	67	graded	grade	VERB
ejpam-4753	512	68	left	leave	VERB
ejpam-4753	512	69	a−module	a−module	ADP
ejpam-4753	512	70	m	m	NOUN
ejpam-4753	512	71	of	of	ADP
ejpam-4753	512	72	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	512	73	)	)	PUNCT
ejpam-4753	512	74	we	we	PRON
ejpam-4753	512	75	correspond	correspond	VERB
ejpam-4753	512	76	the	the	DET
ejpam-4753	512	77	associate	associate	ADJ
ejpam-4753	512	78	complex	complex	ADJ
ejpam-4753	512	79	sequence	sequence	NOUN
ejpam-4753	512	80	(	(	PUNCT
ejpam-4753	512	81	ch	ch	NOUN
ejpam-4753	512	82	◦	◦	NOUN
ejpam-4753	512	83	s−1)(m	s−1)(m	NUM
ejpam-4753	512	84	)	)	PUNCT
ejpam-4753	513	1	=	=	PRON
ejpam-4753	513	2	(	(	PUNCT
ejpam-4753	513	3	s−1m)∗	s−1m)∗	NOUN
ejpam-4753	513	4	to	to	ADP
ejpam-4753	513	5	a	a	DET
ejpam-4753	513	6	graded	grade	VERB
ejpam-4753	513	7	a−module	a−module	ADP
ejpam-4753	513	8	m	m	NOUN
ejpam-4753	513	9	and	and	CCONJ
ejpam-4753	513	10	for	for	ADP
ejpam-4753	513	11	all	all	DET
ejpam-4753	513	12	graded	grade	VERB
ejpam-4753	513	13	morphism	morphism	NOUN
ejpam-4753	513	14	of	of	ADP
ejpam-4753	513	15	graded	grade	VERB
ejpam-4753	513	16	left	leave	VERB
ejpam-4753	513	17	a−modules	a−module	NOUN
ejpam-4753	513	18	f	f	X
ejpam-4753	513	19	:	:	PUNCT
ejpam-4753	513	20	m	m	VERB
ejpam-4753	513	21	−→	−→	ADJ
ejpam-4753	513	22	n	n	PRON
ejpam-4753	513	23	of	of	ADP
ejpam-4753	513	24	degree	degree	NOUN
ejpam-4753	514	1	k	k	NOUN
ejpam-4753	514	2	we	we	PRON
ejpam-4753	514	3	correspond	correspond	VERB
ejpam-4753	514	4	the	the	DET
ejpam-4753	514	5	associate	associate	ADJ
ejpam-4753	514	6	complex	complex	NOUN
ejpam-4753	514	7	chain	chain	NOUN
ejpam-4753	514	8	(	(	PUNCT
ejpam-4753	514	9	ch	ch	NOUN
ejpam-4753	514	10	◦	◦	NOUN
ejpam-4753	514	11	s−1)(f	s−1)(f	PROPN
ejpam-4753	514	12	)	)	PUNCT
ejpam-4753	514	13	=	=	PUNCT
ejpam-4753	515	1	(	(	PUNCT
ejpam-4753	515	2	s−1f)k∗	s−1f)k∗	NOUN
ejpam-4753	515	3	to	to	ADP
ejpam-4753	515	4	a	a	DET
ejpam-4753	515	5	morphism	morphism	NOUN
ejpam-4753	515	6	of	of	ADP
ejpam-4753	515	7	graded	grade	VERB
ejpam-4753	515	8	left	leave	VERB
ejpam-4753	515	9	a−module	a−module	ADP
ejpam-4753	515	10	f	f	X
ejpam-4753	515	11	:	:	PUNCT
ejpam-4753	515	12	m	m	VERB
ejpam-4753	515	13	−→	−→	ADJ
ejpam-4753	515	14	n	n	NOUN
ejpam-4753	515	15	is	be	AUX
ejpam-4753	516	1	additively	additively	ADV
ejpam-4753	516	2	exact	exact	ADJ
ejpam-4753	516	3	covariant	covariant	ADJ
ejpam-4753	516	4	functor	functor	PROPN
ejpam-4753	516	5	.	.	PUNCT
ejpam-4753	516	6	proof	proof	NOUN
ejpam-4753	516	7	.	.	PUNCT
ejpam-4753	517	1	similarly	similarly	ADV
ejpam-4753	517	2	to	to	ADP
ejpam-4753	517	3	the	the	DET
ejpam-4753	517	4	proof	proof	NOUN
ejpam-4753	517	5	of	of	ADP
ejpam-4753	517	6	theorem	theorem	ADJ
ejpam-4753	517	7	precedent	precedent	NOUN
ejpam-4753	517	8	7	7	NUM
ejpam-4753	517	9	theorem	theorem	NOUN
ejpam-4753	517	10	11	11	NUM
ejpam-4753	517	11	.	.	PUNCT
ejpam-4753	518	1	let	let	VERB
ejpam-4753	518	2	a	a	DET
ejpam-4753	518	3	=	=	SYM
ejpam-4753	518	4	⊕	⊕	PROPN
ejpam-4753	518	5	n∈z	n∈z	VERB
ejpam-4753	518	6	an	an	DET
ejpam-4753	518	7	be	be	AUX
ejpam-4753	518	8	a	a	DET
ejpam-4753	518	9	graded	grade	VERB
ejpam-4753	518	10	duo	duo	NOUN
ejpam-4753	518	11	-	-	PUNCT
ejpam-4753	518	12	ring	ring	NOUN
ejpam-4753	518	13	,	,	PUNCT
ejpam-4753	518	14	sh	sh	PROPN
ejpam-4753	518	15	is	be	AUX
ejpam-4753	518	16	a	a	DET
ejpam-4753	518	17	part	part	NOUN
ejpam-4753	518	18	formed	form	VERB
ejpam-4753	518	19	of	of	ADP
ejpam-4753	518	20	regulars	regular	NOUN
ejpam-4753	518	21	homogeneous	homogeneous	ADJ
ejpam-4753	518	22	elements	element	NOUN
ejpam-4753	518	23	of	of	ADP
ejpam-4753	518	24	a	a	DET
ejpam-4753	518	25	and	and	CCONJ
ejpam-4753	518	26	gr(s	gr(s	NOUN
ejpam-4753	518	27	−1	−1	NOUN
ejpam-4753	518	28	h	h	NOUN
ejpam-4753	518	29	a−mod	a−mod	NOUN
ejpam-4753	518	30	)	)	PUNCT
ejpam-4753	518	31	the	the	DET
ejpam-4753	518	32	category	category	NOUN
ejpam-4753	518	33	of	of	ADP
ejpam-4753	518	34	graded	grade	VERB
ejpam-4753	518	35	left	leave	VERB
ejpam-4753	518	36	s	s	PRON
ejpam-4753	518	37	−1	−1	NOUN
ejpam-4753	518	38	h	h	NOUN
ejpam-4753	518	39	a−modules	a−module	NOUN
ejpam-4753	518	40	,	,	PUNCT
ejpam-4753	518	41	then	then	ADV
ejpam-4753	518	42	the	the	DET
ejpam-4753	518	43	relation	relation	NOUN
ejpam-4753	518	44	(	(	PUNCT
ejpam-4753	518	45	ch	ch	NOUN
ejpam-4753	518	46	◦	◦	NOUN
ejpam-4753	518	47	s−1	s−1	PROPN
ejpam-4753	518	48	h	h	NOUN
ejpam-4753	518	49	)	)	PUNCT
ejpam-4753	518	50	(	(	PUNCT
ejpam-4753	518	51	−	−	NOUN
ejpam-4753	518	52	)	)	PUNCT
ejpam-4753	518	53	:	:	PUNCT
ejpam-4753	518	54	gr(a−mod	gr(a−mod	X
ejpam-4753	518	55	)	)	PUNCT
ejpam-4753	518	56	−→	−→	NOUN
ejpam-4753	518	57	comp	comp	NOUN
ejpam-4753	518	58	(	(	PUNCT
ejpam-4753	518	59	gr(s	gr(s	NOUN
ejpam-4753	518	60	−1	−1	NOUN
ejpam-4753	518	61	h	h	NOUN
ejpam-4753	518	62	a−mod	a−mod	NOUN
ejpam-4753	518	63	)	)	PUNCT
ejpam-4753	518	64	)	)	PUNCT
ejpam-4753	518	65	which	which	PRON
ejpam-4753	518	66	that	that	SCONJ
ejpam-4753	518	67	for	for	ADP
ejpam-4753	518	68	all	all	PRON
ejpam-4753	518	69	graded	grade	VERB
ejpam-4753	518	70	left	leave	VERB
ejpam-4753	518	71	a−module	a−module	ADP
ejpam-4753	518	72	m	m	NOUN
ejpam-4753	518	73	of	of	ADP
ejpam-4753	518	74	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	518	75	)	)	PUNCT
ejpam-4753	519	1	we	we	PRON
ejpam-4753	519	2	correspond	correspond	VERB
ejpam-4753	519	3	the	the	DET
ejpam-4753	519	4	associate	associate	ADJ
ejpam-4753	519	5	complex	complex	ADJ
ejpam-4753	519	6	sequence	sequence	NOUN
ejpam-4753	519	7	(	(	PUNCT
ejpam-4753	519	8	ch	ch	NOUN
ejpam-4753	519	9	◦	◦	NOUN
ejpam-4753	519	10	s−1	s−1	PROPN
ejpam-4753	519	11	h	h	NOUN
ejpam-4753	519	12	)	)	PUNCT
ejpam-4753	519	13	(	(	PUNCT
ejpam-4753	519	14	m	m	NOUN
ejpam-4753	519	15	)	)	PUNCT
ejpam-4753	519	16	=	=	SYM
ejpam-4753	520	1	(	(	PUNCT
ejpam-4753	520	2	s	s	NOUN
ejpam-4753	520	3	−1	−1	NOUN
ejpam-4753	520	4	h	h	NOUN
ejpam-4753	520	5	m)∗	m)∗	VERB
ejpam-4753	520	6	to	to	ADP
ejpam-4753	520	7	a	a	DET
ejpam-4753	520	8	graded	grade	VERB
ejpam-4753	520	9	a−module	a−module	ADP
ejpam-4753	520	10	m	m	NOUN
ejpam-4753	520	11	and	and	CCONJ
ejpam-4753	520	12	for	for	ADP
ejpam-4753	520	13	all	all	DET
ejpam-4753	520	14	graded	grade	VERB
ejpam-4753	520	15	morphism	morphism	NOUN
ejpam-4753	520	16	of	of	ADP
ejpam-4753	520	17	graded	grade	VERB
ejpam-4753	520	18	left	leave	VERB
ejpam-4753	520	19	a−modules	a−module	NOUN
ejpam-4753	520	20	f	f	X
ejpam-4753	520	21	:	:	PUNCT
ejpam-4753	520	22	m	m	VERB
ejpam-4753	520	23	−→	−→	ADJ
ejpam-4753	520	24	n	n	PRON
ejpam-4753	520	25	of	of	ADP
ejpam-4753	520	26	degree	degree	NOUN
ejpam-4753	521	1	k	k	NOUN
ejpam-4753	521	2	we	we	PRON
ejpam-4753	521	3	correspond	correspond	VERB
ejpam-4753	521	4	the	the	DET
ejpam-4753	521	5	associate	associate	ADJ
ejpam-4753	521	6	complex	complex	NOUN
ejpam-4753	521	7	chain	chain	NOUN
ejpam-4753	521	8	(	(	PUNCT
ejpam-4753	521	9	ch	ch	NOUN
ejpam-4753	521	10	◦	◦	NOUN
ejpam-4753	521	11	s−1	s−1	PROPN
ejpam-4753	521	12	h	h	NOUN
ejpam-4753	521	13	)	)	PUNCT
ejpam-4753	521	14	(	(	PUNCT
ejpam-4753	521	15	f	f	X
ejpam-4753	521	16	)	)	PUNCT
ejpam-4753	521	17	=	=	SYM
ejpam-4753	522	1	(	(	PUNCT
ejpam-4753	522	2	s	s	AUX
ejpam-4753	522	3	−1	−1	NOUN
ejpam-4753	522	4	h	h	NOUN
ejpam-4753	522	5	f)k∗	f)k∗	ADJ
ejpam-4753	522	6	to	to	ADP
ejpam-4753	522	7	a	a	DET
ejpam-4753	522	8	morphism	morphism	NOUN
ejpam-4753	522	9	of	of	ADP
ejpam-4753	522	10	graded	grade	VERB
ejpam-4753	522	11	left	leave	VERB
ejpam-4753	522	12	a−module	a−module	ADP
ejpam-4753	522	13	f	f	X
ejpam-4753	522	14	:	:	PUNCT
ejpam-4753	522	15	m	m	VERB
ejpam-4753	522	16	−→	−→	ADJ
ejpam-4753	523	1	n	n	NOUN
ejpam-4753	523	2	is	be	AUX
ejpam-4753	523	3	additively	additively	ADV
ejpam-4753	523	4	exact	exact	ADJ
ejpam-4753	523	5	covariant	covariant	ADJ
ejpam-4753	523	6	functor	functor	PROPN
