id	sid	tid	token	lemma	pos
ejpam-3889	1	1	european	european	PROPN
ejpam-3889	1	2	journal	journal	PROPN
ejpam-3889	1	3	of	of	ADP
ejpam-3889	1	4	pure	pure	ADJ
ejpam-3889	1	5	and	and	CCONJ
ejpam-3889	1	6	applied	apply	VERB
ejpam-3889	1	7	mathematics	mathematic	NOUN
ejpam-3889	1	8	vol	vol	NOUN
ejpam-3889	1	9	.	.	PUNCT
ejpam-3889	2	1	14	14	NUM
ejpam-3889	2	2	,	,	PUNCT
ejpam-3889	2	3	no	no	INTJ
ejpam-3889	2	4	.	.	NOUN
ejpam-3889	2	5	2	2	NUM
ejpam-3889	2	6	,	,	PUNCT
ejpam-3889	2	7	2021	2021	NUM
ejpam-3889	2	8	,	,	PUNCT
ejpam-3889	2	9	404	404	NUM
ejpam-3889	2	10	-	-	SYM
ejpam-3889	2	11	422	422	NUM
ejpam-3889	2	12	issn	issn	PROPN
ejpam-3889	2	13	1307	1307	NUM
ejpam-3889	2	14	-	-	SYM
ejpam-3889	2	15	5543	5543	NUM
ejpam-3889	2	16	–	–	PUNCT
ejpam-3889	2	17	ejpam.com	ejpam.com	X
ejpam-3889	2	18	published	publish	VERB
ejpam-3889	2	19	by	by	ADP
ejpam-3889	2	20	new	new	PROPN
ejpam-3889	2	21	york	york	PROPN
ejpam-3889	2	22	business	business	PROPN
ejpam-3889	2	23	global	global	ADJ
ejpam-3889	2	24	localization	localization	NOUN
ejpam-3889	2	25	of	of	ADP
ejpam-3889	2	26	hopfian	hopfian	ADJ
ejpam-3889	2	27	and	and	CCONJ
ejpam-3889	2	28	cohopfian	cohopfian	ADJ
ejpam-3889	2	29	objects	object	NOUN
ejpam-3889	2	30	in	in	ADP
ejpam-3889	2	31	the	the	DET
ejpam-3889	2	32	categories	category	NOUN
ejpam-3889	2	33	of	of	ADP
ejpam-3889	2	34	a−mod	a−mod	NOUN
ejpam-3889	2	35	,	,	PUNCT
ejpam-3889	2	36	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	2	37	)	)	PUNCT
ejpam-3889	2	38	and	and	CCONJ
ejpam-3889	2	39	comp	comp	NOUN
ejpam-3889	2	40	(	(	PUNCT
ejpam-3889	2	41	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	2	42	)	)	PUNCT
ejpam-3889	2	43	)	)	PUNCT
ejpam-3889	2	44	seydina	seydina	PROPN
ejpam-3889	2	45	ababacar	ababacar	PROPN
ejpam-3889	2	46	balde1,∗	balde1,∗	PROPN
ejpam-3889	2	47	,	,	PUNCT
ejpam-3889	2	48	mohamed	mohamed	PROPN
ejpam-3889	2	49	ben	ben	PROPN
ejpam-3889	2	50	faraj	faraj	PROPN
ejpam-3889	2	51	ben	ben	PROPN
ejpam-3889	2	52	maaouia2	maaouia2	PROPN
ejpam-3889	2	53	,	,	PUNCT
ejpam-3889	2	54	ahmed	ahmed	PROPN
ejpam-3889	2	55	ould	ould	PROPN
ejpam-3889	2	56	chbih3	chbih3	PROPN
ejpam-3889	2	57	1	1	NUM
ejpam-3889	2	58	applied	apply	VERB
ejpam-3889	2	59	mathematics	mathematic	NOUN
ejpam-3889	2	60	,	,	PUNCT
ejpam-3889	2	61	ufr	ufr	NOUN
ejpam-3889	2	62	-	-	PUNCT
ejpam-3889	2	63	sat	sit	VERB
ejpam-3889	2	64	/	/	SYM
ejpam-3889	2	65	gaston	gaston	PROPN
ejpam-3889	2	66	berger	berger	PROPN
ejpam-3889	2	67	,	,	PUNCT
ejpam-3889	2	68	university	university	NOUN
ejpam-3889	2	69	,	,	PUNCT
ejpam-3889	2	70	saint	saint	NOUN
ejpam-3889	2	71	-	-	PUNCT
ejpam-3889	2	72	louis	louis	NOUN
ejpam-3889	2	73	,	,	PUNCT
ejpam-3889	2	74	senegal	senegal	ADJ
ejpam-3889	2	75	abstract	abstract	NOUN
ejpam-3889	2	76	.	.	PUNCT
ejpam-3889	3	1	the	the	DET
ejpam-3889	3	2	aim	aim	NOUN
ejpam-3889	3	3	of	of	ADP
ejpam-3889	3	4	this	this	DET
ejpam-3889	3	5	paper	paper	NOUN
ejpam-3889	3	6	is	be	AUX
ejpam-3889	3	7	to	to	PART
ejpam-3889	3	8	study	study	VERB
ejpam-3889	3	9	the	the	DET
ejpam-3889	3	10	localization	localization	NOUN
ejpam-3889	3	11	of	of	ADP
ejpam-3889	3	12	hopfian	hopfian	ADJ
ejpam-3889	3	13	and	and	CCONJ
ejpam-3889	3	14	cohopfian	cohopfian	ADJ
ejpam-3889	3	15	objects	object	NOUN
ejpam-3889	3	16	in	in	ADP
ejpam-3889	3	17	the	the	DET
ejpam-3889	3	18	categories	category	NOUN
ejpam-3889	3	19	a	a	DET
ejpam-3889	3	20	−	−	PROPN
ejpam-3889	3	21	mod	mod	NOUN
ejpam-3889	3	22	of	of	ADP
ejpam-3889	3	23	left	leave	VERB
ejpam-3889	3	24	a	a	DET
ejpam-3889	3	25	-	-	PUNCT
ejpam-3889	3	26	modules	module	NOUN
ejpam-3889	3	27	,	,	PUNCT
ejpam-3889	3	28	agr(a	agr(a	PROPN
ejpam-3889	3	29	−	−	PROPN
ejpam-3889	3	30	mod	mod	NOUN
ejpam-3889	3	31	)	)	PUNCT
ejpam-3889	3	32	of	of	ADP
ejpam-3889	3	33	graded	grade	VERB
ejpam-3889	3	34	left	leave	VERB
ejpam-3889	3	35	a	a	DET
ejpam-3889	3	36	-	-	PUNCT
ejpam-3889	3	37	modules	module	NOUN
ejpam-3889	3	38	and	and	CCONJ
ejpam-3889	3	39	comp	comp	NOUN
ejpam-3889	3	40	(	(	PUNCT
ejpam-3889	3	41	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	3	42	)	)	PUNCT
ejpam-3889	3	43	)	)	PUNCT
ejpam-3889	3	44	of	of	ADP
ejpam-3889	3	45	complex	complex	ADJ
ejpam-3889	3	46	sequences	sequence	NOUN
ejpam-3889	3	47	associated	associate	VERB
ejpam-3889	3	48	to	to	AUX
ejpam-3889	3	49	graded	grade	VERB
ejpam-3889	3	50	left	leave	VERB
ejpam-3889	3	51	a	a	DET
ejpam-3889	3	52	-	-	PUNCT
ejpam-3889	3	53	modules	module	NOUN
ejpam-3889	3	54	.	.	PUNCT
ejpam-3889	4	1	we	we	PRON
ejpam-3889	4	2	have	have	VERB
ejpam-3889	4	3	among	among	ADP
ejpam-3889	4	4	others	other	NOUN
ejpam-3889	4	5	the	the	DET
ejpam-3889	4	6	main	main	ADJ
ejpam-3889	4	7	following	follow	VERB
ejpam-3889	4	8	results	result	NOUN
ejpam-3889	4	9	(	(	PUNCT
ejpam-3889	4	10	i	i	NOUN
ejpam-3889	4	11	)	)	PUNCT
ejpam-3889	4	12	let	let	VERB
ejpam-3889	4	13	m	m	PRON
ejpam-3889	4	14	be	be	AUX
ejpam-3889	4	15	a	a	DET
ejpam-3889	4	16	noetherian	noetherian	ADJ
ejpam-3889	4	17	graded	grade	VERB
ejpam-3889	4	18	left	leave	VERB
ejpam-3889	4	19	a	a	DET
ejpam-3889	4	20	-	-	PUNCT
ejpam-3889	4	21	module	module	NOUN
ejpam-3889	4	22	,	,	PUNCT
ejpam-3889	4	23	s	s	VERB
ejpam-3889	4	24	a	a	DET
ejpam-3889	4	25	saturated	saturate	VERB
ejpam-3889	4	26	multiplicative	multiplicative	ADJ
ejpam-3889	4	27	part	part	NOUN
ejpam-3889	4	28	formed	form	VERB
ejpam-3889	4	29	by	by	ADP
ejpam-3889	4	30	the	the	DET
ejpam-3889	4	31	non	non	ADJ
ejpam-3889	4	32	-	-	ADJ
ejpam-3889	4	33	zero	zero	ADJ
ejpam-3889	4	34	homogeneous	homogeneous	ADJ
ejpam-3889	4	35	elements	element	NOUN
ejpam-3889	4	36	of	of	ADP
ejpam-3889	4	37	a	a	DET
ejpam-3889	4	38	verifying	verifying	NOUN
ejpam-3889	4	39	the	the	DET
ejpam-3889	4	40	left	left	ADJ
ejpam-3889	4	41	ore	ore	NOUN
ejpam-3889	4	42	conditions	condition	NOUN
ejpam-3889	4	43	,	,	PUNCT
ejpam-3889	4	44	n	n	CCONJ
ejpam-3889	4	45	a	a	DET
ejpam-3889	4	46	submodule	submodule	NOUN
ejpam-3889	4	47	of	of	ADP
ejpam-3889	4	48	m	m	PROPN
ejpam-3889	4	49	,	,	PUNCT
ejpam-3889	4	50	m∗	m∗	PROPN
ejpam-3889	4	51	is	be	AUX
ejpam-3889	4	52	a	a	DET
ejpam-3889	4	53	noetherian	noetherian	ADJ
ejpam-3889	4	54	quasi	quasi	ADJ
ejpam-3889	4	55	-	-	ADJ
ejpam-3889	4	56	injective	injective	ADJ
ejpam-3889	4	57	complex	complex	ADJ
ejpam-3889	4	58	sequence	sequence	NOUN
ejpam-3889	4	59	associated	associate	VERB
ejpam-3889	4	60	with	with	ADP
ejpam-3889	4	61	m	m	NOUN
ejpam-3889	4	62	and	and	CCONJ
ejpam-3889	4	63	n∗	n∗	PROPN
ejpam-3889	4	64	is	be	AUX
ejpam-3889	4	65	an	an	DET
ejpam-3889	4	66	essential	essential	ADJ
ejpam-3889	4	67	and	and	CCONJ
ejpam-3889	4	68	completely	completely	ADV
ejpam-3889	4	69	invariant	invariant	ADJ
ejpam-3889	4	70	complex	complex	ADJ
ejpam-3889	4	71	sub	sub	NOUN
ejpam-3889	4	72	-	-	NOUN
ejpam-3889	4	73	sequence	sequence	NOUN
ejpam-3889	4	74	of	of	ADP
ejpam-3889	4	75	m∗.	m∗.	PROPN
ejpam-3889	4	76	then	then	ADV
ejpam-3889	4	77	,	,	PUNCT
ejpam-3889	4	78	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	4	79	)	)	PUNCT
ejpam-3889	4	80	the	the	DET
ejpam-3889	4	81	complex	complex	ADJ
ejpam-3889	4	82	sequence	sequence	NOUN
ejpam-3889	4	83	of	of	ADP
ejpam-3889	4	84	morphisms	morphism	NOUN
ejpam-3889	4	85	of	of	ADP
ejpam-3889	4	86	left	left	ADJ
ejpam-3889	4	87	s−1a	s−1a	NOUN
ejpam-3889	4	88	-	-	PUNCT
ejpam-3889	4	89	modules	module	NOUN
ejpam-3889	4	90	is	be	AUX
ejpam-3889	4	91	cohopfian	cohopfian	ADJ
ejpam-3889	4	92	if	if	SCONJ
ejpam-3889	4	93	,	,	PUNCT
ejpam-3889	4	94	and	and	CCONJ
ejpam-3889	4	95	only	only	ADV
ejpam-3889	4	96	,	,	PUNCT
ejpam-3889	4	97	if	if	SCONJ
ejpam-3889	4	98	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	4	99	)	)	PUNCT
ejpam-3889	4	100	is	be	AUX
ejpam-3889	4	101	cohopfian	cohopfian	ADJ
ejpam-3889	4	102	;	;	PUNCT
ejpam-3889	4	103	(	(	PUNCT
ejpam-3889	4	104	ii	ii	NOUN
ejpam-3889	4	105	)	)	PUNCT
ejpam-3889	4	106	let	let	VERB
ejpam-3889	4	107	m	m	PRON
ejpam-3889	4	108	be	be	AUX
ejpam-3889	4	109	a	a	DET
ejpam-3889	4	110	graded	grade	VERB
ejpam-3889	4	111	left	leave	VERB
ejpam-3889	4	112	a	a	DET
ejpam-3889	4	113	-	-	PUNCT
ejpam-3889	4	114	module	module	NOUN
ejpam-3889	4	115	and	and	CCONJ
ejpam-3889	4	116	s	s	VERB
ejpam-3889	4	117	a	a	DET
ejpam-3889	4	118	saturated	saturate	VERB
ejpam-3889	4	119	multiplicative	multiplicative	ADJ
ejpam-3889	4	120	part	part	NOUN
ejpam-3889	4	121	formed	form	VERB
ejpam-3889	4	122	by	by	ADP
ejpam-3889	4	123	the	the	DET
ejpam-3889	4	124	non	non	ADJ
ejpam-3889	4	125	-	-	ADJ
ejpam-3889	4	126	zero	zero	ADJ
ejpam-3889	4	127	homogeneous	homogeneous	ADJ
ejpam-3889	4	128	elements	element	NOUN
ejpam-3889	4	129	of	of	ADP
ejpam-3889	4	130	a	a	DET
ejpam-3889	4	131	verifying	verifying	NOUN
ejpam-3889	4	132	the	the	DET
ejpam-3889	4	133	left	left	ADJ
ejpam-3889	4	134	ore	ore	NOUN
ejpam-3889	4	135	conditions	condition	NOUN
ejpam-3889	4	136	.	.	PUNCT
ejpam-3889	5	1	if	if	SCONJ
ejpam-3889	5	2	m∗	m∗	PROPN
ejpam-3889	5	3	is	be	AUX
ejpam-3889	5	4	a	a	DET
ejpam-3889	5	5	hopfian	hopfian	ADJ
ejpam-3889	5	6	,	,	PUNCT
ejpam-3889	5	7	noetherian	noetherian	ADJ
ejpam-3889	5	8	and	and	CCONJ
ejpam-3889	5	9	quasi	quasi	ADJ
ejpam-3889	5	10	-	-	ADJ
ejpam-3889	5	11	injective	injective	ADJ
ejpam-3889	5	12	complex	complex	ADJ
ejpam-3889	5	13	sequence	sequence	NOUN
ejpam-3889	5	14	associated	associate	VERB
ejpam-3889	5	15	with	with	ADP
ejpam-3889	5	16	m	m	PROPN
ejpam-3889	5	17	,	,	PUNCT
ejpam-3889	5	18	then	then	ADV