ejpam-4753	523	7	.	.	PUNCT
ejpam-4753	523	8	proof	proof	NOUN
ejpam-4753	523	9	.	.	PUNCT
ejpam-4753	524	1	similarly	similarly	ADV
ejpam-4753	524	2	to	to	ADP
ejpam-4753	524	3	the	the	DET
ejpam-4753	524	4	proof	proof	NOUN
ejpam-4753	524	5	of	of	ADP
ejpam-4753	524	6	theorem	theorem	ADJ
ejpam-4753	524	7	precedent7	precedent7	NOUN
ejpam-4753	524	8	lemma	lemma	PROPN
ejpam-4753	524	9	1	1	X
ejpam-4753	524	10	.	.	PUNCT
ejpam-4753	524	11	let	let	VERB
ejpam-4753	524	12	a	a	DET
ejpam-4753	524	13	=	=	SYM
ejpam-4753	524	14	⊕	⊕	PROPN
ejpam-4753	524	15	n∈z	n∈z	VERB
ejpam-4753	524	16	an	an	DET
ejpam-4753	524	17	be	be	AUX
ejpam-4753	524	18	a	a	DET
ejpam-4753	524	19	graded	grade	VERB
ejpam-4753	524	20	ring	ring	NOUN
ejpam-4753	524	21	,	,	PUNCT
ejpam-4753	524	22	m	m	VERB
ejpam-4753	524	23	=	=	ADJ
ejpam-4753	524	24	⊕	⊕	PROPN
ejpam-4753	524	25	n∈z	n∈z	VERB
ejpam-4753	524	26	mn	mn	PROPN
ejpam-4753	524	27	and	and	CCONJ
ejpam-4753	524	28	a	a	DET
ejpam-4753	524	29	graded	grade	VERB
ejpam-4753	524	30	left	leave	VERB
ejpam-4753	524	31	a−module	a−module	ADP
ejpam-4753	524	32	then	then	ADV
ejpam-4753	524	33	for	for	ADP
ejpam-4753	524	34	all	all	PRON
ejpam-4753	524	35	n	n	DET
ejpam-4753	524	36	∈	∈	PROPN
ejpam-4753	524	37	z	z	NOUN
ejpam-4753	524	38	hn(m∗	hn(m∗	NOUN
ejpam-4753	524	39	)	)	PUNCT
ejpam-4753	525	1	∼=	∼=	PART
ejpam-4753	525	2	m(n+	m(n+	NOUN
ejpam-4753	525	3	2	2	NUM
ejpam-4753	525	4	)	)	PUNCT
ejpam-4753	525	5	.	.	PUNCT
ejpam-4753	526	1	proof	proof	NOUN
ejpam-4753	526	2	.	.	PUNCT
ejpam-4753	527	1	let	let	VERB
ejpam-4753	527	2	m∗	m∗	NOUN
ejpam-4753	527	3	:	:	PUNCT
ejpam-4753	527	4	·	·	PUNCT
ejpam-4753	527	5	·	·	PUNCT
ejpam-4753	527	6	·	·	PUNCT
ejpam-4753	527	7	→	→	PUNCT
ejpam-4753	527	8	m(n+1	m(n+1	NUM
ejpam-4753	527	9	)	)	PUNCT
ejpam-4753	527	10	dn+1→	dn+1→	NOUN
ejpam-4753	527	11	m(n	m(n	PROPN
ejpam-4753	527	12	)	)	PUNCT
ejpam-4753	527	13	dn→	dn→	NOUN
ejpam-4753	527	14	m(n−	m(n−	NOUN
ejpam-4753	527	15	1	1	NUM
ejpam-4753	527	16	)	)	PUNCT
ejpam-4753	527	17	→	→	PUNCT
ejpam-4753	527	18	·	·	PUNCT
ejpam-4753	527	19	·	·	PUNCT
ejpam-4753	527	20	·	·	PUNCT
ejpam-4753	528	1	the	the	DET
ejpam-4753	528	2	complex	complex	ADJ
ejpam-4753	528	3	sequence	sequence	NOUN
ejpam-4753	528	4	,	,	PUNCT
ejpam-4753	528	5	then	then	ADV
ejpam-4753	528	6	ker(dn	ker(dn	X
ejpam-4753	528	7	)	)	PUNCT
ejpam-4753	528	8	=	=	SYM
ejpam-4753	528	9	m(n+	m(n+	NOUN
ejpam-4753	528	10	1	1	NUM
ejpam-4753	528	11	)	)	PUNCT
ejpam-4753	528	12	and	and	CCONJ
ejpam-4753	528	13	im(dn	im(dn	PROPN
ejpam-4753	528	14	)	)	PUNCT
ejpam-4753	529	1	=	=	SYM
ejpam-4753	529	2	mn	mn	PROPN
ejpam-4753	529	3	so	so	ADV
ejpam-4753	529	4	hn(m∗	hn(m∗	ADV
ejpam-4753	529	5	)	)	PUNCT
ejpam-4753	529	6	=	=	SYM
ejpam-4753	529	7	ker(dn)/im(dn+1	ker(dn)/im(dn+1	X
ejpam-4753	529	8	)	)	PUNCT
ejpam-4753	529	9	=	=	SYM
ejpam-4753	529	10	m(n+1)/mn+1	m(n+1)/mn+1	NOUN
ejpam-4753	529	11	=	=	SYM
ejpam-4753	529	12	(	(	PUNCT
ejpam-4753	529	13	mn+1⊕m(n+2))/mn+1	mn+1⊕m(n+2))/mn+1	VERB
ejpam-4753	529	14	∼=	∼=	ADV
ejpam-4753	529	15	m(n+2	m(n+2	NUM
ejpam-4753	529	16	)	)	PUNCT
ejpam-4753	529	17	.	.	PUNCT
ejpam-4753	530	1	theorem	theorem	NOUN
ejpam-4753	530	2	12	12	NUM
ejpam-4753	530	3	.	.	PUNCT
ejpam-4753	531	1	let	let	VERB
ejpam-4753	531	2	a	a	DET
ejpam-4753	531	3	=	=	SYM
ejpam-4753	531	4	⊕	⊕	PROPN
ejpam-4753	531	5	n∈z	n∈z	VERB
ejpam-4753	531	6	an	an	DET
ejpam-4753	531	7	be	be	AUX
ejpam-4753	531	8	a	a	DET
ejpam-4753	531	9	graded	grade	VERB
ejpam-4753	531	10	ring	ring	NOUN
ejpam-4753	531	11	,	,	PUNCT
ejpam-4753	531	12	we	we	PRON
ejpam-4753	531	13	have	have	VERB
ejpam-4753	531	14	the	the	DET
ejpam-4753	531	15	induced	induce	VERB
ejpam-4753	531	16	functor	functor	NOUN
ejpam-4753	531	17	of	of	ADP
ejpam-4753	531	18	hn	hn	PROPN
ejpam-4753	531	19	:	:	PUNCT
ejpam-4753	531	20	comp	comp	NOUN
ejpam-4753	531	21	(	(	PUNCT
ejpam-4753	531	22	gr(a	gr(a	NOUN
ejpam-4753	531	23	−	−	PROPN
ejpam-4753	531	24	mod	mod	PROPN
ejpam-4753	531	25	)	)	PUNCT
ejpam-4753	531	26	)	)	PUNCT
ejpam-4753	532	1	−→	−→	NOUN
ejpam-4753	532	2	gr(a	gr(a	PUNCT
ejpam-4753	532	3	−	−	PROPN
ejpam-4753	532	4	mod	mod	PROPN
ejpam-4753	532	5	)	)	PUNCT
ejpam-4753	532	6	which	which	PRON
ejpam-4753	532	7	that	that	SCONJ
ejpam-4753	532	8	for	for	ADP
ejpam-4753	532	9	all	all	DET
ejpam-4753	532	10	associate	associate	ADJ
ejpam-4753	532	11	complex	complex	ADJ
ejpam-4753	532	12	sequence	sequence	NOUN
ejpam-4753	532	13	m∗	m∗	VERB
ejpam-4753	532	14	to	to	ADP
ejpam-4753	532	15	a	a	DET
ejpam-4753	532	16	graded	grade	VERB
ejpam-4753	532	17	a−module	a−module	ADP
ejpam-4753	532	18	m	m	PROPN
ejpam-4753	532	19	=	=	PROPN
ejpam-4753	532	20	⊕	⊕	PROPN
ejpam-4753	532	21	n∈z	n∈z	ADJ
ejpam-4753	533	1	mn	mn	PROPN
ejpam-4753	534	1	we	we	PRON
ejpam-4753	534	2	correspond	correspond	VERB
ejpam-4753	534	3	hn(m∗	hn(m∗	ADV
ejpam-4753	534	4	)	)	PUNCT
ejpam-4753	535	1	=	=	VERB
ejpam-4753	536	1	m(n	m(n	NOUN
ejpam-4753	536	2	+	+	PUNCT
ejpam-4753	536	3	2	2	NUM
ejpam-4753	536	4	)	)	PUNCT
ejpam-4753	536	5	and	and	CCONJ
ejpam-4753	536	6	for	for	ADP
ejpam-4753	536	7	all	all	DET
ejpam-4753	536	8	associate	associate	ADJ
ejpam-4753	536	9	complex	complex	ADJ
ejpam-4753	536	10	chain	chain	NOUN
ejpam-4753	536	11	fk	fk	INTJ
ejpam-4753	536	12	∗	∗	NOUN
ejpam-4753	536	13	to	to	ADP
ejpam-4753	536	14	a	a	DET
ejpam-4753	536	15	morphism	morphism	NOUN
ejpam-4753	536	16	of	of	ADP
ejpam-4753	536	17	graded	grade	VERB
ejpam-4753	536	18	left	leave	VERB
ejpam-4753	536	19	a−module	a−module	ADP
ejpam-4753	536	20	f	f	X
ejpam-4753	536	21	:	:	PUNCT
ejpam-4753	536	22	m	m	VERB
ejpam-4753	536	23	=	=	PROPN
ejpam-4753	536	24	⊕	⊕	PROPN
ejpam-4753	536	25	n∈z	n∈z	VERB
ejpam-4753	536	26	mn	mn	PROPN
ejpam-4753	536	27	−→	−→	NOUN
ejpam-4753	536	28	n	n	PROPN
ejpam-4753	536	29	=	=	SYM
ejpam-4753	536	30	⊕	⊕	PROPN
ejpam-4753	536	31	n∈z	n∈z	VERB
ejpam-4753	536	32	nn	nn	INTJ
ejpam-4753	536	33	we	we	PRON
ejpam-4753	536	34	correspond	correspond	VERB
ejpam-4753	536	35	hn(f∗	hn(f∗	NUM
ejpam-4753	536	36	)	)	PUNCT
ejpam-4753	537	1	=	=	SYM
ejpam-4753	537	2	fk(n+	fk(n+	NOUN
ejpam-4753	537	3	2	2	NUM
ejpam-4753	537	4	)	)	PUNCT
ejpam-4753	537	5	,	,	PUNCT
ejpam-4753	537	6	is	be	AUX
ejpam-4753	537	7	a	a	DET
ejpam-4753	537	8	covariant	covariant	ADJ
ejpam-4753	537	9	functor	functor	NOUN
ejpam-4753	537	10	.	.	PUNCT
ejpam-4753	537	11	a.	a.	PROPN
ejpam-4753	537	12	o.	o.	PROPN
ejpam-4753	537	13	chbih	chbih	PROPN
ejpam-4753	537	14	,	,	PUNCT
ejpam-4753	537	15	m.	m.	PROPN
ejpam-4753	537	16	b.	b.	PROPN
ejpam-4753	537	17	maaouia	maaouia	PROPN
ejpam-4753	537	18	,	,	PUNCT
ejpam-4753	537	19	m.	m.	NOUN
ejpam-4753	537	20	sanghare	sanghare	PROPN
ejpam-4753	537	21	/	/	SYM
ejpam-4753	537	22	eur	eur	PROPN
ejpam-4753	537	23	.	.	PUNCT
ejpam-4753	538	1	j.	j.	PROPN
ejpam-4753	538	2	pure	pure	PROPN
ejpam-4753	538	3	appl	appl	PROPN
ejpam-4753	538	4	.	.	PROPN
ejpam-4753	538	5	math	math	PROPN
ejpam-4753	538	6	,	,	PUNCT
ejpam-4753	538	7	16	16	NUM
ejpam-4753	538	8	(	(	PUNCT
ejpam-4753	538	9	3	3	NUM
ejpam-4753	538	10	)	)	PUNCT
ejpam-4753	538	11	(	(	PUNCT
ejpam-4753	538	12	2023	2023	NUM
ejpam-4753	538	13	)	)	PUNCT
ejpam-4753	538	14	,	,	PUNCT
ejpam-4753	538	15	1913	1913	NUM
ejpam-4753	538	16	-	-	SYM
ejpam-4753	538	17	1939	1939	NUM
ejpam-4753	538	18	1936	1936	NUM
ejpam-4753	538	19	theorem	theorem	VERB
ejpam-4753	538	20	13	13	NUM
ejpam-4753	538	21	.	.	PUNCT
ejpam-4753	539	1	let	let	VERB
ejpam-4753	539	2	a	a	DET
ejpam-4753	539	3	=	=	SYM
ejpam-4753	539	4	⊕	⊕	PROPN
ejpam-4753	539	5	n∈z	n∈z	VERB
ejpam-4753	539	6	an	an	DET
ejpam-4753	539	7	be	be	AUX
ejpam-4753	539	8	a	a	DET
ejpam-4753	539	9	graded	grade	VERB
ejpam-4753	539	10	ring	ring	NOUN
ejpam-4753	539	11	,	,	PUNCT
ejpam-4753	539	12	for	for	ADP
ejpam-4753	539	13	all	all	PRON
ejpam-4753	539	14	n	n	PRON
ejpam-4753	539	15	∈	∈	PROPN