ejpam-3889	5	19	the	the	DET
ejpam-3889	5	20	complex	complex	ADJ
ejpam-3889	5	21	sequence	sequence	NOUN
ejpam-3889	5	22	of	of	ADP
ejpam-3889	5	23	morphisms	morphism	NOUN
ejpam-3889	5	24	of	of	ADP
ejpam-3889	5	25	left	left	ADJ
ejpam-3889	5	26	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	5	27	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	5	28	)	)	PUNCT
ejpam-3889	5	29	has	have	VERB
ejpam-3889	5	30	the	the	DET
ejpam-3889	5	31	following	follow	VERB
ejpam-3889	5	32	property	property	NOUN
ejpam-3889	5	33	:	:	PUNCT
ejpam-3889	5	34	�	�	VERB
ejpam-3889	5	35	any	any	DET
ejpam-3889	5	36	epimorphism	epimorphism	NOUN
ejpam-3889	5	37	of	of	ADP
ejpam-3889	5	38	sub	sub	ADJ
ejpam-3889	5	39	-	-	ADJ
ejpam-3889	5	40	complex	complex	ADJ
ejpam-3889	5	41	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	5	42	)	)	PUNCT
ejpam-3889	5	43	of	of	ADP
ejpam-3889	5	44	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	5	45	)	)	PUNCT
ejpam-3889	5	46	is	be	AUX
ejpam-3889	5	47	an	an	DET
ejpam-3889	5	48	isomorphism	isomorphism	NOUN
ejpam-3889	5	49	�	�	PROPN
ejpam-3889	5	50	;	;	PUNCT
ejpam-3889	5	51	(	(	PUNCT
ejpam-3889	5	52	iii	iii	X
ejpam-3889	5	53	)	)	PUNCT
ejpam-3889	5	54	let	let	VERB
ejpam-3889	5	55	m	m	PRON
ejpam-3889	5	56	be	be	AUX
ejpam-3889	5	57	a	a	DET
ejpam-3889	5	58	graded	grade	VERB
ejpam-3889	5	59	left	leave	VERB
ejpam-3889	5	60	a	a	DET
ejpam-3889	5	61	-	-	PUNCT
ejpam-3889	5	62	module	module	NOUN
ejpam-3889	5	63	,	,	PUNCT
ejpam-3889	5	64	n	n	CCONJ
ejpam-3889	5	65	a	a	DET
ejpam-3889	5	66	graded	grade	VERB
ejpam-3889	5	67	submodule	submodule	NOUN
ejpam-3889	5	68	of	of	ADP
ejpam-3889	5	69	m	m	PROPN
ejpam-3889	5	70	,	,	PUNCT
ejpam-3889	5	71	s	s	VERB
ejpam-3889	5	72	a	a	DET
ejpam-3889	5	73	saturated	saturate	VERB
ejpam-3889	5	74	multiplicative	multiplicative	ADJ
ejpam-3889	5	75	part	part	NOUN
ejpam-3889	5	76	formed	form	VERB
ejpam-3889	5	77	by	by	ADP
ejpam-3889	5	78	the	the	DET
ejpam-3889	5	79	non	non	ADJ
ejpam-3889	5	80	-	-	ADJ
ejpam-3889	5	81	zero	zero	ADJ
ejpam-3889	5	82	homogeneous	homogeneous	ADJ
ejpam-3889	5	83	elements	element	NOUN
ejpam-3889	5	84	of	of	ADP
ejpam-3889	5	85	a	a	DET
ejpam-3889	5	86	verifying	verifying	NOUN
ejpam-3889	5	87	the	the	DET
ejpam-3889	5	88	left	left	ADJ
ejpam-3889	5	89	ore	ore	NOUN
ejpam-3889	5	90	conditions	condition	NOUN
ejpam-3889	5	91	.	.	PUNCT
ejpam-3889	6	1	m∗	m∗	VERB
ejpam-3889	6	2	the	the	DET
ejpam-3889	6	3	quasi	quasi	ADJ
ejpam-3889	6	4	-	-	ADJ
ejpam-3889	6	5	projective	projective	ADJ
ejpam-3889	6	6	complex	complex	ADJ
ejpam-3889	6	7	sequence	sequence	NOUN
ejpam-3889	6	8	associated	associate	VERB
ejpam-3889	6	9	with	with	ADP
ejpam-3889	6	10	m	m	PROPN
ejpam-3889	6	11	and	and	CCONJ
ejpam-3889	6	12	n∗	n∗	VERB
ejpam-3889	6	13	a	a	DET
ejpam-3889	6	14	superfluous	superfluous	ADJ
ejpam-3889	6	15	and	and	CCONJ
ejpam-3889	6	16	completely	completely	ADV
ejpam-3889	6	17	invariant	invariant	ADJ
ejpam-3889	6	18	complex	complex	ADJ
ejpam-3889	6	19	sub	sub	NOUN
ejpam-3889	6	20	-	-	NOUN
ejpam-3889	6	21	sequence	sequence	NOUN
ejpam-3889	6	22	of	of	ADP
ejpam-3889	6	23	m∗.	m∗.	PROPN
ejpam-3889	6	24	then	then	ADV
ejpam-3889	6	25	the	the	DET
ejpam-3889	6	26	complex	complex	ADJ
ejpam-3889	6	27	morphism	morphism	NOUN
ejpam-3889	6	28	sequence	sequence	NOUN
ejpam-3889	6	29	of	of	ADP
ejpam-3889	6	30	left	left	ADJ
ejpam-3889	6	31	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	6	32	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	6	33	)	)	PUNCT
ejpam-3889	6	34	is	be	AUX
ejpam-3889	6	35	hopfian	hopfian	ADJ
ejpam-3889	6	36	if	if	SCONJ
ejpam-3889	6	37	,	,	PUNCT
ejpam-3889	6	38	and	and	CCONJ
ejpam-3889	6	39	only	only	ADV
ejpam-3889	6	40	if	if	SCONJ
ejpam-3889	6	41	,	,	PUNCT
ejpam-3889	6	42	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	6	43	)	)	PUNCT
ejpam-3889	6	44	the	the	DET
ejpam-3889	6	45	complex	complex	ADJ
ejpam-3889	6	46	sequence	sequence	NOUN
ejpam-3889	6	47	associated	associate	VERB
ejpam-3889	6	48	with	with	ADP
ejpam-3889	6	49	s−1(m	s−1(m	PROPN
ejpam-3889	6	50	/	/	SYM
ejpam-3889	6	51	n	n	CCONJ
ejpam-3889	6	52	)	)	PUNCT
ejpam-3889	6	53	is	be	AUX
ejpam-3889	6	54	hopfian	hopfian	ADJ
ejpam-3889	6	55	.	.	PUNCT
ejpam-3889	7	1	2020	2020	NUM
ejpam-3889	7	2	mathematics	mathematic	NOUN
ejpam-3889	7	3	subject	subject	NOUN
ejpam-3889	7	4	classifications	classification	NOUN
ejpam-3889	7	5	:	:	PUNCT
ejpam-3889	7	6	mathematics	mathematic	NOUN
ejpam-3889	7	7	subject	subject	ADJ
ejpam-3889	7	8	classification	classification	NOUN
ejpam-3889	7	9	codes	code	NOUN
ejpam-3889	7	10	key	key	ADJ
ejpam-3889	7	11	words	word	NOUN
ejpam-3889	7	12	and	and	CCONJ
ejpam-3889	7	13	phrases	phrase	NOUN
ejpam-3889	7	14	:	:	PUNCT
ejpam-3889	7	15	graded	grade	VERB
ejpam-3889	7	16	ring	ring	NOUN
ejpam-3889	7	17	,	,	PUNCT
ejpam-3889	7	18	a	a	DET
ejpam-3889	7	19	saturated	saturate	VERB
ejpam-3889	7	20	multiplicative	multiplicative	ADJ
ejpam-3889	7	21	part	part	NOUN
ejpam-3889	7	22	formed	form	VERB
ejpam-3889	7	23	by	by	ADP
ejpam-3889	7	24	the	the	DET
ejpam-3889	7	25	non	non	ADJ
ejpam-3889	7	26	-	-	ADJ
ejpam-3889	7	27	zero	zero	ADJ
ejpam-3889	7	28	homogeneous	homogeneous	ADJ
ejpam-3889	7	29	elements	element	NOUN
ejpam-3889	7	30	of	of	ADP
ejpam-3889	7	31	a	a	DET
ejpam-3889	7	32	,	,	PUNCT
ejpam-3889	7	33	ore	ore	NOUN
ejpam-3889	7	34	conditions	condition	NOUN
ejpam-3889	7	35	,	,	PUNCT
ejpam-3889	7	36	hopfian	hopfian	ADJ
ejpam-3889	7	37	,	,	PUNCT
ejpam-3889	7	38	cohopfian	cohopfian	ADJ
ejpam-3889	7	39	,	,	PUNCT
ejpam-3889	7	40	sequence	sequence	NOUN
ejpam-3889	7	41	complex	complex	NOUN
ejpam-3889	7	42	,	,	PUNCT
ejpam-3889	7	43	chain	chain	NOUN
ejpam-3889	7	44	complex	complex	NOUN
ejpam-3889	7	45	,	,	PUNCT
ejpam-3889	7	46	quasi	quasi	ADJ
ejpam-3889	7	47	-	-	ADJ
ejpam-3889	7	48	injective	injective	ADJ
ejpam-3889	7	49	and	and	CCONJ
ejpam-3889	7	50	quasi	quasi	ADJ
ejpam-3889	7	51	-	-	ADJ
ejpam-3889	7	52	projective	projective	ADJ
ejpam-3889	7	53	.	.	PUNCT
ejpam-3889	8	1	∗corresponding	∗corresponde	VERB
ejpam-3889	8	2	author	author	NOUN
ejpam-3889	8	3	.	.	PUNCT
ejpam-3889	9	1	doi	doi	NOUN
ejpam-3889	9	2	:	:	PUNCT
ejpam-3889	9	3	https://doi.org/10.29020/nybg.ejpam.v14i2.3889	https://doi.org/10.29020/nybg.ejpam.v14i2.3889	NOUN
ejpam-3889	9	4	email	email	NOUN
ejpam-3889	9	5	addresses	address	NOUN
ejpam-3889	9	6	:	:	PUNCT
ejpam-3889	9	7	sbalde878@gmail.com	sbalde878@gmail.com	X
ejpam-3889	9	8	(	(	PUNCT
ejpam-3889	9	9	s.a	s.a	PROPN
ejpam-3889	9	10	.	.	PROPN
ejpam-3889	9	11	balde	balde	PROPN
ejpam-3889	9	12	)	)	PUNCT
ejpam-3889	10	1	maaouiaalg@hotmail.com	maaouiaalg@hotmail.com	PROPN
ejpam-3889	10	2	(	(	PUNCT
ejpam-3889	10	3	m.	m.	NOUN
ejpam-3889	10	4	ben	ben	PROPN
ejpam-3889	10	5	maaouia	maaouia	PROPN
ejpam-3889	10	6	)	)	PUNCT
ejpam-3889	10	7	achbih@gmail.com	achbih@gmail.com	X
ejpam-3889	11	1	(	(	PUNCT
ejpam-3889	11	2	a.	a.	PROPN
ejpam-3889	11	3	o.	o.	PROPN
ejpam-3889	11	4	chbih	chbih	PROPN
ejpam-3889	11	5	)	)	PUNCT
ejpam-3889	12	1	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3889	13	1	404	404	NUM
ejpam-3889	14	1	c	c	X
ejpam-3889	14	2	©	©	PROPN
ejpam-3889	14	3	2021	2021	NUM
ejpam-3889	14	4	ejpam	ejpam	VERB
ejpam-3889	14	5	all	all	DET
ejpam-3889	14	6	rights	right	NOUN
ejpam-3889	14	7	reserved	reserve	VERB
ejpam-3889	14	8	.	.	PUNCT
ejpam-3889	15	1	s.	s.	PROPN
ejpam-3889	15	2	a.	a.	PROPN
ejpam-3889	15	3	balde	balde	PROPN
ejpam-3889	15	4	,	,	PUNCT
ejpam-3889	15	5	m.	m.	PROPN
ejpam-3889	15	6	b.	b.	PROPN
ejpam-3889	15	7	maaouia	maaouia	PROPN
ejpam-3889	15	8	,	,	PUNCT
ejpam-3889	15	9	a.	a.	NOUN
ejpam-3889	15	10	o.	o.	NOUN
ejpam-3889	15	11	chbih	chbih	PROPN
ejpam-3889	15	12	/	/	SYM
ejpam-3889	15	13	eur	eur	PROPN
ejpam-3889	15	14	.	.	PUNCT
ejpam-3889	16	1	j.	j.	PROPN
ejpam-3889	16	2	pure	pure	PROPN
ejpam-3889	16	3	appl	appl	PROPN
ejpam-3889	16	4	.	.	PROPN
ejpam-3889	16	5	math	math	PROPN
ejpam-3889	16	6	,	,	PUNCT
ejpam-3889	16	7	14	14	NUM
ejpam-3889	16	8	(	(	PUNCT
ejpam-3889	16	9	2	2	NUM
ejpam-3889	16	10	)	)	PUNCT
ejpam-3889	16	11	(	(	PUNCT
ejpam-3889	16	12	2021	2021	NUM
ejpam-3889	16	13	)	)	PUNCT
ejpam-3889	16	14	,	,	PUNCT
ejpam-3889	16	15	404	404	NUM
ejpam-3889	16	16	-	-	SYM
ejpam-3889	16	17	422	422	NUM
ejpam-3889	16	18	405	405	NUM
ejpam-3889	16	19	1	1	NUM
ejpam-3889	16	20	.	.	PUNCT
ejpam-3889	17	1	introduction	introduction	NOUN
ejpam-3889	17	2	in	in	ADP
ejpam-3889	17	3	this	this	DET
ejpam-3889	17	4	paper	paper	NOUN
ejpam-3889	17	5	,	,	PUNCT
ejpam-3889	17	6	the	the	DET
ejpam-3889	17	7	ring	ring	NOUN
ejpam-3889	17	8	a	a	PRON
ejpam-3889	17	9	is	be	AUX
ejpam-3889	17	10	supposed	suppose	VERB
ejpam-3889	17	11	to	to	PART
ejpam-3889	17	12	be	be	AUX
ejpam-3889	17	13	associative	associative	ADJ
ejpam-3889	17	14	,	,	PUNCT
ejpam-3889	17	15	unitary	unitary	ADJ
ejpam-3889	17	16	,	,	PUNCT
ejpam-3889	17	17	not	not	PART
ejpam-3889	17	18	necessairly	necessairly	ADV
ejpam-3889	17	19	commutative	commutative	ADJ
ejpam-3889	17	20	,	,	PUNCT
ejpam-3889	17	21	and	and	CCONJ
ejpam-3889	17	22	any	any	DET
ejpam-3889	17	23	left	left	ADJ
ejpam-3889	17	24	a	a	PRON
ejpam-3889	17	25	-	-	PUNCT
ejpam-3889	17	26	modules	module	NOUN
ejpam-3889	17	27	is	be	AUX
ejpam-3889	17	28	unifary	unifary	ADJ
ejpam-3889	17	29	.	.	PUNCT
ejpam-3889	18	1	in	in	ADP
ejpam-3889	18	2	this	this	DET
ejpam-3889	18	3	article	article	NOUN
ejpam-3889	18	4	,	,	PUNCT