ejpam-4753	539	16	z	z	NOUN
ejpam-4753	539	17	and	and	CCONJ
ejpam-4753	539	18	for	for	ADP
ejpam-4753	539	19	all	all	DET
ejpam-4753	539	20	short	short	ADJ
ejpam-4753	539	21	exact	exact	ADJ
ejpam-4753	539	22	sequence	sequence	NOUN
ejpam-4753	539	23	0	0	NUM
ejpam-4753	539	24	−→	−→	NOUN
ejpam-4753	539	25	m	m	VERB
ejpam-4753	539	26	f−→	f−→	NOUN
ejpam-4753	539	27	n	n	NOUN
ejpam-4753	539	28	g−→	g−→	NOUN
ejpam-4753	539	29	l	l	NOUN
ejpam-4753	539	30	−→	−→	NOUN
ejpam-4753	539	31	0	0	NUM
ejpam-4753	539	32	of	of	ADP
ejpam-4753	539	33	a	a	DET
ejpam-4753	539	34	graded	grade	VERB
ejpam-4753	539	35	left	leave	VERB
ejpam-4753	539	36	a−modules	a−module	NOUN
ejpam-4753	539	37	of	of	ADP
ejpam-4753	539	38	graded	grade	VERB
ejpam-4753	539	39	morphism	morphism	NOUN
ejpam-4753	539	40	of	of	ADP
ejpam-4753	539	41	degree	degree	NOUN
ejpam-4753	539	42	k	k	PROPN
ejpam-4753	539	43	∈	∈	PROPN
ejpam-4753	540	1	z	z	NOUN
ejpam-4753	540	2	we	we	PRON
ejpam-4753	540	3	have	have	VERB
ejpam-4753	540	4	the	the	DET
ejpam-4753	540	5	following	follow	VERB
ejpam-4753	540	6	long	long	ADJ
ejpam-4753	540	7	exact	exact	ADJ
ejpam-4753	540	8	sequence	sequence	NOUN
ejpam-4753	540	9	·	·	PUNCT
ejpam-4753	540	10	·	·	PUNCT
ejpam-4753	540	11	·	·	PUNCT
ejpam-4753	541	1	−→	−→	NOUN
ejpam-4753	541	2	m(n+2	m(n+2	NUM
ejpam-4753	541	3	)	)	PUNCT
ejpam-4753	541	4	fk(n+2)−→	fk(n+2)−→	X
ejpam-4753	541	5	n(n+2	n(n+2	NUM
ejpam-4753	541	6	)	)	PUNCT
ejpam-4753	541	7	gk(n+2)−→	gk(n+2)−→	NOUN
ejpam-4753	541	8	l(n+2	l(n+2	X
ejpam-4753	541	9	)	)	PUNCT
ejpam-4753	541	10	δn−→	δn−→	NUM
ejpam-4753	541	11	m(n+1	m(n+1	NUM
ejpam-4753	541	12	)	)	PUNCT
ejpam-4753	541	13	fk(n+1)−→	fk(n+1)−→	X
ejpam-4753	541	14	n(n+1	n(n+1	NOUN
ejpam-4753	541	15	)	)	PUNCT
ejpam-4753	541	16	−→	−→	NOUN
ejpam-4753	541	17	·	·	PUNCT
ejpam-4753	541	18	·	·	PUNCT
ejpam-4753	541	19	·	·	PUNCT
ejpam-4753	541	20	of	of	ADP
ejpam-4753	541	21	a	a	DET
ejpam-4753	541	22	graded	grade	VERB
ejpam-4753	541	23	left	leave	VERB
ejpam-4753	541	24	a−modules	a−module	NOUN
ejpam-4753	541	25	of	of	ADP
ejpam-4753	541	26	graded	grade	VERB
ejpam-4753	541	27	morphism	morphism	NOUN
ejpam-4753	541	28	of	of	ADP
ejpam-4753	541	29	degree	degree	NOUN
ejpam-4753	541	30	k	k	PROPN
ejpam-4753	541	31	∈	∈	PROPN
ejpam-4753	541	32	z.	z.	PROPN
ejpam-4753	542	1	furthermore	furthermore	ADV
ejpam-4753	542	2	,	,	PUNCT
ejpam-4753	542	3	if	if	SCONJ
ejpam-4753	542	4	s	s	VERB
ejpam-4753	542	5	is	be	AUX
ejpam-4753	542	6	a	a	DET
ejpam-4753	542	7	multiplicatively	multiplicatively	ADV
ejpam-4753	542	8	closed	close	VERB
ejpam-4753	542	9	subset	subset	NOUN
ejpam-4753	542	10	satisfying	satisfy	VERB
ejpam-4753	542	11	the	the	DET
ejpam-4753	542	12	left	left	ADJ
ejpam-4753	542	13	conditions	condition	NOUN
ejpam-4753	542	14	of	of	ADP
ejpam-4753	542	15	ore	ore	NOUN
ejpam-4753	542	16	formed	form	VERB
ejpam-4753	542	17	of	of	ADP
ejpam-4753	542	18	homogeneous	homogeneous	ADJ
ejpam-4753	542	19	elements	element	NOUN
ejpam-4753	542	20	of	of	ADP
ejpam-4753	542	21	a	a	PRON
ejpam-4753	542	22	,	,	PUNCT
ejpam-4753	542	23	we	we	PRON
ejpam-4753	542	24	have	have	VERB
ejpam-4753	542	25	the	the	DET
ejpam-4753	542	26	following	follow	VERB
ejpam-4753	542	27	longs	long	NOUN
ejpam-4753	542	28	exacts	exact	VERB
ejpam-4753	542	29	sequences	sequence	NOUN
ejpam-4753	542	30	of	of	ADP
ejpam-4753	542	31	a	a	DET
ejpam-4753	542	32	graded	grade	VERB
ejpam-4753	542	33	left	leave	VERB
ejpam-4753	542	34	s−1a−modules	s−1a−module	NOUN
ejpam-4753	542	35	·	·	PUNCT
ejpam-4753	542	36	·	·	PUNCT
ejpam-4753	542	37	·	·	PUNCT
ejpam-4753	542	38	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	542	39	2	2	NUM
ejpam-4753	542	40	)	)	PUNCT
ejpam-4753	542	41	s−1fk(n+2)−→	s−1fk(n+2)−→	NOUN
ejpam-4753	542	42	s−1n(n+	s−1n(n+	VERB
ejpam-4753	542	43	2	2	NUM
ejpam-4753	542	44	)	)	PUNCT
ejpam-4753	542	45	s−1gk(n+2)−→	s−1gk(n+2)−→	X
ejpam-4753	542	46	s−1l(n+	s−1l(n+	NOUN
ejpam-4753	542	47	2	2	NUM
ejpam-4753	542	48	)	)	PUNCT
ejpam-4753	542	49	s−1δn−→	s−1δn−→	NOUN
ejpam-4753	542	50	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	542	51	1	1	NUM
ejpam-4753	542	52	)	)	PUNCT
ejpam-4753	542	53	·	·	PUNCT
ejpam-4753	542	54	·	·	PUNCT
ejpam-4753	542	55	·	·	PUNCT
ejpam-4753	542	56	·	·	PUNCT
ejpam-4753	542	57	·	·	PUNCT
ejpam-4753	542	58	·	·	PUNCT
ejpam-4753	542	59	s−1⊗m(n+2	s−1⊗m(n+2	PROPN
ejpam-4753	542	60	)	)	PUNCT
ejpam-4753	542	61	s−1⊗fk(n+2)−→	s−1⊗fk(n+2)−→	PUNCT
ejpam-4753	542	62	s−1⊗n(n+2	s−1⊗n(n+2	PROPN
ejpam-4753	542	63	)	)	PUNCT
ejpam-4753	542	64	s−1⊗gk(n+2)−→	s−1⊗gk(n+2)−→	PROPN
ejpam-4753	542	65	s−1⊗l(n+2	s−1⊗l(n+2	PROPN
ejpam-4753	542	66	)	)	PUNCT
ejpam-4753	542	67	s−1⊗δn−→	s−1⊗δn−→	X
ejpam-4753	542	68	s−1⊗m(n+1	s−1⊗m(n+1	NOUN
ejpam-4753	542	69	)	)	PUNCT
ejpam-4753	542	70	·	·	PUNCT
ejpam-4753	542	71	·	·	PUNCT
ejpam-4753	542	72	·	·	PUNCT
ejpam-4753	542	73	proof	proof	NOUN
ejpam-4753	542	74	.	.	PUNCT
ejpam-4753	543	1	let	let	VERB
ejpam-4753	543	2	0	0	NUM
ejpam-4753	543	3	−→	−→	NOUN
ejpam-4753	543	4	m	m	ADP
ejpam-4753	543	5	f−→	f−→	NOUN
ejpam-4753	543	6	n	n	NOUN
ejpam-4753	543	7	g−→	g−→	NOUN
ejpam-4753	543	8	l	l	NOUN
ejpam-4753	543	9	−→	−→	NOUN
ejpam-4753	543	10	0	0	NUM
ejpam-4753	543	11	be	be	AUX
ejpam-4753	543	12	the	the	DET
ejpam-4753	543	13	short	short	ADJ
ejpam-4753	543	14	exact	exact	ADJ
ejpam-4753	543	15	sequence	sequence	NOUN
ejpam-4753	543	16	of	of	ADP
ejpam-4753	543	17	graded	grade	VERB
ejpam-4753	543	18	left	leave	VERB
ejpam-4753	543	19	a−modules	a−module	NOUN
ejpam-4753	543	20	then	then	ADV
ejpam-4753	543	21	we	we	PRON
ejpam-4753	543	22	make	make	VERB
ejpam-4753	543	23	the	the	DET
ejpam-4753	543	24	functor	functor	PROPN
ejpam-4753	543	25	c	c	PROPN
ejpam-4753	543	26	(	(	PUNCT
ejpam-4753	543	27	)	)	PUNCT
ejpam-4753	543	28	to	to	ADP
ejpam-4753	543	29	the	the	DET
ejpam-4753	543	30	short	short	ADJ
ejpam-4753	543	31	exact	exact	ADJ
ejpam-4753	543	32	sequence	sequence	NOUN
ejpam-4753	543	33	of	of	ADP
ejpam-4753	543	34	graded	grade	VERB
ejpam-4753	543	35	a−modules	a−module	NOUN
ejpam-4753	543	36	then	then	ADV
ejpam-4753	543	37	we	we	PRON
ejpam-4753	543	38	have	have	VERB
ejpam-4753	543	39	0	0	NUM
ejpam-4753	543	40	−→	−→	NOUN
ejpam-4753	543	41	m∗	m∗	NOUN
ejpam-4753	543	42	fk	fk	INTJ
ejpam-4753	543	43	∗−→	∗−→	NUM
ejpam-4753	543	44	n∗	n∗	PROPN
ejpam-4753	543	45	gk∗−→	gk∗−→	VERB
ejpam-4753	543	46	l∗	l∗	VERB
ejpam-4753	543	47	−→	−→	ADV
ejpam-4753	543	48	0	0	PUNCT
ejpam-4753	544	1	the	the	DET
ejpam-4753	544	2	associate	associate	ADJ
ejpam-4753	544	3	short	short	ADJ
ejpam-4753	544	4	exact	exact	ADJ
ejpam-4753	544	5	complex	complex	NOUN
ejpam-4753	544	6	to	to	ADP
ejpam-4753	544	7	a	a	PRON
ejpam-4753	544	8	of	of	ADP
ejpam-4753	544	9	graded	grade	VERB
ejpam-4753	544	10	a−modules	a−module	NOUN
ejpam-4753	544	11	or	or	CCONJ
ejpam-4753	544	12	since	since	SCONJ
ejpam-4753	544	13	precedent	precedent	NOUN
ejpam-4753	544	14	theorem	theorem	NOUN
ejpam-4753	544	15	12	12	NUM
ejpam-4753	544	16	,	,	PUNCT
ejpam-4753	544	17	it	it	PRON
ejpam-4753	544	18	exist	exist	VERB
ejpam-4753	544	19	a	a	DET
ejpam-4753	544	20	morphism	morphism	NOUN
ejpam-4753	544	21	of	of	ADP
ejpam-4753	544	22	left	left	ADJ
ejpam-4753	544	23	a−module	a−module	ADP
ejpam-4753	544	24	hn(l∗	hn(l∗	NOUN
ejpam-4753	544	25	)	)	PUNCT
ejpam-4753	544	26	δn−→	δn−→	NUM
ejpam-4753	544	27	hn−1(m∗	hn−1(m∗	ADV
ejpam-4753	544	28	)	)	PUNCT
ejpam-4753	544	29	such	such	ADJ
ejpam-4753	544	30	that	that	SCONJ
ejpam-4753	544	31	we	we	PRON
ejpam-4753	544	32	have	have	VERB
ejpam-4753	544	33	the	the	DET
ejpam-4753	544	34	following	follow	VERB
ejpam-4753	544	35	long	long	ADJ
ejpam-4753	544	36	exact	exact	ADJ
ejpam-4753	544	37	sequence	sequence	NOUN
ejpam-4753	544	38	of	of	ADP
ejpam-4753	544	39	graded	grade	VERB
ejpam-4753	544	40	left	leave	VERB
ejpam-4753	544	41	a−modules	a−module	NOUN
ejpam-4753	544	42	·	·	PUNCT