ejpam-3889	18	5	we	we	PRON
ejpam-3889	18	6	study	study	VERB
ejpam-3889	18	7	the	the	DET
ejpam-3889	18	8	localization	localization	NOUN
ejpam-3889	18	9	of	of	ADP
ejpam-3889	18	10	hopfian	hopfian	ADJ
ejpam-3889	18	11	and	and	CCONJ
ejpam-3889	18	12	cohopfian	cohopfian	ADJ
ejpam-3889	18	13	objects	object	NOUN
ejpam-3889	18	14	in	in	ADP
ejpam-3889	18	15	the	the	DET
ejpam-3889	18	16	categories	category	NOUN
ejpam-3889	18	17	a−mod	a−mod	NOUN
ejpam-3889	18	18	of	of	ADP
ejpam-3889	18	19	left	leave	VERB
ejpam-3889	18	20	a	a	PRON
ejpam-3889	18	21	-	-	PUNCT
ejpam-3889	18	22	modules	module	NOUN
ejpam-3889	18	23	,	,	PUNCT
ejpam-3889	18	24	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	18	25	)	)	PUNCT
ejpam-3889	18	26	of	of	ADP
ejpam-3889	18	27	graded	grade	VERB
ejpam-3889	18	28	left	leave	VERB
ejpam-3889	18	29	a	a	DET
ejpam-3889	18	30	-	-	PUNCT
ejpam-3889	18	31	modules	module	NOUN
ejpam-3889	18	32	and	and	CCONJ
ejpam-3889	18	33	comp	comp	NOUN
ejpam-3889	18	34	(	(	PUNCT
ejpam-3889	18	35	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	18	36	)	)	PUNCT
ejpam-3889	18	37	)	)	PUNCT
ejpam-3889	18	38	of	of	ADP
ejpam-3889	18	39	complex	complex	ADJ
ejpam-3889	18	40	sequences	sequence	NOUN
ejpam-3889	18	41	associated	associate	VERB
ejpam-3889	18	42	to	to	AUX
ejpam-3889	18	43	graded	grade	VERB
ejpam-3889	18	44	left	leave	VERB
ejpam-3889	18	45	a	a	DET
ejpam-3889	18	46	-	-	PUNCT
ejpam-3889	18	47	modules	module	NOUN
ejpam-3889	18	48	.	.	PUNCT
ejpam-3889	19	1	we	we	PRON
ejpam-3889	19	2	rely	rely	VERB
ejpam-3889	19	3	on	on	ADP
ejpam-3889	19	4	the	the	DET
ejpam-3889	19	5	articles	article	NOUN
ejpam-3889	19	6	,	,	PUNCT
ejpam-3889	19	7	�	�	PROPN
ejpam-3889	19	8	graduation	graduation	NOUN
ejpam-3889	19	9	of	of	ADP
ejpam-3889	19	10	module	module	NOUN
ejpam-3889	19	11	of	of	ADP
ejpam-3889	19	12	fraction	fraction	NOUN
ejpam-3889	19	13	on	on	ADP
ejpam-3889	19	14	a	a	DET
ejpam-3889	19	15	graded	grade	VERB
ejpam-3889	19	16	domain	domain	NOUN
ejpam-3889	19	17	ring	ring	NOUN
ejpam-3889	19	18	not	not	PART
ejpam-3889	19	19	necessarily	necessarily	ADV
ejpam-3889	19	20	commutative	commutative	ADJ
ejpam-3889	19	21	�	�	PROPN
ejpam-3889	19	22	[2	[2	X
ejpam-3889	19	23	]	]	PUNCT
ejpam-3889	19	24	,	,	PUNCT
ejpam-3889	19	25	�	�	PROPN
ejpam-3889	19	26	factorization	factorization	NOUN
ejpam-3889	19	27	of	of	ADP
ejpam-3889	19	28	graded	grade	VERB
ejpam-3889	19	29	modules	module	NOUN
ejpam-3889	19	30	of	of	ADP
ejpam-3889	19	31	fractions	fraction	NOUN
ejpam-3889	19	32	�	�	PROPN
ejpam-3889	20	1	[	[	X
ejpam-3889	20	2	3	3	NUM
ejpam-3889	20	3	]	]	PUNCT
ejpam-3889	20	4	,	,	PUNCT
ejpam-3889	20	5	�	�	PROPN
ejpam-3889	20	6	module	module	NOUN
ejpam-3889	20	7	de	de	NOUN
ejpam-3889	20	8	fractions	fraction	NOUN
ejpam-3889	20	9	,	,	PUNCT
ejpam-3889	20	10	sous	sous	ADJ
ejpam-3889	20	11	-	-	PUNCT
ejpam-3889	20	12	modules	module	NOUN
ejpam-3889	20	13	s−saturée	s−saturée	PROPN
ejpam-3889	20	14	et	et	NOUN
ejpam-3889	20	15	foncteur	foncteur	NOUN
ejpam-3889	21	1	s−1	s−1	PROPN
ejpam-3889	21	2	�	�	PROPN
ejpam-3889	22	1	[	[	X
ejpam-3889	22	2	18	18	NUM
ejpam-3889	22	3	]	]	PUNCT
ejpam-3889	22	4	and	and	CCONJ
ejpam-3889	22	5	�	�	PROPN
ejpam-3889	22	6	hopfian	hopfian	ADJ
ejpam-3889	22	7	and	and	CCONJ
ejpam-3889	22	8	cohopfian	cohopfian	ADJ
ejpam-3889	22	9	objects	object	NOUN
ejpam-3889	22	10	in	in	ADP
ejpam-3889	22	11	the	the	DET
ejpam-3889	22	12	categories	category	NOUN
ejpam-3889	22	13	of	of	ADP
ejpam-3889	22	14	gr(a−mod	gr(a−mod	NOUN
ejpam-3889	22	15	)	)	PUNCT
ejpam-3889	22	16	and	and	CCONJ
ejpam-3889	22	17	comp	comp	NOUN
ejpam-3889	22	18	(	(	PUNCT
ejpam-3889	22	19	gr(a−mod))	gr(a−mod))	NOUN
ejpam-3889	22	20	�	�	PROPN
ejpam-3889	22	21	[23	[23	NOUN
ejpam-3889	22	22	]	]	X
ejpam-3889	22	23	,	,	PUNCT
ejpam-3889	22	24	which	which	PRON
ejpam-3889	22	25	are	be	AUX
ejpam-3889	22	26	used	use	VERB
ejpam-3889	22	27	as	as	ADP
ejpam-3889	22	28	a	a	DET
ejpam-3889	22	29	basis	basis	NOUN
ejpam-3889	22	30	for	for	ADP
ejpam-3889	22	31	studying	study	VERB
ejpam-3889	22	32	the	the	DET
ejpam-3889	22	33	notions	notion	NOUN
ejpam-3889	22	34	of	of	ADP
ejpam-3889	22	35	localization	localization	NOUN
ejpam-3889	22	36	,	,	PUNCT
ejpam-3889	22	37	hopficity	hopficity	NOUN
ejpam-3889	22	38	and	and	CCONJ
ejpam-3889	22	39	cohopficity	cohopficity	NOUN
ejpam-3889	22	40	.	.	PUNCT
ejpam-3889	23	1	the	the	DET
ejpam-3889	23	2	transition	transition	NOUN
ejpam-3889	23	3	to	to	ADP
ejpam-3889	23	4	localization	localization	NOUN
ejpam-3889	23	5	and	and	CCONJ
ejpam-3889	23	6	the	the	DET
ejpam-3889	23	7	study	study	NOUN
ejpam-3889	23	8	of	of	ADP
ejpam-3889	23	9	hopficity	hopficity	NOUN
ejpam-3889	23	10	and	and	CCONJ