ejpam-4753	544	43	·	·	PUNCT
ejpam-4753	544	44	·	·	PUNCT
ejpam-4753	545	1	−→	−→	NOUN
ejpam-4753	545	2	hn(m∗	hn(m∗	NOUN
ejpam-4753	545	3	)	)	PUNCT
ejpam-4753	545	4	hn(fk	hn(fk	PROPN
ejpam-4753	545	5	∗	∗	NOUN
ejpam-4753	545	6	)	)	PUNCT
ejpam-4753	545	7	−→	−→	NOUN
ejpam-4753	545	8	hn(n∗	hn(n∗	NOUN
ejpam-4753	545	9	)	)	PUNCT
ejpam-4753	546	1	hn(gk∗	hn(gk∗	NOUN
ejpam-4753	546	2	)	)	PUNCT
ejpam-4753	546	3	−→	−→	NOUN
ejpam-4753	546	4	hn(l∗	hn(l∗	NOUN
ejpam-4753	546	5	)	)	PUNCT
ejpam-4753	546	6	δn−→	δn−→	NUM
ejpam-4753	546	7	hn−1(m∗	hn−1(m∗	NOUN
ejpam-4753	546	8	)	)	PUNCT
ejpam-4753	546	9	hn−1(fk	hn−1(fk	PROPN
ejpam-4753	546	10	∗	∗	NOUN
ejpam-4753	546	11	)	)	PUNCT
ejpam-4753	546	12	−→	−→	NOUN
ejpam-4753	546	13	hn−1(n∗	hn−1(n∗	NOUN
ejpam-4753	546	14	)	)	PUNCT
ejpam-4753	546	15	−→	−→	NOUN
ejpam-4753	546	16	·	·	PUNCT
ejpam-4753	546	17	·	·	PUNCT
ejpam-4753	546	18	·	·	PUNCT
ejpam-4753	547	1	i.e	i.e	X
ejpam-4753	547	2	·	·	PUNCT
ejpam-4753	547	3	·	·	PUNCT
ejpam-4753	547	4	·	·	PUNCT
ejpam-4753	547	5	−→	−→	NOUN
ejpam-4753	547	6	m(n+2	m(n+2	NUM
ejpam-4753	547	7	)	)	PUNCT
ejpam-4753	547	8	fk(n+2)−→	fk(n+2)−→	X
ejpam-4753	547	9	n(n+2	n(n+2	NUM
ejpam-4753	547	10	)	)	PUNCT
ejpam-4753	547	11	gk(n+2)−→	gk(n+2)−→	NOUN
ejpam-4753	547	12	l(n+2	l(n+2	X
ejpam-4753	547	13	)	)	PUNCT
ejpam-4753	547	14	δn−→	δn−→	NUM
ejpam-4753	547	15	m(n+1	m(n+1	NUM
ejpam-4753	547	16	)	)	PUNCT
ejpam-4753	547	17	fk(n+1)−→	fk(n+1)−→	X
ejpam-4753	547	18	nk(n+1	nk(n+1	PROPN
ejpam-4753	547	19	)	)	PUNCT
ejpam-4753	547	20	−→	−→	NOUN
ejpam-4753	547	21	·	·	PUNCT
ejpam-4753	547	22	·	·	PUNCT
ejpam-4753	547	23	·	·	PUNCT
ejpam-4753	547	24	we	we	PRON
ejpam-4753	547	25	have	have	VERB
ejpam-4753	547	26	the	the	DET
ejpam-4753	547	27	functor	functor	PROPN
ejpam-4753	547	28	s−1	s−1	PROPN
ejpam-4753	547	29	(	(	PUNCT
ejpam-4753	547	30	)	)	PUNCT
ejpam-4753	547	31	is	be	AUX
ejpam-4753	547	32	exact	exact	ADJ
ejpam-4753	547	33	then	then	ADV
ejpam-4753	547	34	we	we	PRON
ejpam-4753	547	35	have	have	AUX
ejpam-4753	547	36	·	·	PUNCT
ejpam-4753	547	37	·	·	PUNCT
ejpam-4753	547	38	·	·	PUNCT
ejpam-4753	547	39	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	547	40	2	2	NUM
ejpam-4753	547	41	)	)	PUNCT
ejpam-4753	547	42	s−1fk(n+2)−→	s−1fk(n+2)−→	NOUN
ejpam-4753	547	43	s−1n(n+	s−1n(n+	VERB
ejpam-4753	547	44	2	2	NUM
ejpam-4753	547	45	)	)	PUNCT
ejpam-4753	547	46	s−1gk(n+2)−→	s−1gk(n+2)−→	X
ejpam-4753	547	47	s−1l(n+	s−1l(n+	NOUN
ejpam-4753	547	48	2	2	NUM
ejpam-4753	547	49	)	)	PUNCT
ejpam-4753	547	50	s−1δn−→	s−1δn−→	NOUN
ejpam-4753	547	51	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	547	52	1	1	NUM
ejpam-4753	547	53	)	)	PUNCT
ejpam-4753	547	54	·	·	PUNCT
ejpam-4753	547	55	·	·	PUNCT
ejpam-4753	547	56	·	·	PUNCT
ejpam-4753	547	57	or	or	CCONJ
ejpam-4753	547	58	the	the	DET
ejpam-4753	547	59	functor	functor	PROPN
ejpam-4753	547	60	s−1	s−1	PROPN
ejpam-4753	547	61	(	(	PUNCT
ejpam-4753	547	62	)	)	PUNCT
ejpam-4753	547	63	and	and	CCONJ
ejpam-4753	547	64	the	the	DET
ejpam-4753	547	65	functor	functor	PROPN
ejpam-4753	547	66	s−1a	s−1a	PROPN
ejpam-4753	547	67	⊗	⊗	PROPN
ejpam-4753	547	68	a	a	PRON
ejpam-4753	547	69	are	be	AUX
ejpam-4753	547	70	isomorph	isomorph	NOUN
ejpam-4753	547	71	then	then	ADV
ejpam-4753	547	72	we	we	PRON
ejpam-4753	547	73	have	have	VERB
ejpam-4753	547	74	also	also	ADV
ejpam-4753	547	75	·	·	PUNCT
ejpam-4753	547	76	·	·	PUNCT
ejpam-4753	547	77	·	·	PUNCT
ejpam-4753	547	78	s−1⊗m(n+2	s−1⊗m(n+2	PROPN
ejpam-4753	547	79	)	)	PUNCT
ejpam-4753	547	80	s−1⊗fk(n+2)−→	s−1⊗fk(n+2)−→	PUNCT
ejpam-4753	547	81	s−1⊗n(n+2	s−1⊗n(n+2	PROPN
ejpam-4753	547	82	)	)	PUNCT
ejpam-4753	547	83	s−1⊗gk(n+2)−→	s−1⊗gk(n+2)−→	PROPN
ejpam-4753	547	84	s−1⊗l(n+2	s−1⊗l(n+2	PROPN
ejpam-4753	547	85	)	)	PUNCT
ejpam-4753	547	86	s−1⊗δn−→	s−1⊗δn−→	X
ejpam-4753	547	87	s−1⊗m(n+1	s−1⊗m(n+1	NOUN
ejpam-4753	547	88	)	)	PUNCT
ejpam-4753	547	89	·	·	PUNCT
ejpam-4753	547	90	·	·	PUNCT
ejpam-4753	547	91	·	·	PUNCT
ejpam-4753	547	92	a.	a.	PROPN
ejpam-4753	547	93	o.	o.	PROPN
ejpam-4753	547	94	chbih	chbih	PROPN
ejpam-4753	547	95	,	,	PUNCT
ejpam-4753	547	96	m.	m.	PROPN
ejpam-4753	547	97	b.	b.	PROPN
ejpam-4753	547	98	maaouia	maaouia	PROPN
ejpam-4753	547	99	,	,	PUNCT
ejpam-4753	547	100	m.	m.	NOUN
ejpam-4753	547	101	sanghare	sanghare	PROPN
ejpam-4753	547	102	/	/	SYM
ejpam-4753	547	103	eur	eur	PROPN
ejpam-4753	547	104	.	.	PUNCT
ejpam-4753	548	1	j.	j.	PROPN
ejpam-4753	548	2	pure	pure	PROPN
ejpam-4753	548	3	appl	appl	PROPN
ejpam-4753	548	4	.	.	PROPN
ejpam-4753	548	5	math	math	PROPN
ejpam-4753	548	6	,	,	PUNCT
ejpam-4753	548	7	16	16	NUM
ejpam-4753	548	8	(	(	PUNCT
ejpam-4753	548	9	3	3	NUM
ejpam-4753	548	10	)	)	PUNCT
ejpam-4753	548	11	(	(	PUNCT
ejpam-4753	548	12	2023	2023	NUM
ejpam-4753	548	13	)	)	PUNCT
ejpam-4753	548	14	,	,	PUNCT
ejpam-4753	548	15	1913	1913	NUM
ejpam-4753	548	16	-	-	SYM
ejpam-4753	548	17	1939	1939	NUM
ejpam-4753	548	18	1937	1937	NUM
ejpam-4753	548	19	proposition	proposition	NOUN
ejpam-4753	548	20	18	18	NUM
ejpam-4753	548	21	.	.	PUNCT
ejpam-4753	549	1	let	let	VERB
ejpam-4753	549	2	a	a	DET
ejpam-4753	549	3	=	=	SYM
ejpam-4753	549	4	⊕	⊕	PROPN
ejpam-4753	549	5	n∈z	n∈z	VERB
ejpam-4753	549	6	an	an	DET
ejpam-4753	549	7	be	be	AUX
ejpam-4753	549	8	a	a	DET
ejpam-4753	549	9	graded	grade	VERB
ejpam-4753	549	10	ring	ring	NOUN
ejpam-4753	549	11	,	,	PUNCT
ejpam-4753	549	12	for	for	ADP
ejpam-4753	549	13	all	all	PRON
ejpam-4753	549	14	n	n	PRON
ejpam-4753	549	15	∈	∈	PROPN
ejpam-4753	549	16	z	z	NOUN
ejpam-4753	549	17	and	and	CCONJ
ejpam-4753	549	18	for	for	ADP
ejpam-4753	549	19	all	all	DET
ejpam-4753	549	20	short	short	ADJ
ejpam-4753	549	21	exact	exact	ADJ
ejpam-4753	549	22	sequence	sequence	NOUN
ejpam-4753	549	23	0	0	NUM
ejpam-4753	549	24	−→	−→	NOUN
ejpam-4753	549	25	m	m	VERB
ejpam-4753	549	26	f−→	f−→	NOUN
ejpam-4753	549	27	n	n	NOUN
ejpam-4753	549	28	g−→	g−→	NOUN
ejpam-4753	549	29	l	l	NOUN
ejpam-4753	549	30	−→	−→	NOUN
ejpam-4753	549	31	0	0	NUM
ejpam-4753	549	32	of	of	ADP
ejpam-4753	549	33	a	a	DET
ejpam-4753	549	34	graded	grade	VERB
ejpam-4753	549	35	left	leave	VERB
ejpam-4753	549	36	a−modules	a−module	NOUN
ejpam-4753	549	37	of	of	ADP
ejpam-4753	549	38	graded	grade	VERB
ejpam-4753	549	39	morphism	morphism	NOUN
ejpam-4753	549	40	of	of	ADP
ejpam-4753	549	41	degree	degree	NOUN
ejpam-4753	549	42	k	k	PROPN
ejpam-4753	549	43	∈	∈	PROPN
ejpam-4753	550	1	z	z	NOUN
ejpam-4753	550	2	we	we	PRON
ejpam-4753	550	3	have	have	VERB
ejpam-4753	550	4	the	the	DET
ejpam-4753	550	5	following	follow	VERB
ejpam-4753	550	6	long	long	ADJ
ejpam-4753	550	7	exact	exact	ADJ
ejpam-4753	550	8	sequence	sequence	NOUN
ejpam-4753	550	9	·	·	PUNCT
ejpam-4753	550	10	·	·	PUNCT
ejpam-4753	550	11	·	·	PUNCT
ejpam-4753	551	1	−→	−→	NOUN
ejpam-4753	551	2	m(n+2	m(n+2	NUM
ejpam-4753	551	3	)	)	PUNCT
ejpam-4753	551	4	fk(n+2)−→	fk(n+2)−→	X
ejpam-4753	551	5	n(n+2	n(n+2	NUM
ejpam-4753	551	6	)	)	PUNCT
ejpam-4753	551	7	gk(n+2)−→	gk(n+2)−→	NOUN
ejpam-4753	551	8	l(n+2	l(n+2	X
ejpam-4753	551	9	)	)	PUNCT
ejpam-4753	551	10	δn−→	δn−→	NUM
ejpam-4753	551	11	m(n+1	m(n+1	NUM
ejpam-4753	551	12	)	)	PUNCT
ejpam-4753	551	13	fk(n+1)−→	fk(n+1)−→	X
ejpam-4753	551	14	n(n+1	n(n+1	NOUN
ejpam-4753	551	15	)	)	PUNCT
ejpam-4753	551	16	−→	−→	NOUN
ejpam-4753	551	17	·	·	PUNCT
ejpam-4753	551	18	·	·	PUNCT
ejpam-4753	551	19	·	·	PUNCT
ejpam-4753	551	20	of	of	ADP
ejpam-4753	551	21	a	a	DET
ejpam-4753	551	22	graded	grade	VERB
ejpam-4753	551	23	left	leave	VERB
ejpam-4753	551	24	a−modules	a−module	NOUN
ejpam-4753	551	25	of	of	ADP
ejpam-4753	551	26	graded	grade	VERB
ejpam-4753	551	27	morphism	morphism	NOUN
ejpam-4753	551	28	of	of	ADP
ejpam-4753	551	29	degree	degree	NOUN
ejpam-4753	551	30	k	k	PROPN
ejpam-4753	551	31	∈	∈	PROPN
ejpam-4753	551	32	z.	z.	PROPN
ejpam-4753	552	1	furthermore	furthermore	ADV
ejpam-4753	552	2	,	,	PUNCT
ejpam-4753	552	3	if	if	SCONJ