ejpam-3889	23	11	cohopficity	cohopficity	NOUN
ejpam-3889	23	12	from	from	ADP
ejpam-3889	23	13	the	the	DET
ejpam-3889	23	14	category	category	NOUN
ejpam-3889	23	15	of	of	ADP
ejpam-3889	23	16	left	left	ADJ
ejpam-3889	23	17	a−mod	a−mod	NOUN
ejpam-3889	23	18	whose	whose	DET
ejpam-3889	23	19	objects	object	NOUN
ejpam-3889	23	20	are	be	AUX
ejpam-3889	23	21	the	the	DET
ejpam-3889	23	22	left	left	ADJ
ejpam-3889	23	23	a	a	PRON
ejpam-3889	23	24	-	-	PUNCT
ejpam-3889	23	25	modules	module	NOUN
ejpam-3889	23	26	and	and	CCONJ
ejpam-3889	23	27	the	the	DET
ejpam-3889	23	28	morphisms	morphism	NOUN
ejpam-3889	23	29	are	be	AUX
ejpam-3889	23	30	the	the	DET
ejpam-3889	23	31	left	left	ADJ
ejpam-3889	23	32	a	a	DET
ejpam-3889	23	33	-	-	PUNCT
ejpam-3889	23	34	module	module	NOUN
ejpam-3889	23	35	morphisms	morphism	NOUN
ejpam-3889	23	36	to	to	ADP
ejpam-3889	23	37	the	the	DET
ejpam-3889	23	38	category	category	NOUN
ejpam-3889	23	39	agr(a	agr(a	NOUN
ejpam-3889	23	40	−mod	−mod	ADP
ejpam-3889	23	41	)	)	PUNCT
ejpam-3889	23	42	whose	whose	DET
ejpam-3889	23	43	objects	object	NOUN
ejpam-3889	23	44	are	be	AUX
ejpam-3889	23	45	the	the	DET
ejpam-3889	23	46	graded	grade	VERB
ejpam-3889	23	47	left	leave	VERB
ejpam-3889	23	48	a	a	DET
ejpam-3889	23	49	-	-	PUNCT
ejpam-3889	23	50	module	module	NOUN
ejpam-3889	23	51	and	and	CCONJ
ejpam-3889	23	52	the	the	DET
ejpam-3889	23	53	morphisms	morphism	NOUN
ejpam-3889	23	54	are	be	AUX
ejpam-3889	23	55	the	the	DET
ejpam-3889	23	56	graded	grade	VERB
ejpam-3889	23	57	left	leave	VERB
ejpam-3889	23	58	a	a	DET
ejpam-3889	23	59	-	-	PUNCT
ejpam-3889	23	60	module	module	NOUN
ejpam-3889	23	61	graded	grade	VERB
ejpam-3889	23	62	morphisms	morphism	NOUN
ejpam-3889	23	63	and	and	CCONJ
ejpam-3889	23	64	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	23	65	)	)	PUNCT
ejpam-3889	23	66	to	to	ADP
ejpam-3889	23	67	the	the	DET
ejpam-3889	23	68	category	category	NOUN
ejpam-3889	23	69	of	of	ADP
ejpam-3889	23	70	comp	comp	NOUN
ejpam-3889	23	71	(	(	PUNCT
ejpam-3889	23	72	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	23	73	)	)	PUNCT
ejpam-3889	23	74	)	)	PUNCT
ejpam-3889	24	1	whose	whose	DET
ejpam-3889	24	2	the	the	DET
ejpam-3889	24	3	objects	object	NOUN
ejpam-3889	24	4	are	be	AUX
ejpam-3889	24	5	the	the	DET
ejpam-3889	24	6	complex	complex	ADJ
ejpam-3889	24	7	sequences	sequence	NOUN
ejpam-3889	24	8	of	of	ADP
ejpam-3889	24	9	graded	grade	VERB
ejpam-3889	24	10	left	leave	VERB
ejpam-3889	24	11	a	a	DET
ejpam-3889	24	12	-	-	PUNCT
ejpam-3889	24	13	modules	module	NOUN
ejpam-3889	24	14	and	and	CCONJ
ejpam-3889	24	15	the	the	DET
ejpam-3889	24	16	morphisms	morphism	NOUN
ejpam-3889	24	17	are	be	AUX
ejpam-3889	24	18	the	the	DET
ejpam-3889	24	19	chain	chain	NOUN
ejpam-3889	24	20	complexes	complex	NOUN
ejpam-3889	24	21	associated	associate	VERB
ejpam-3889	24	22	to	to	ADP
ejpam-3889	24	23	the	the	DET
ejpam-3889	24	24	graded	grade	VERB
ejpam-3889	24	25	morphims	morphim	NOUN
ejpam-3889	24	26	of	of	ADP
ejpam-3889	24	27	graded	grade	VERB
ejpam-3889	24	28	left	leave	VERB
ejpam-3889	24	29	a	a	DET
ejpam-3889	24	30	-	-	PUNCT
ejpam-3889	24	31	modules	module	NOUN
ejpam-3889	24	32	is	be	AUX
ejpam-3889	24	33	not	not	PART
ejpam-3889	24	34	easy	easy	ADJ
ejpam-3889	24	35	.	.	PUNCT
ejpam-3889	25	1	in	in	ADP
ejpam-3889	25	2	our	our	PRON
ejpam-3889	25	3	opinion	opinion	NOUN
ejpam-3889	25	4	,	,	PUNCT
ejpam-3889	25	5	these	these	DET
ejpam-3889	25	6	reasons	reason	NOUN
ejpam-3889	25	7	justify	justify	VERB
ejpam-3889	25	8	our	our	PRON
ejpam-3889	25	9	work	work	NOUN
ejpam-3889	25	10	.	.	PUNCT
ejpam-3889	26	1	thus	thus	ADV
ejpam-3889	26	2	,	,	PUNCT
ejpam-3889	26	3	the	the	DET
ejpam-3889	26	4	paper	paper	NOUN
ejpam-3889	26	5	is	be	AUX
ejpam-3889	26	6	organized	organize	VERB
ejpam-3889	26	7	as	as	SCONJ
ejpam-3889	26	8	follows	follow	VERB
ejpam-3889	26	9	in	in	ADP
ejpam-3889	26	10	the	the	DET
ejpam-3889	26	11	section	section	NOUN
ejpam-3889	26	12	2	2	NUM
ejpam-3889	26	13	,	,	PUNCT
ejpam-3889	26	14	we	we	PRON
ejpam-3889	26	15	study	study	VERB
ejpam-3889	26	16	the	the	DET
ejpam-3889	26	17	localization	localization	NOUN
ejpam-3889	26	18	of	of	ADP
ejpam-3889	26	19	hopfian	hopfian	ADJ
ejpam-3889	26	20	and	and	CCONJ
ejpam-3889	26	21	cohopfian	cohopfian	ADJ
ejpam-3889	26	22	objects	object	NOUN
ejpam-3889	26	23	in	in	ADP
ejpam-3889	26	24	the	the	DET
ejpam-3889	26	25	category	category	NOUN
ejpam-3889	26	26	a−mod	a−mod	NOUN
ejpam-3889	26	27	and	and	CCONJ
ejpam-3889	26	28	we	we	PRON
ejpam-3889	26	29	show	show	VERB
ejpam-3889	26	30	the	the	DET
ejpam-3889	26	31	following	follow	VERB
ejpam-3889	26	32	results	result	NOUN
ejpam-3889	26	33	:	:	PUNCT
ejpam-3889	26	34	(	(	PUNCT