ejpam-4753	552	4	s	s	VERB
ejpam-4753	552	5	is	be	AUX
ejpam-4753	552	6	a	a	DET
ejpam-4753	552	7	multiplicatively	multiplicatively	ADV
ejpam-4753	552	8	closed	close	VERB
ejpam-4753	552	9	subset	subset	NOUN
ejpam-4753	552	10	satisfying	satisfy	VERB
ejpam-4753	552	11	the	the	DET
ejpam-4753	552	12	left	left	ADJ
ejpam-4753	552	13	conditions	condition	NOUN
ejpam-4753	552	14	of	of	ADP
ejpam-4753	552	15	ore	ore	NOUN
ejpam-4753	552	16	formed	form	VERB
ejpam-4753	552	17	of	of	ADP
ejpam-4753	552	18	homogeneous	homogeneous	ADJ
ejpam-4753	552	19	elements	element	NOUN
ejpam-4753	552	20	of	of	ADP
ejpam-4753	552	21	a	a	PRON
ejpam-4753	552	22	,	,	PUNCT
ejpam-4753	552	23	we	we	PRON
ejpam-4753	552	24	have	have	VERB
ejpam-4753	552	25	the	the	DET
ejpam-4753	552	26	following	follow	VERB
ejpam-4753	552	27	longs	long	NOUN
ejpam-4753	552	28	exacts	exact	VERB
ejpam-4753	552	29	sequences	sequence	NOUN
ejpam-4753	552	30	of	of	ADP
ejpam-4753	552	31	a	a	DET
ejpam-4753	552	32	graded	grade	VERB
ejpam-4753	552	33	left	leave	VERB
ejpam-4753	552	34	s−1a−modules	s−1a−module	NOUN
ejpam-4753	552	35	·	·	PUNCT
ejpam-4753	552	36	·	·	PUNCT
ejpam-4753	552	37	·	·	PUNCT
ejpam-4753	552	38	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	552	39	2	2	NUM
ejpam-4753	552	40	)	)	PUNCT
ejpam-4753	552	41	s−1fk(n+2)−→	s−1fk(n+2)−→	NOUN
ejpam-4753	552	42	s−1n(n+	s−1n(n+	VERB
ejpam-4753	552	43	2	2	NUM
ejpam-4753	552	44	)	)	PUNCT
ejpam-4753	552	45	s−1gk(n+2)−→	s−1gk(n+2)−→	X
ejpam-4753	552	46	s−1l(n+	s−1l(n+	NOUN
ejpam-4753	552	47	2	2	NUM
ejpam-4753	552	48	)	)	PUNCT
ejpam-4753	552	49	s−1δn−→	s−1δn−→	NOUN
ejpam-4753	552	50	s−1m(n+	s−1m(n+	ADJ
ejpam-4753	552	51	1	1	NUM
ejpam-4753	552	52	)	)	PUNCT
ejpam-4753	552	53	·	·	PUNCT
ejpam-4753	552	54	·	·	PUNCT
ejpam-4753	552	55	·	·	PUNCT
ejpam-4753	552	56	·	·	PUNCT
ejpam-4753	552	57	·	·	PUNCT
ejpam-4753	552	58	·	·	PUNCT
ejpam-4753	552	59	s−1⊗m(n+2	s−1⊗m(n+2	PROPN
ejpam-4753	552	60	)	)	PUNCT
ejpam-4753	552	61	s−1⊗fk(n+2)−→	s−1⊗fk(n+2)−→	PUNCT
ejpam-4753	552	62	s−1⊗n(n+2	s−1⊗n(n+2	PROPN
ejpam-4753	552	63	)	)	PUNCT
ejpam-4753	552	64	s−1⊗gk(n+2)−→	s−1⊗gk(n+2)−→	PROPN
ejpam-4753	552	65	s−1⊗l(n+2	s−1⊗l(n+2	PROPN
ejpam-4753	552	66	)	)	PUNCT
ejpam-4753	552	67	s−1⊗δn−→	s−1⊗δn−→	X
ejpam-4753	552	68	s−1⊗m(n+1	s−1⊗m(n+1	NOUN
ejpam-4753	552	69	)	)	PUNCT
ejpam-4753	552	70	·	·	PUNCT
ejpam-4753	552	71	·	·	PUNCT
ejpam-4753	552	72	·	·	PUNCT
ejpam-4753	552	73	proof	proof	NOUN
ejpam-4753	552	74	.	.	PUNCT
ejpam-4753	553	1	similarly	similarly	ADV
ejpam-4753	553	2	to	to	ADP
ejpam-4753	553	3	the	the	DET
ejpam-4753	553	4	proof	proof	NOUN
ejpam-4753	553	5	of	of	ADP
ejpam-4753	553	6	theorem	theorem	ADJ
ejpam-4753	553	7	precedent13	precedent13	NOUN
ejpam-4753	553	8	corollary	corollary	ADJ
ejpam-4753	553	9	11	11	NUM
ejpam-4753	553	10	.	.	PUNCT
ejpam-4753	554	1	let	let	VERB
ejpam-4753	554	2	a	a	DET
ejpam-4753	554	3	=	=	SYM
ejpam-4753	554	4	⊕	⊕	PROPN
ejpam-4753	554	5	n∈z	n∈z	VERB
ejpam-4753	554	6	an	an	DET
ejpam-4753	554	7	be	be	AUX
ejpam-4753	554	8	a	a	DET
ejpam-4753	554	9	graded	grade	VERB
ejpam-4753	554	10	duo	duo	NOUN
ejpam-4753	554	11	-	-	PUNCT
ejpam-4753	554	12	ring	ring	NOUN
ejpam-4753	554	13	,	,	PUNCT
ejpam-4753	554	14	for	for	ADP
ejpam-4753	554	15	all	all	PRON
ejpam-4753	554	16	n	n	PRON
ejpam-4753	554	17	∈	∈	PROPN
ejpam-4753	554	18	z	z	NOUN
ejpam-4753	554	19	and	and	CCONJ
ejpam-4753	554	20	for	for	ADP
ejpam-4753	554	21	all	all	DET
ejpam-4753	554	22	short	short	ADJ
ejpam-4753	554	23	exact	exact	ADJ
ejpam-4753	554	24	sequence	sequence	NOUN
ejpam-4753	554	25	0	0	NUM
ejpam-4753	554	26	−→	−→	NOUN
ejpam-4753	554	27	m	m	VERB
ejpam-4753	554	28	f−→	f−→	NOUN
ejpam-4753	554	29	n	n	NOUN
ejpam-4753	554	30	g−→	g−→	NOUN
ejpam-4753	554	31	l	l	NOUN
ejpam-4753	554	32	−→	−→	NOUN
ejpam-4753	554	33	0	0	NUM
ejpam-4753	554	34	of	of	ADP
ejpam-4753	554	35	a	a	DET
ejpam-4753	554	36	graded	grade	VERB
ejpam-4753	554	37	left	leave	VERB
ejpam-4753	554	38	a−modules	a−module	NOUN
ejpam-4753	554	39	we	we	PRON
ejpam-4753	554	40	have	have	VERB
ejpam-4753	554	41	the	the	DET
ejpam-4753	554	42	following	follow	VERB
ejpam-4753	554	43	long	long	ADJ
ejpam-4753	554	44	exact	exact	ADJ
ejpam-4753	554	45	sequence	sequence	NOUN
ejpam-4753	554	46	·	·	PUNCT
ejpam-4753	554	47	·	·	PUNCT
ejpam-4753	554	48	·	·	PUNCT
ejpam-4753	555	1	−→	−→	NOUN
ejpam-4753	555	2	m(n+2	m(n+2	NUM
ejpam-4753	555	3	)	)	PUNCT
ejpam-4753	555	4	fk(n+2)−→	fk(n+2)−→	X
ejpam-4753	555	5	n(n+2	n(n+2	NUM
ejpam-4753	555	6	)	)	PUNCT
ejpam-4753	555	7	gk(n+2)−→	gk(n+2)−→	NOUN
ejpam-4753	555	8	l(n+2	l(n+2	X
ejpam-4753	555	9	)	)	PUNCT
ejpam-4753	555	10	δn−→	δn−→	NUM
ejpam-4753	555	11	m(n+1	m(n+1	NUM
ejpam-4753	555	12	)	)	PUNCT
ejpam-4753	555	13	fk(n+1)−→	fk(n+1)−→	X
ejpam-4753	555	14	n(n+1	n(n+1	NOUN
ejpam-4753	555	15	)	)	PUNCT
ejpam-4753	555	16	−→	−→	NOUN
ejpam-4753	555	17	·	·	PUNCT
ejpam-4753	555	18	·	·	PUNCT
ejpam-4753	555	19	·	·	PUNCT
ejpam-4753	555	20	of	of	ADP
ejpam-4753	555	21	a	a	DET
ejpam-4753	555	22	graded	grade	VERB
ejpam-4753	555	23	left	leave	VERB
ejpam-4753	555	24	a−modules	a−module	NOUN
ejpam-4753	555	25	.	.	PUNCT
ejpam-4753	556	1	furthermore	furthermore	ADV
ejpam-4753	556	2	,	,	PUNCT
ejpam-4753	556	3	if	if	SCONJ
ejpam-4753	556	4	sh	sh	PROPN
ejpam-4753	556	5	is	be	AUX
ejpam-4753	556	6	the	the	DET
ejpam-4753	556	7	part	part	NOUN
ejpam-4753	556	8	of	of	ADP
ejpam-4753	556	9	regulars	regular	NOUN
ejpam-4753	556	10	homogeneous	homogeneous	ADJ
ejpam-4753	556	11	elements	element	NOUN
ejpam-4753	556	12	of	of	ADP
ejpam-4753	556	13	a	a	PRON
ejpam-4753	556	14	,	,	PUNCT
ejpam-4753	556	15	we	we	PRON
ejpam-4753	556	16	have	have	VERB
ejpam-4753	556	17	the	the	DET
ejpam-4753	556	18	following	follow	VERB
ejpam-4753	556	19	longs	long	NOUN
ejpam-4753	556	20	exacts	exact	VERB
ejpam-4753	556	21	sequences	sequence	NOUN
ejpam-4753	556	22	of	of	ADP
ejpam-4753	556	23	a	a	DET
ejpam-4753	556	24	graded	grade	VERB
ejpam-4753	556	25	left	leave	VERB
ejpam-4753	556	26	s	s	PRON
ejpam-4753	556	27	−1	−1	NOUN
ejpam-4753	556	28	h	h	NOUN
ejpam-4753	556	29	a−modules	a−modules	X
ejpam-4753	556	30	·	·	PUNCT
ejpam-4753	556	31	·	·	PUNCT
ejpam-4753	556	32	·	·	PUNCT
ejpam-4753	556	33	s−1	s−1	ADJ
ejpam-4753	556	34	h	h	NOUN
ejpam-4753	556	35	m(n+	m(n+	NOUN
ejpam-4753	556	36	2	2	NUM
ejpam-4753	556	37	)	)	PUNCT
ejpam-4753	556	38	s	s	VERB
ejpam-4753	556	39	−1	−1	NOUN
ejpam-4753	556	40	h	h	NOUN
ejpam-4753	557	1	fk(n+2)−→	fk(n+2)−→	NOUN
ejpam-4753	557	2	s	s	NOUN
ejpam-4753	557	3	−1	−1	NOUN
ejpam-4753	557	4	h	h	NOUN
ejpam-4753	557	5	n(n+	n(n+	NOUN
ejpam-4753	557	6	2	2	NUM
ejpam-4753	557	7	)	)	PUNCT
ejpam-4753	557	8	s	s	PART
ejpam-4753	557	9	−1	−1	NOUN
ejpam-4753	557	10	h	h	NOUN
ejpam-4753	557	11	gk(n+2)−→	gk(n+2)−→	PROPN
ejpam-4753	557	12	s	s	NOUN
ejpam-4753	557	13	−1	−1	NOUN
ejpam-4753	557	14	h	h	NOUN
ejpam-4753	557	15	l(n+	l(n+	ADJ
ejpam-4753	557	16	2	2	NUM
ejpam-4753	557	17	)	)	PUNCT
ejpam-4753	557	18	s	s	VERB
ejpam-4753	557	19	−1	−1	NOUN
ejpam-4753	557	20	h	h	NOUN
ejpam-4753	557	21	δn−→	δn−→	NUM
ejpam-4753	557	22	s	s	NOUN
ejpam-4753	557	23	−1	−1	NOUN
ejpam-4753	557	24	h	h	NOUN
ejpam-4753	557	25	m(n+	m(n+	NOUN
ejpam-4753	557	26	1	1	NUM
ejpam-4753	557	27	)	)	PUNCT
ejpam-4753	557	28	·	·	PUNCT
ejpam-4753	557	29	·	·	PUNCT
ejpam-4753	557	30	·	·	PUNCT
ejpam-4753	557	31	·	·	PUNCT
ejpam-4753	557	32	·	·	PUNCT
ejpam-4753	558	1	·	·	PUNCT
ejpam-4753	558	2	s−1	s−1	PROPN
ejpam-4753	558	3	h	h	NOUN
ejpam-4753	558	4	⊗m(n+2	⊗m(n+2	NUM
ejpam-4753	558	5	)	)	PUNCT
ejpam-4753	558	6	s	s	PART
ejpam-4753	558	7	−1	−1	NOUN
ejpam-4753	558	8	h	h	NOUN
ejpam-4753	559	1	⊗fk(n+2)−→	⊗fk(n+2)−→	NOUN
ejpam-4753	559	2	s	s	NOUN
ejpam-4753	559	3	−1	−1	NOUN
ejpam-4753	559	4	h	h	NOUN
ejpam-4753	559	5	⊗n(n+2	⊗n(n+2	NUM
ejpam-4753	559	6	)	)	PUNCT
ejpam-4753	559	7	s	s	VERB
ejpam-4753	559	8	−1	−1	NOUN
ejpam-4753	559	9	h	h	NOUN
ejpam-4753	559	10	⊗gk(n+2)−→	⊗gk(n+2)−→	PROPN
ejpam-4753	559	11	s	s	PART
ejpam-4753	559	12	−1	−1	NOUN
ejpam-4753	559	13	h	h	NOUN
ejpam-4753	559	14	⊗l(n+2	⊗l(n+2	NUM
ejpam-4753	559	15	)	)	PUNCT
ejpam-4753	559	16	s	s	VERB
ejpam-4753	559	17	−1	−1	NOUN
ejpam-4753	559	18	h	h	NOUN
ejpam-4753	559	19	⊗δn−→	⊗δn−→	VERB
ejpam-4753	559	20	s	s	NOUN
ejpam-4753	559	21	−1	−1	NOUN
ejpam-4753	559	22	h	h	NOUN
ejpam-4753	559	23	⊗m(n+1	⊗m(n+1	PROPN
ejpam-4753	559	24	)	)	PUNCT
ejpam-4753	559	25	·	·	PUNCT
ejpam-4753	559	26	·	·	PUNCT
ejpam-4753	559	27	·	·	PUNCT
ejpam-4753	560	1	proof	proof	NOUN
ejpam-4753	560	2	.	.	PUNCT
ejpam-4753	561	1	it	it	PRON
ejpam-4753	561	2	is	be	AUX
ejpam-4753	561	3	sufficient	sufficient	ADJ
ejpam-4753	561	4	to	to	PART
ejpam-4753	561	5	note	note	VERB
ejpam-4753	561	6	that	that	SCONJ
ejpam-4753	561	7	sh	sh	PROPN
ejpam-4753	561	8	is	be	AUX
ejpam-4753	561	9	a	a	DET
ejpam-4753	561	10	multiplicatively	multiplicatively	ADV
ejpam-4753	561	11	closed	close	VERB
ejpam-4753	561	12	subset	subset	NOUN
ejpam-4753	561	13	satisfying	satisfy	VERB
ejpam-4753	561	14	the	the	DET
ejpam-4753	561	15	left	left	ADJ
ejpam-4753	561	16	conditions	condition	NOUN
ejpam-4753	561	17	of	of	ADP
ejpam-4753	561	18	ore	ore	NOUN
ejpam-4753	561	19	formed	form	VERB
ejpam-4753	561	20	of	of	ADP
ejpam-4753	561	21	homogeneous	homogeneous	ADJ
ejpam-4753	561	22	elements	element	NOUN
ejpam-4753	561	23	of	of	ADP
ejpam-4753	561	24	a	a	PRON
ejpam-4753	561	25	and	and	CCONJ
ejpam-4753	561	26	according	accord	VERB
ejpam-4753	561	27	to	to	ADP
ejpam-4753	561	28	proposition	proposition	NOUN
ejpam-4753	561	29	18	18	NUM
ejpam-4753	561	30	references	reference	NOUN
ejpam-4753	561	31	1938	1938	NUM
ejpam-4753	561	32	proposition	proposition	NOUN
ejpam-4753	561	33	19	19	NUM
ejpam-4753	561	34	.	.	PUNCT
ejpam-4753	562	1	let	let	VERB
ejpam-4753	562	2	a	a	DET
ejpam-4753	562	3	=	=	SYM
ejpam-4753	562	4	⊕	⊕	PROPN
ejpam-4753	562	5	n∈z	n∈z	VERB
ejpam-4753	562	6	an	an	DET
ejpam-4753	562	7	be	be	AUX
ejpam-4753	562	8	a	a	DET
ejpam-4753	562	9	graded	grade	VERB
ejpam-4753	562	10	duo	duo	NOUN
ejpam-4753	562	11	-	-	PUNCT
ejpam-4753	562	12	ring	ring	NOUN
ejpam-4753	562	13	,	,	PUNCT
ejpam-4753	562	14	m	m	VERB
ejpam-4753	562	15	=	=	ADJ
ejpam-4753	562	16	⊕	⊕	PROPN
ejpam-4753	562	17	n∈z	n∈z	VERB
ejpam-4753	562	18	mn	mn	PROPN
ejpam-4753	562	19	a	a	DET
ejpam-4753	562	20	graded	grade	VERB
ejpam-4753	562	21	left	leave	VERB
ejpam-4753	562	22	a−module	a−module	NOUN
ejpam-4753	562	23	and	and	CCONJ
ejpam-4753	562	24	sh	sh	INTJ
ejpam-4753	562	25	be	be	AUX
ejpam-4753	562	26	the	the	DET
ejpam-4753	562	27	part	part	NOUN
ejpam-4753	562	28	formed	form	VERB
ejpam-4753	562	29	of	of	ADP
ejpam-4753	562	30	regulars	regular	NOUN
ejpam-4753	562	31	homogeneous	homogeneous	ADJ
ejpam-4753	562	32	elements	element	NOUN
ejpam-4753	562	33	of	of	ADP
ejpam-4753	562	34	a	a	PRON
ejpam-4753	562	35	,	,	PUNCT
ejpam-4753	562	36	then	then	ADV
ejpam-4753	562	37	for	for	ADP
ejpam-4753	562	38	all	all	DET
ejpam-4753	562	39	n	n	PRON
ejpam-4753	562	40	∈	∈	PROPN
ejpam-4753	562	41	z	z	NOUN
ejpam-4753	562	42	s	s	PART
ejpam-4753	562	43	−1	−1	NOUN
ejpam-4753	562	44	h	h	NOUN
ejpam-4753	562	45	(	(	PUNCT
ejpam-4753	562	46	hn(m∗	hn(m∗	PROPN
ejpam-4753	562	47	)	)	PUNCT
ejpam-4753	562	48	)	)	PUNCT
ejpam-4753	563	1	∼=	∼=	PROPN
ejpam-4753	563	2	s	s	PART
ejpam-4753	563	3	−1	−1	NOUN
ejpam-4753	563	4	h	h	NOUN
ejpam-4753	563	5	(	(	PUNCT
ejpam-4753	563	6	a	a	X
ejpam-4753	563	7	)	)	PUNCT
ejpam-4753	563	8	⊗	⊗	PROPN
ejpam-4753	563	9	m(n+	m(n+	PROPN
ejpam-4753	563	10	2	2	NUM
ejpam-4753	563	11	)	)	PUNCT
ejpam-4753	563	12	.	.	PUNCT
ejpam-4753	564	1	moreover	moreover	ADV
ejpam-4753	564	2	s	s	VERB
ejpam-4753	564	3	−1	−1	NOUN
ejpam-4753	564	4	h	h	NOUN
ejpam-4753	564	5	(	(	PUNCT
ejpam-4753	564	6	hn(m∗	hn(m∗	PROPN
ejpam-4753	564	7	)	)	PUNCT
ejpam-4753	564	8	)	)	PUNCT
ejpam-4753	565	1	∼=	∼=	VERB
ejpam-4753	565	2	hn((s	hn((s	ADJ
ejpam-4753	565	3	−1	−1	NOUN
ejpam-4753	565	4	h	h	NOUN
ejpam-4753	565	5	(	(	PUNCT
ejpam-4753	565	6	m)∗	m)∗	PROPN
ejpam-4753	565	7	)	)	PUNCT
ejpam-4753	565	8	.	.	PUNCT
ejpam-4753	566	1	proof	proof	NOUN
ejpam-4753	566	2	.	.	PUNCT
ejpam-4753	567	1	we	we	PRON
ejpam-4753	567	2	have	have	VERB
ejpam-4753	567	3	hn(m∗	hn(m∗	NOUN
ejpam-4753	567	4	)	)	PUNCT
ejpam-4753	567	5	∼=	∼=	PROPN
ejpam-4753	567	6	m(n+2	m(n+2	X
ejpam-4753	567	7	)	)	PUNCT
ejpam-4753	567	8	and	and	CCONJ
ejpam-4753	567	9	as	as	SCONJ
ejpam-4753	567	10	sh	sh	PROPN
ejpam-4753	567	11	is	be	AUX
ejpam-4753	567	12	a	a	DET
ejpam-4753	567	13	multiplicatively	multiplicatively	ADV
ejpam-4753	567	14	closed	close	VERB
ejpam-4753	567	15	subset	subset	NOUN
ejpam-4753	567	16	satisfying	satisfy	VERB
ejpam-4753	567	17	the	the	DET
ejpam-4753	567	18	left	left	ADJ
ejpam-4753	567	19	conditions	condition	NOUN
ejpam-4753	567	20	of	of	ADP
ejpam-4753	567	21	ore	ore	NOUN
ejpam-4753	567	22	formed	form	VERB
ejpam-4753	567	23	of	of	ADP
ejpam-4753	567	24	homogeneous	homogeneous	ADJ
ejpam-4753	567	25	elements	element	NOUN
ejpam-4753	567	26	of	of	ADP
ejpam-4753	567	27	a	a	DET
ejpam-4753	567	28	then	then	ADV
ejpam-4753	567	29	s	s	PART
ejpam-4753	567	30	−1	−1	NOUN
ejpam-4753	567	31	h	h	NOUN
ejpam-4753	567	32	(	(	PUNCT
ejpam-4753	567	33	hn(m∗	hn(m∗	PROPN
ejpam-4753	567	34	)	)	PUNCT
ejpam-4753	567	35	)	)	PUNCT
ejpam-4753	568	1	∼=	∼=	PROPN
ejpam-4753	568	2	s	s	PART
ejpam-4753	568	3	−1	−1	NOUN
ejpam-4753	568	4	h	h	NOUN
ejpam-4753	568	5	(	(	PUNCT
ejpam-4753	568	6	m(n+	m(n+	PROPN
ejpam-4753	568	7	2	2	NUM
ejpam-4753	568	8	)	)	PUNCT
ejpam-4753	568	9	)	)	PUNCT
ejpam-4753	568	10	or	or	CCONJ
ejpam-4753	568	11	s	s	VERB
ejpam-4753	568	12	−1	−1	NOUN
ejpam-4753	568	13	h	h	NOUN
ejpam-4753	568	14	a	a	DET
ejpam-4753	568	15	⊗	⊗	PROPN
ejpam-4753	568	16	m(n	m(n	X
ejpam-4753	568	17	)	)	PUNCT
ejpam-4753	568	18	∼=	∼=	PROPN
ejpam-4753	568	19	s	s	PART
ejpam-4753	568	20	−1	−1	NOUN
ejpam-4753	568	21	h	h	NOUN
ejpam-4753	568	22	m(n	m(n	NOUN
ejpam-4753	568	23	)	)	PUNCT
ejpam-4753	569	1	so	so	SCONJ
ejpam-4753	569	2	s	s	VERB
ejpam-4753	569	3	−1	−1	NOUN
ejpam-4753	569	4	h	h	NOUN
ejpam-4753	569	5	(	(	PUNCT
ejpam-4753	569	6	hn(m∗	hn(m∗	PROPN
ejpam-4753	569	7	)	)	PUNCT
ejpam-4753	569	8	)	)	PUNCT
ejpam-4753	570	1	∼=	∼=	PROPN
ejpam-4753	570	2	s	s	PART
ejpam-4753	570	3	−1	−1	NOUN
ejpam-4753	570	4	h	h	NOUN
ejpam-4753	570	5	a	a	DET
ejpam-4753	570	6	⊗	⊗	PROPN
ejpam-4753	570	7	(	(	PUNCT
ejpam-4753	570	8	m(n+	m(n+	NOUN
ejpam-4753	570	9	2	2	NUM
ejpam-4753	570	10	)	)	PUNCT
ejpam-4753	570	11	)	)	PUNCT
ejpam-4753	570	12	.	.	PUNCT
ejpam-4753	571	1	on	on	ADP
ejpam-4753	571	2	other	other	ADJ
ejpam-4753	571	3	hand	hand	NOUN
ejpam-4753	571	4	hn((s	hn((	VERB
ejpam-4753	571	5	−1	−1	NOUN
ejpam-4753	571	6	h	h	NOUN
ejpam-4753	571	7	(	(	PUNCT
ejpam-4753	571	8	m))∗	m))∗	PROPN
ejpam-4753	571	9	)	)	PUNCT
ejpam-4753	571	10	∼=	∼=	PROPN
ejpam-4753	571	11	(	(	PUNCT
ejpam-4753	571	12	s	s	NOUN
ejpam-4753	571	13	−1	−1	NOUN
ejpam-4753	571	14	h	h	NOUN
ejpam-4753	571	15	(	(	PUNCT
ejpam-4753	571	16	m))(n	m))(n	PROPN
ejpam-4753	571	17	+	+	CCONJ
ejpam-4753	571	18	2	2	X
ejpam-4753	571	19	)	)	PUNCT
ejpam-4753	571	20	∼=	∼=	PROPN
ejpam-4753	571	21	s	s	PART
ejpam-4753	571	22	−1	−1	NOUN
ejpam-4753	571	23	h	h	NOUN
ejpam-4753	571	24	(	(	PUNCT
ejpam-4753	571	25	m(n	m(n	PROPN
ejpam-4753	571	26	+	+	PROPN
ejpam-4753	571	27	2	2	NUM
ejpam-4753	571	28	)	)	PUNCT
ejpam-4753	571	29	)	)	PUNCT
ejpam-4753	572	1	∼=	∼=	PROPN
ejpam-4753	572	2	s	s	PART
ejpam-4753	572	3	−1	−1	NOUN
ejpam-4753	572	4	h	h	NOUN
ejpam-4753	572	5	(	(	PUNCT
ejpam-4753	572	6	hn(m∗	hn(m∗	PROPN
ejpam-4753	572	7	)	)	PUNCT
ejpam-4753	572	8	)	)	PUNCT
ejpam-4753	572	9	thus	thus	ADV
ejpam-4753	572	10	s	s	VERB
ejpam-4753	572	11	−1	−1	NOUN
ejpam-4753	572	12	h	h	NOUN
ejpam-4753	572	13	(	(	PUNCT