ejpam-3889	26	35	i	i	NOUN
ejpam-3889	26	36	)	)	PUNCT
ejpam-3889	26	37	let	let	VERB
ejpam-3889	26	38	a	a	PRON
ejpam-3889	26	39	be	be	AUX
ejpam-3889	26	40	a	a	DET
ejpam-3889	26	41	ring	ring	NOUN
ejpam-3889	26	42	,	,	PUNCT
ejpam-3889	26	43	s	s	VERB
ejpam-3889	26	44	a	a	DET
ejpam-3889	26	45	saturated	saturate	VERB
ejpam-3889	26	46	multiplicative	multiplicative	ADJ
ejpam-3889	26	47	part	part	NOUN
ejpam-3889	26	48	of	of	ADP
ejpam-3889	26	49	a	a	DET
ejpam-3889	26	50	verifying	verifying	NOUN
ejpam-3889	26	51	the	the	DET
ejpam-3889	26	52	left	left	ADJ
ejpam-3889	26	53	ore	ore	NOUN
ejpam-3889	26	54	conditions	condition	NOUN
ejpam-3889	26	55	,	,	PUNCT
ejpam-3889	26	56	m	m	VERB
ejpam-3889	26	57	a	a	DET
ejpam-3889	26	58	left	left	ADJ
ejpam-3889	26	59	a	a	DET
ejpam-3889	26	60	-	-	PUNCT
ejpam-3889	26	61	module	module	NOUN
ejpam-3889	26	62	.	.	PUNCT
ejpam-3889	27	1	if	if	SCONJ
ejpam-3889	27	2	s−1	s−1	PROPN
ejpam-3889	27	3	m	m	VERB
ejpam-3889	27	4	is	be	AUX
ejpam-3889	27	5	hopfian	hopfian	ADJ
ejpam-3889	27	6	,	,	PUNCT
ejpam-3889	27	7	then	then	ADV
ejpam-3889	27	8	m	m	VERB
ejpam-3889	27	9	is	be	AUX
ejpam-3889	27	10	hopfian	hopfian	ADJ
ejpam-3889	27	11	;	;	PUNCT
ejpam-3889	27	12	(	(	PUNCT
ejpam-3889	27	13	ii	ii	NOUN
ejpam-3889	27	14	)	)	PUNCT
ejpam-3889	27	15	let	let	VERB
ejpam-3889	27	16	a	a	PRON
ejpam-3889	27	17	be	be	AUX
ejpam-3889	27	18	a	a	DET
ejpam-3889	27	19	ring	ring	NOUN
ejpam-3889	27	20	,	,	PUNCT
ejpam-3889	27	21	s	s	VERB
ejpam-3889	27	22	a	a	DET
ejpam-3889	27	23	saturated	saturate	VERB
ejpam-3889	27	24	multiplicative	multiplicative	ADJ
ejpam-3889	27	25	part	part	NOUN
ejpam-3889	27	26	of	of	ADP
ejpam-3889	27	27	a	a	DET
ejpam-3889	27	28	verifying	verifying	NOUN
ejpam-3889	27	29	the	the	DET
ejpam-3889	27	30	left	left	ADJ
ejpam-3889	27	31	ore	ore	NOUN
ejpam-3889	27	32	conditions	condition	NOUN
ejpam-3889	27	33	,	,	PUNCT
ejpam-3889	27	34	m	m	VERB
ejpam-3889	27	35	a	a	DET
ejpam-3889	27	36	left	left	ADJ
ejpam-3889	27	37	a	a	DET
ejpam-3889	27	38	-	-	PUNCT
ejpam-3889	27	39	module	module	NOUN
ejpam-3889	27	40	.	.	PUNCT
ejpam-3889	28	1	if	if	SCONJ
ejpam-3889	28	2	m	m	NOUN
ejpam-3889	28	3	is	be	AUX
ejpam-3889	28	4	a	a	DET
ejpam-3889	28	5	cohopfian	cohopfian	ADJ
ejpam-3889	28	6	and	and	CCONJ
ejpam-3889	28	7	completely	completely	ADV
ejpam-3889	28	8	invariant	invariant	ADJ
ejpam-3889	28	9	submodule	submodule	NOUN
ejpam-3889	28	10	of	of	ADP
ejpam-3889	28	11	left	leave	VERB
ejpam-3889	28	12	a	a	DET
ejpam-3889	28	13	-	-	PUNCT
ejpam-3889	28	14	module	module	NOUN
ejpam-3889	28	15	s−1(m	s−1(m	PROPN
ejpam-3889	28	16	)	)	PUNCT
ejpam-3889	28	17	,	,	PUNCT
ejpam-3889	28	18	then	then	ADV
ejpam-3889	28	19	s−1(m	s−1(m	PROPN
ejpam-3889	28	20	)	)	PUNCT
ejpam-3889	28	21	is	be	AUX
ejpam-3889	28	22	a	a	DET
ejpam-3889	28	23	cohopfian	cohopfian	ADJ
ejpam-3889	28	24	left	leave	VERB
ejpam-3889	28	25	s−1(a)(respectively	s−1(a)(respectively	ADV
ejpam-3889	28	26	a)module	a)module	NOUN
ejpam-3889	28	27	;	;	PUNCT
ejpam-3889	28	28	(	(	PUNCT
ejpam-3889	28	29	iii	iii	X
ejpam-3889	28	30	)	)	PUNCT
ejpam-3889	28	31	let	let	VERB
ejpam-3889	28	32	m	m	PRON
ejpam-3889	28	33	be	be	AUX
ejpam-3889	28	34	a	a	DET
ejpam-3889	28	35	noetherian	noetherian	ADJ
ejpam-3889	28	36	quasi	quasi	ADJ
ejpam-3889	28	37	-	-	ADJ
ejpam-3889	28	38	injective	injective	ADJ
ejpam-3889	28	39	left	leave	VERB
ejpam-3889	28	40	a	a	DET
ejpam-3889	28	41	-	-	PUNCT
ejpam-3889	28	42	module	module	NOUN
ejpam-3889	28	43	,	,	PUNCT
ejpam-3889	28	44	n	n	PRON
ejpam-3889	28	45	be	be	VERB
ejpam-3889	28	46	an	an	DET
ejpam-3889	28	47	essential	essential	ADJ
ejpam-3889	28	48	and	and	CCONJ
ejpam-3889	28	49	completely	completely	ADV
ejpam-3889	28	50	invariant	invariant	ADJ
ejpam-3889	28	51	submodule	submodule	NOUN
ejpam-3889	28	52	of	of	ADP
ejpam-3889	28	53	m	m	PROPN
ejpam-3889	28	54	and	and	CCONJ
ejpam-3889	28	55	s	s	VERB
ejpam-3889	28	56	a	a	DET
ejpam-3889	28	57	saturated	saturate	VERB
ejpam-3889	28	58	multiplicative	multiplicative	ADJ
ejpam-3889	28	59	part	part	NOUN
ejpam-3889	28	60	of	of	ADP
ejpam-3889	28	61	a	a	DET
ejpam-3889	28	62	verifying	verifying	NOUN
ejpam-3889	28	63	the	the	DET
ejpam-3889	28	64	left	left	ADJ
ejpam-3889	28	65	ore	ore	NOUN
ejpam-3889	28	66	conditions	condition	NOUN
ejpam-3889	28	67	.	.	PUNCT
ejpam-3889	29	1	then	then	ADV
ejpam-3889	29	2	,	,	PUNCT
ejpam-3889	29	3	the	the	DET
ejpam-3889	29	4	left	leave	VERB
ejpam-3889	29	5	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	29	6	s−1(n	s−1(n	NOUN
ejpam-3889	29	7	)	)	PUNCT
ejpam-3889	29	8	is	be	AUX
ejpam-3889	29	9	cohopfian	cohopfian	ADJ
ejpam-3889	29	10	if	if	SCONJ
ejpam-3889	29	11	,	,	PUNCT
ejpam-3889	29	12	and	and	CCONJ
ejpam-3889	29	13	only	only	ADV