ejpam-4753	572	14	hn(m∗	hn(m∗	PROPN
ejpam-4753	572	15	)	)	PUNCT
ejpam-4753	572	16	)	)	PUNCT
ejpam-4753	573	1	∼=	∼=	VERB
ejpam-4753	573	2	hn((s	hn((s	ADJ
ejpam-4753	573	3	−1	−1	NOUN
ejpam-4753	573	4	h	h	NOUN
ejpam-4753	573	5	(	(	PUNCT
ejpam-4753	573	6	m))∗	m))∗	PROPN
ejpam-4753	573	7	)	)	PUNCT
ejpam-4753	573	8	.	.	PUNCT
ejpam-4753	574	1	corollary	corollary	ADJ
ejpam-4753	574	2	12	12	NUM
ejpam-4753	574	3	.	.	PUNCT
ejpam-4753	575	1	let	let	VERB
ejpam-4753	575	2	a	a	DET
ejpam-4753	575	3	=	=	SYM
ejpam-4753	575	4	⊕	⊕	PROPN
ejpam-4753	575	5	n∈z	n∈z	VERB
ejpam-4753	575	6	an	an	DET
ejpam-4753	575	7	be	be	AUX
ejpam-4753	575	8	a	a	DET
ejpam-4753	575	9	graded	grade	VERB
ejpam-4753	575	10	duo	duo	NOUN
ejpam-4753	575	11	-	-	PUNCT
ejpam-4753	575	12	ring	ring	NOUN
ejpam-4753	575	13	,	,	PUNCT
ejpam-4753	575	14	m	m	VERB
ejpam-4753	575	15	=	=	ADJ
ejpam-4753	575	16	⊕	⊕	PROPN
ejpam-4753	575	17	n∈z	n∈z	VERB
ejpam-4753	575	18	mn	mn	PROPN
ejpam-4753	575	19	and	and	CCONJ
ejpam-4753	575	20	be	be	AUX
ejpam-4753	575	21	a	a	DET
ejpam-4753	575	22	graded	grade	VERB
ejpam-4753	575	23	left	leave	VERB
ejpam-4753	575	24	a−modules	a−module	NOUN
ejpam-4753	575	25	and	and	CCONJ
ejpam-4753	575	26	sh	sh	INTJ
ejpam-4753	575	27	be	be	AUX
ejpam-4753	575	28	the	the	DET
ejpam-4753	575	29	set	set	NOUN
ejpam-4753	575	30	of	of	ADP
ejpam-4753	575	31	all	all	DET
ejpam-4753	575	32	regular	regular	ADJ
ejpam-4753	575	33	homogeneous	homogeneous	ADJ
ejpam-4753	575	34	of	of	ADP
ejpam-4753	575	35	a	a	PRON
ejpam-4753	575	36	,	,	PUNCT
ejpam-4753	575	37	then	then	ADV
ejpam-4753	575	38	for	for	ADP
ejpam-4753	575	39	all	all	DET
ejpam-4753	575	40	n	n	PRON
ejpam-4753	575	41	∈	∈	NOUN
ejpam-4753	575	42	z	z	NOUN
ejpam-4753	575	43	s−1	s−1	PROPN
ejpam-4753	575	44	h	h	NOUN
ejpam-4753	575	45	(	(	PUNCT
ejpam-4753	575	46	hn(m∗	hn(m∗	PROPN
ejpam-4753	575	47	)	)	PUNCT
ejpam-4753	575	48	)	)	PUNCT
ejpam-4753	576	1	∼=	∼=	ADP
ejpam-4753	576	2	s−1	s−1	ADJ
ejpam-4753	576	3	h	h	NOUN
ejpam-4753	576	4	(	(	PUNCT
ejpam-4753	576	5	a	a	NOUN
ejpam-4753	576	6	)	)	PUNCT
ejpam-4753	576	7	⊗	⊗	PROPN
ejpam-4753	576	8	m(n+	m(n+	PROPN
ejpam-4753	576	9	2	2	NUM
ejpam-4753	576	10	)	)	PUNCT
ejpam-4753	576	11	.	.	PUNCT
ejpam-4753	577	1	moreover	moreover	ADV
ejpam-4753	577	2	s−1	s−1	PROPN
ejpam-4753	577	3	h	h	NOUN
ejpam-4753	577	4	(	(	PUNCT
ejpam-4753	577	5	hn(m∗	hn(m∗	PROPN
ejpam-4753	577	6	)	)	PUNCT
ejpam-4753	577	7	)	)	PUNCT
ejpam-4753	578	1	∼=	∼=	VERB
ejpam-4753	578	2	hn((s	hn((s	ADJ
ejpam-4753	578	3	−1	−1	NOUN
ejpam-4753	578	4	h	h	NOUN
ejpam-4753	578	5	(	(	PUNCT
ejpam-4753	578	6	m)∗	m)∗	PROPN
ejpam-4753	578	7	)	)	PUNCT
ejpam-4753	578	8	.	.	PUNCT
ejpam-4753	579	1	proof	proof	NOUN
ejpam-4753	579	2	.	.	PUNCT
ejpam-4753	580	1	it	it	PRON
ejpam-4753	580	2	is	be	AUX
ejpam-4753	580	3	sufficient	sufficient	ADJ
ejpam-4753	580	4	to	to	PART
ejpam-4753	580	5	note	note	VERB
ejpam-4753	580	6	that	that	SCONJ
ejpam-4753	581	1	sh	sh	INTJ
ejpam-4753	581	2	=	=	SYM
ejpam-4753	581	3	sh	sh	INTJ
ejpam-4753	581	4	.	.	PUNCT
ejpam-4753	582	1	5	5	X
ejpam-4753	582	2	.	.	X
ejpam-4753	582	3	conclusion	conclusion	NOUN
ejpam-4753	582	4	in	in	ADP
ejpam-4753	582	5	this	this	DET
ejpam-4753	582	6	article	article	NOUN
ejpam-4753	582	7	,	,	PUNCT
ejpam-4753	582	8	we	we	PRON
ejpam-4753	582	9	study	study	VERB
ejpam-4753	582	10	the	the	DET
ejpam-4753	582	11	localization	localization	NOUN
ejpam-4753	582	12	in	in	ADP
ejpam-4753	582	13	the	the	DET
ejpam-4753	582	14	category	category	NOUN
ejpam-4753	582	15	comp	comp	NOUN
ejpam-4753	582	16	(	(	PUNCT
ejpam-4753	582	17	gr(a	gr(a	NOUN
ejpam-4753	582	18	−mod	−mod	NOUN
ejpam-4753	582	19	)	)	PUNCT
ejpam-4753	582	20	)	)	PUNCT
ejpam-4753	582	21	and	and	CCONJ
ejpam-4753	582	22	we	we	PRON
ejpam-4753	582	23	used	use	VERB
ejpam-4753	582	24	the	the	DET
ejpam-4753	582	25	localization	localization	NOUN
ejpam-4753	582	26	in	in	ADP
ejpam-4753	582	27	the	the	DET
ejpam-4753	582	28	category	category	NOUN
ejpam-4753	582	29	gr(a	gr(a	PUNCT
ejpam-4753	582	30	−	−	PROPN
ejpam-4753	582	31	mod	mod	PROPN
ejpam-4753	582	32	)	)	PUNCT
ejpam-4753	582	33	,	,	PUNCT
ejpam-4753	582	34	and	and	CCONJ
ejpam-4753	582	35	we	we	PRON
ejpam-4753	582	36	proof	proof	VERB
ejpam-4753	582	37	that	that	SCONJ
ejpam-4753	582	38	for	for	SCONJ
ejpam-4753	582	39	all	all	DET
ejpam-4753	582	40	n	n	PRON
ejpam-4753	582	41	∈	∈	PROPN
ejpam-4753	582	42	z	z	NOUN
ejpam-4753	582	43	fixed	fix	VERB
ejpam-4753	582	44	and	and	CCONJ
ejpam-4753	582	45	for	for	ADP
ejpam-4753	582	46	all	all	DET
ejpam-4753	582	47	m	m	NOUN
ejpam-4753	582	48	∈	∈	NOUN
ejpam-4753	582	49	gr(a−mod	gr(a−mod	NOUN
ejpam-4753	582	50	)	)	PUNCT
ejpam-4753	582	51	we	we	PRON
ejpam-4753	582	52	have	have	VERB
ejpam-4753	582	53	:	:	PUNCT
ejpam-4753	583	1	s	s	VERB
ejpam-4753	583	2	−1	−1	NOUN
ejpam-4753	583	3	h	h	NOUN
ejpam-4753	583	4	(	(	PUNCT
ejpam-4753	583	5	(	(	PUNCT
ejpam-4753	583	6	hn	hn	PROPN
ejpam-4753	583	7	◦	◦	NOUN
ejpam-4753	583	8	c)(m	c)(m	VERB
ejpam-4753	583	9	)	)	PUNCT
ejpam-4753	583	10	)	)	PUNCT
ejpam-4753	584	1	∼=	∼=	PROPN
ejpam-4753	584	2	hn(ch	hn(ch	NOUN
ejpam-4753	584	3	◦	◦	VERB
ejpam-4753	584	4	s−1	s−1	PROPN
ejpam-4753	584	5	h	h	NOUN
ejpam-4753	584	6	)	)	PUNCT
ejpam-4753	584	7	(	(	PUNCT
ejpam-4753	584	8	m	m	NOUN
ejpam-4753	584	9	)	)	PUNCT
ejpam-4753	584	10	)	)	PUNCT
ejpam-4753	584	11	.	.	PUNCT
ejpam-4753	585	1	references	reference	NOUN
ejpam-4753	585	2	[	[	X
ejpam-4753	585	3	1	1	NUM
ejpam-4753	585	4	]	]	X
ejpam-4753	585	5	o	o	X
ejpam-4753	585	6	c	c	NOUN
ejpam-4753	585	7	ahmed	ahmed	PROPN
ejpam-4753	585	8	.	.	PUNCT
ejpam-4753	586	1	graduation	graduation	NOUN
ejpam-4753	586	2	et	et	PROPN
ejpam-4753	586	3	filtration	filtration	NOUN
ejpam-4753	586	4	des	des	X
ejpam-4753	586	5	modules	module	NOUN
ejpam-4753	586	6	de	de	X
ejpam-4753	586	7	fractions	fraction	NOUN
ejpam-4753	586	8	sur	sur	PROPN
ejpam-4753	586	9	des	des	X
ejpam-4753	586	10	anneaux	anneaux	PROPN
ejpam-4753	586	11	non	non	PROPN
ejpam-4753	586	12	nécessairement	nécessairement	PROPN
ejpam-4753	586	13	commutatifs	commutatifs	PROPN
ejpam-4753	586	14	.	.	PUNCT
ejpam-4753	587	1	phd	phd	NOUN
ejpam-4753	587	2	thesis	thesis	PROPN
ejpam-4753	587	3	,	,	PUNCT
ejpam-4753	587	4	2016	2016	NUM
ejpam-4753	587	5	.	.	PUNCT
ejpam-4753	588	1	[	[	X
ejpam-4753	588	2	2	2	X
ejpam-4753	588	3	]	]	X
ejpam-4753	588	4	o	o	X
ejpam-4753	588	5	c	c	NOUN
ejpam-4753	588	6	ahmed	ahmed	PROPN
ejpam-4753	588	7	,	,	PUNCT
ejpam-4753	588	8	m	m	PROPN
ejpam-4753	588	9	f	f	NOUN
ejpam-4753	588	10	maaouia	maaouia	NOUN
ejpam-4753	588	11	,	,	PUNCT
ejpam-4753	588	12	and	and	CCONJ
ejpam-4753	588	13	m	m	PROPN
ejpam-4753	588	14	sanghare	sanghare	ADJ
ejpam-4753	588	15	.	.	PUNCT
ejpam-4753	589	1	graduation	graduation	NOUN
ejpam-4753	589	2	of	of	ADP
ejpam-4753	589	3	module	module	NOUN
ejpam-4753	589	4	of	of	ADP
ejpam-4753	589	5	fraction	fraction	NOUN
ejpam-4753	589	6	on	on	ADP
ejpam-4753	589	7	a	a	DET
ejpam-4753	589	8	graded	grade	VERB
ejpam-4753	589	9	domain	domain	NOUN
ejpam-4753	589	10	ring	ring	NOUN
ejpam-4753	589	11	not	not	PART
ejpam-4753	589	12	necessarily	necessarily	ADV
ejpam-4753	589	13	commutative	commutative	ADJ
ejpam-4753	589	14	.	.	PUNCT
ejpam-4753	590	1	international	international	ADJ
ejpam-4753	590	2	journal	journal	PROPN
ejpam-4753	590	3	of	of	ADP
ejpam-4753	590	4	algebra	algebra	PROPN
ejpam-4753	590	5	,	,	PUNCT
ejpam-4753	590	6	10:457–474	10:457–474	PROPN
ejpam-4753	590	7	,	,	PUNCT
ejpam-4753	590	8	2015	2015	NUM
ejpam-4753	590	9	.	.	PUNCT
ejpam-4753	591	1	references	reference	NOUN
ejpam-4753	591	2	1939	1939	NUM
ejpam-4753	592	1	[	[	X
ejpam-4753	592	2	3	3	NUM
ejpam-4753	592	3	]	]	PUNCT
ejpam-4753	592	4	e	e	PROPN
ejpam-4753	592	5	c	c	PROPN
ejpam-4753	592	6	dade	dade	PROPN
ejpam-4753	592	7	.	.	PUNCT
ejpam-4753	592	8	group	group	PROPN
ejpam-4753	592	9	graded	grade	VERB
ejpam-4753	592	10	rings	ring	NOUN
ejpam-4753	592	11	and	and	CCONJ
ejpam-4753	592	12	modules	module	NOUN
ejpam-4753	592	13	.	.	PUNCT
ejpam-4753	593	1	math	math	NOUN
ejpam-4753	593	2	.	.	PUNCT
ejpam-4753	594	1	z.	z.	PROPN
ejpam-4753	594	2	,	,	PUNCT
ejpam-4753	594	3	pages	page	VERB
ejpam-4753	594	4	241–262	241–262	NUM
ejpam-4753	594	5	,	,	PUNCT
ejpam-4753	594	6	1980	1980	NUM
ejpam-4753	594	7	.	.	PUNCT
ejpam-4753	595	1	[	[	X
ejpam-4753	595	2	4	4	NUM
ejpam-4753	595	3	]	]	X
ejpam-4753	595	4	d	d	X
ejpam-4753	595	5	faye	faye	PROPN
ejpam-4753	595	6	,	,	PUNCT
ejpam-4753	595	7	m	m	PROPN
ejpam-4753	595	8	f	f	NOUN
ejpam-4753	595	9	maaouia	maaouia	NOUN
ejpam-4753	595	10	,	,	PUNCT
ejpam-4753	595	11	and	and	CCONJ
ejpam-4753	595	12	m	m	PROPN
ejpam-4753	595	13	sanghare	sanghare	ADJ
ejpam-4753	595	14	.	.	PUNCT
ejpam-4753	596	1	localization	localization	NOUN
ejpam-4753	596	2	in	in	ADP
ejpam-4753	596	3	a	a	DET
ejpam-4753	596	4	duo	duo	NOUN
ejpam-4753	596	5	-	-	PUNCT
ejpam-4753	596	6	ring	ring	NOUN
ejpam-4753	596	7	and	and	CCONJ
ejpam-4753	596	8	polynomials	polynomial	NOUN
ejpam-4753	596	9	algebra	algebra	NOUN
ejpam-4753	596	10	.	.	PUNCT
ejpam-4753	597	1	non	non	ADJ
ejpam-4753	597	2	-	-	ADJ
ejpam-4753	597	3	associative	associative	ADJ
ejpam-4753	597	4	algebra	algebra	NOUN
ejpam-4753	597	5	and	and	CCONJ
ejpam-4753	597	6	operator	operator	NOUN
ejpam-4753	597	7	theory	theory	NOUN
ejpam-4753	597	8	,	,	PUNCT
ejpam-4753	597	9	springer	springer	NOUN
ejpam-4753	597	10	proceedings	proceeding	NOUN
ejpam-4753	597	11	in	in	ADP
ejpam-4753	597	12	mathematics	mathematic	NOUN
ejpam-4753	597	13	and	and	CCONJ
ejpam-4753	597	14	statistics	statistic	NOUN
ejpam-4753	597	15	,	,	PUNCT
ejpam-4753	597	16	switzerland	switzerland	PROPN
ejpam-4753	597	17	,	,	PUNCT
ejpam-4753	597	18	160:183–191	160:183–191	NUM
ejpam-4753	597	19	,	,	PUNCT
ejpam-4753	597	20	2016	2016	NUM
ejpam-4753	597	21	.	.	PUNCT
ejpam-4753	598	1	[	[	X
ejpam-4753	598	2	5	5	NUM
ejpam-4753	598	3	]	]	PUNCT
ejpam-4753	598	4	m	m	PROPN
ejpam-4753	598	5	f	f	NOUN
ejpam-4753	598	6	maaouia	maaouia	NOUN
ejpam-4753	598	7	.	.	PUNCT
ejpam-4753	599	1	localisation	localisation	NOUN
ejpam-4753	599	2	et	et	PROPN
ejpam-4753	599	3	enveloppe	enveloppe	PROPN
ejpam-4753	599	4	plate	plate	PROPN
ejpam-4753	599	5	dans	dans	PROPN
ejpam-4753	599	6	un	un	PROPN
ejpam-4753	599	7	duo	duo	PROPN
ejpam-4753	599	8	-	-	PUNCT
ejpam-4753	599	9	anneau	anneau	NOUN
ejpam-4753	599	10	.	.	PUNCT
ejpam-4753	600	1	phd	phd	NOUN
ejpam-4753	600	2	thesis	thesis	NOUN
ejpam-4753	600	3	,	,	PUNCT
ejpam-4753	600	4	2003	2003	NUM
ejpam-4753	600	5	.	.	PUNCT
ejpam-4753	601	1	[	[	X
ejpam-4753	601	2	6	6	NUM
ejpam-4753	601	3	]	]	PUNCT
ejpam-4753	601	4	m	m	PROPN
ejpam-4753	601	5	f	f	NOUN
ejpam-4753	601	6	maaouia	maaouia	NOUN
ejpam-4753	601	7	.	.	PUNCT
ejpam-4753	602	1	les	le	NOUN
ejpam-4753	602	2	anneaux	anneaux	ADV
ejpam-4753	602	3	et	et	PROPN
ejpam-4753	602	4	modules	module	NOUN
ejpam-4753	602	5	de	de	X
ejpam-4753	602	6	fractions	fraction	NOUN
ejpam-4753	602	7	,	,	PUNCT
ejpam-4753	602	8	enveloppe	enveloppe	PROPN
ejpam-4753	602	9	et	et	PROPN
ejpam-4753	602	10	couverture	couverture	PROPN
ejpam-4753	602	11	plate	plate	PROPN
ejpam-4753	602	12	dans	dans	PROPN
ejpam-4753	602	13	un	un	PROPN
ejpam-4753	602	14	duo	duo	NOUN
ejpam-4753	602	15	-	-	PUNCT
ejpam-4753	602	16	anneaux	anneaux	ADV
ejpam-4753	602	17	.	.	PUNCT
ejpam-4753	603	1	phd	phd	NOUN
ejpam-4753	603	2	thesis	thesis	PROPN
ejpam-4753	603	3	,	,	PUNCT
ejpam-4753	603	4	faculté	faculté	PROPN
ejpam-4753	603	5	des	des	PROPN
ejpam-4753	603	6	sciences	sciences	PROPN
ejpam-4753	603	7	et	et	PROPN
ejpam-4753	603	8	techniques	technique	NOUN
ejpam-4753	603	9	,	,	PUNCT
ejpam-4753	603	10	ucad	ucad	ADJ
ejpam-4753	603	11	,	,	PUNCT
ejpam-4753	603	12	dakar	dakar	NOUN
ejpam-4753	603	13	,	,	PUNCT
ejpam-4753	603	14	thèse	thèse	PROPN
ejpam-4753	603	15	d’état	d’état	PROPN
ejpam-4753	603	16	,	,	PUNCT
ejpam-4753	603	17	2011	2011	NUM
ejpam-4753	603	18	.	.	PUNCT
ejpam-4753	604	1	[	[	X
ejpam-4753	604	2	7	7	X
ejpam-4753	604	3	]	]	X
ejpam-4753	604	4	m	m	PROPN
ejpam-4753	604	5	f	f	NOUN
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ejpam-4753	604	7	and	and	CCONJ
ejpam-4753	604	8	m	m	AUX
ejpam-4753	604	9	sanghare	sanghare	ADJ
ejpam-4753	604	10	.	.	PUNCT
ejpam-4753	605	1	localisation	localisation	NOUN
ejpam-4753	605	2	dans	dan	NOUN
ejpam-4753	605	3	les	les	VERB
ejpam-4753	605	4	duo	duo	NOUN
ejpam-4753	605	5	-	-	PUNCT
ejpam-4753	605	6	anneaux	anneaux	ADV
ejpam-4753	605	7	.	.	PUNCT
ejpam-4753	606	1	afrika	afrika	PROPN
ejpam-4753	606	2	mathematika	mathematika	PROPN
ejpam-4753	606	3	,	,	PUNCT
ejpam-4753	606	4	20:163–179	20:163–179	PROPN
ejpam-4753	606	5	,	,	PUNCT
ejpam-4753	606	6	2009	2009	NUM
ejpam-4753	606	7	.	.	PUNCT
ejpam-4753	607	1	[	[	X
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ejpam-4753	607	3	]	]	X
ejpam-4753	607	4	m	m	PROPN
ejpam-4753	607	5	f	f	NOUN
ejpam-4753	607	6	maaouia	maaouia	PROPN
ejpam-4753	607	7	and	and	CCONJ
ejpam-4753	607	8	m	m	NOUN
ejpam-4753	607	9	sanghare	sanghare	ADJ
ejpam-4753	607	10	.	.	PUNCT
ejpam-4753	608	1	module	module	NOUN
ejpam-4753	608	2	de	de	X
ejpam-4753	608	3	fractions	fraction	NOUN
ejpam-4753	608	4	,	,	PUNCT
ejpam-4753	608	5	sous	sous	ADJ
ejpam-4753	608	6	-	-	PUNCT
ejpam-4753	608	7	modules	module	NOUN
ejpam-4753	608	8	s−saturée	s−saturée	PROPN
ejpam-4753	608	9	et	et	NOUN
ejpam-4753	608	10	foncteur	foncteur	NOUN
ejpam-4753	608	11	s−1	s−1	PROPN
ejpam-4753	608	12	.	.	PUNCT
ejpam-4753	609	1	international	international	ADJ
ejpam-4753	609	2	journal	journal	PROPN
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ejpam-4753	609	4	algebra	algebra	PROPN
ejpam-4753	609	5	,	,	PUNCT
ejpam-4753	609	6	6	6	NUM
ejpam-4753	609	7	,	,	PUNCT
ejpam-4753	609	8	2012	2012	NUM
ejpam-4753	609	9	.	.	PUNCT
ejpam-4753	610	1	[	[	X
ejpam-4753	610	2	9	9	NUM
ejpam-4753	610	3	]	]	X
ejpam-4753	610	4	m	m	PROPN
ejpam-4753	610	5	mendelson	mendelson	PROPN
ejpam-4753	610	6	.	.	PUNCT
ejpam-4753	610	7	graded	grade	VERB
ejpam-4753	610	8	rings	ring	NOUN
ejpam-4753	610	9	,	,	PUNCT
ejpam-4753	610	10	modules	module	NOUN
ejpam-4753	610	11	and	and	CCONJ
ejpam-4753	610	12	algebras	algebra	NOUN
ejpam-4753	610	13	.	.	PROPN
ejpam-4753	610	14	1970	1970	NUM
ejpam-4753	610	15	.	.	PUNCT
ejpam-4753	611	1	[	[	X
ejpam-4753	611	2	10	10	NUM
ejpam-4753	611	3	]	]	X
ejpam-4753	611	4	c	c	NOUN
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ejpam-4753	611	6	.	.	PUNCT
ejpam-4753	612	1	strongly	strongly	ADV
ejpam-4753	612	2	graded	grade	VERB
ejpam-4753	612	3	rings	ring	NOUN
ejpam-4753	612	4	of	of	ADP
ejpam-4753	612	5	finite	finite	ADJ
ejpam-4753	612	6	groups	group	NOUN
ejpam-4753	612	7	.	.	PUNCT
ejpam-4753	613	1	comm	comm	NOUN
ejpam-4753	613	2	.	.	PUNCT
ejpam-4753	614	1	algebra	algebra	PROPN
ejpam-4753	614	2	,	,	PUNCT
ejpam-4753	614	3	10:1033–1071	10:1033–1071	NUM
ejpam-4753	614	4	,	,	PUNCT
ejpam-4753	614	5	1981	1981	NUM
ejpam-4753	614	6	.	.	PUNCT
ejpam-4753	615	1	[	[	X
ejpam-4753	615	2	11	11	NUM
ejpam-4753	615	3	]	]	X
ejpam-4753	615	4	c	c	X
ejpam-4753	615	5	nastasescu	nastasescu	NOUN
ejpam-4753	615	6	and	and	CCONJ
ejpam-4753	615	7	f	f	PROPN
ejpam-4753	615	8	v	v	NUM
ejpam-4753	615	9	oystaeyen	oystaeyen	NOUN
ejpam-4753	615	10	.	.	PUNCT
ejpam-4753	616	1	graded	grade	VERB
ejpam-4753	616	2	ring	ring	NOUN
ejpam-4753	616	3	theory	theory	NOUN
ejpam-4753	616	4	.	.	PUNCT
ejpam-4753	617	1	mathematical	mathematical	ADJ
ejpam-4753	617	2	library	library	PROPN
ejpam-4753	617	3	28	28	NUM
ejpam-4753	617	4	,	,	PUNCT
ejpam-4753	617	5	north	north	NOUN
ejpam-4753	617	6	holland	holland	PROPN
ejpam-4753	617	7	,	,	PUNCT
ejpam-4753	617	8	amesterdam	amesterdam	PROPN
ejpam-4753	617	9	,	,	PUNCT
ejpam-4753	617	10	,	,	PUNCT
ejpam-4753	617	11	1982	1982	NUM
ejpam-4753	617	12	.	.	PUNCT