ejpam-3889	29	14	,	,	PUNCT
ejpam-3889	29	15	if	if	SCONJ
ejpam-3889	29	16	s−1(m	s−1(m	PROPN
ejpam-3889	29	17	)	)	PUNCT
ejpam-3889	29	18	is	be	AUX
ejpam-3889	29	19	cohopfian	cohopfian	ADJ
ejpam-3889	29	20	left	leave	VERB
ejpam-3889	29	21	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	29	22	;	;	PUNCT
ejpam-3889	29	23	(	(	PUNCT
ejpam-3889	29	24	iv	iv	X
ejpam-3889	29	25	)	)	PUNCT
ejpam-3889	29	26	let	let	VERB
ejpam-3889	29	27	m	m	PRON
ejpam-3889	29	28	be	be	AUX
ejpam-3889	29	29	a	a	DET
ejpam-3889	29	30	quasi	quasi	NOUN
ejpam-3889	29	31	-	-	ADJ
ejpam-3889	29	32	projective	projective	ADJ
ejpam-3889	29	33	left	leave	VERB
ejpam-3889	29	34	a	a	DET
ejpam-3889	29	35	-	-	PUNCT
ejpam-3889	29	36	module	module	NOUN
ejpam-3889	29	37	,	,	PUNCT
ejpam-3889	29	38	n	n	CCONJ
ejpam-3889	29	39	a	a	DET
ejpam-3889	29	40	superfluous	superfluous	ADJ
ejpam-3889	29	41	and	and	CCONJ
ejpam-3889	29	42	completely	completely	ADV
ejpam-3889	29	43	invariant	invariant	ADJ
ejpam-3889	29	44	sub	sub	NOUN
ejpam-3889	29	45	-	-	NOUN
ejpam-3889	29	46	module	module	NOUN
ejpam-3889	29	47	of	of	ADP
ejpam-3889	29	48	m	m	PRON
ejpam-3889	29	49	,	,	PUNCT
ejpam-3889	29	50	s	s	VERB
ejpam-3889	29	51	a	a	DET
ejpam-3889	29	52	saturated	saturate	VERB
ejpam-3889	29	53	multiplicative	multiplicative	ADJ
ejpam-3889	29	54	part	part	NOUN
ejpam-3889	29	55	of	of	ADP
ejpam-3889	29	56	a	a	DET
ejpam-3889	29	57	verifying	verifying	NOUN
ejpam-3889	29	58	the	the	DET
ejpam-3889	29	59	left	left	ADJ
ejpam-3889	29	60	ore	ore	NOUN
ejpam-3889	29	61	conditions	condition	NOUN
ejpam-3889	29	62	.	.	PUNCT
ejpam-3889	30	1	then	then	ADV
ejpam-3889	30	2	the	the	DET
ejpam-3889	30	3	left	left	ADJ
ejpam-3889	30	4	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	30	5	s−1(n	s−1(n	NOUN
ejpam-3889	30	6	)	)	PUNCT
ejpam-3889	30	7	is	be	AUX
ejpam-3889	30	8	hopfian	hopfian	ADJ
ejpam-3889	30	9	if	if	SCONJ
ejpam-3889	30	10	,	,	PUNCT
ejpam-3889	30	11	and	and	CCONJ
ejpam-3889	30	12	only	only	ADV
ejpam-3889	30	13	if	if	SCONJ
ejpam-3889	30	14	,	,	PUNCT
ejpam-3889	30	15	s−1(m	s−1(m	PROPN
ejpam-3889	30	16	/	/	SYM
ejpam-3889	30	17	n	n	CCONJ
ejpam-3889	30	18	)	)	PUNCT
ejpam-3889	30	19	is	be	AUX
ejpam-3889	30	20	hopfian	hopfian	ADJ
ejpam-3889	30	21	.	.	PUNCT
ejpam-3889	31	1	in	in	ADP
ejpam-3889	31	2	section	section	NOUN
ejpam-3889	31	3	3	3	NUM
ejpam-3889	31	4	,	,	PUNCT
ejpam-3889	31	5	we	we	PRON
ejpam-3889	31	6	study	study	VERB
ejpam-3889	31	7	the	the	DET
ejpam-3889	31	8	localization	localization	NOUN
ejpam-3889	31	9	of	of	ADP
ejpam-3889	31	10	hopfian	hopfian	ADJ
ejpam-3889	31	11	and	and	CCONJ
ejpam-3889	31	12	cohopfian	cohopfian	ADJ
ejpam-3889	31	13	objects	object	NOUN
ejpam-3889	31	14	in	in	ADP
ejpam-3889	31	15	the	the	DET
ejpam-3889	31	16	category	category	NOUN
ejpam-3889	31	17	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	31	18	)	)	PUNCT
ejpam-3889	31	19	and	and	CCONJ
ejpam-3889	31	20	we	we	PRON
ejpam-3889	31	21	prove	prove	VERB
ejpam-3889	31	22	the	the	DET
ejpam-3889	31	23	following	follow	VERB
ejpam-3889	31	24	results	result	NOUN
ejpam-3889	31	25	:	:	PUNCT
ejpam-3889	31	26	s.	s.	PROPN
ejpam-3889	31	27	a.	a.	PROPN
ejpam-3889	31	28	balde	balde	PROPN
ejpam-3889	31	29	,	,	PUNCT
ejpam-3889	31	30	m.	m.	PROPN
ejpam-3889	31	31	b.	b.	PROPN
ejpam-3889	31	32	maaouia	maaouia	PROPN
ejpam-3889	31	33	,	,	PUNCT
ejpam-3889	31	34	a.	a.	NOUN
ejpam-3889	31	35	o.	o.	NOUN
ejpam-3889	31	36	chbih	chbih	PROPN
ejpam-3889	31	37	/	/	SYM
ejpam-3889	31	38	eur	eur	PROPN
ejpam-3889	31	39	.	.	PUNCT
ejpam-3889	32	1	j.	j.	PROPN
ejpam-3889	32	2	pure	pure	PROPN
ejpam-3889	32	3	appl	appl	PROPN
ejpam-3889	32	4	.	.	PROPN
ejpam-3889	32	5	math	math	PROPN
ejpam-3889	32	6	,	,	PUNCT
ejpam-3889	32	7	14	14	NUM
ejpam-3889	32	8	(	(	PUNCT
ejpam-3889	32	9	2	2	NUM
ejpam-3889	32	10	)	)	PUNCT
ejpam-3889	32	11	(	(	PUNCT
ejpam-3889	32	12	2021	2021	NUM
ejpam-3889	32	13	)	)	PUNCT
ejpam-3889	32	14	,	,	PUNCT
ejpam-3889	32	15	404	404	NUM
ejpam-3889	32	16	-	-	SYM
ejpam-3889	32	17	422	422	NUM
ejpam-3889	32	18	406	406	NUM
ejpam-3889	32	19	(	(	PUNCT
ejpam-3889	32	20	i	i	NOUN
ejpam-3889	32	21	)	)	PUNCT
ejpam-3889	32	22	let	let	VERB
ejpam-3889	32	23	a	a	DET
ejpam-3889	32	24	=	=	SYM
ejpam-3889	32	25	⊕	⊕	PROPN
ejpam-3889	32	26	n∈z	n∈z	VERB
ejpam-3889	32	27	an	an	DET
ejpam-3889	32	28	be	be	AUX
ejpam-3889	32	29	a	a	DET
ejpam-3889	32	30	graded	grade	VERB
ejpam-3889	32	31	ring	ring	NOUN
ejpam-3889	32	32	,	,	PUNCT
ejpam-3889	32	33	s	s	VERB
ejpam-3889	32	34	a	a	DET
ejpam-3889	32	35	saturated	saturate	VERB
ejpam-3889	32	36	multiplicative	multiplicative	ADJ
ejpam-3889	32	37	part	part	NOUN
ejpam-3889	32	38	formed	form	VERB
ejpam-3889	32	39	by	by	ADP
ejpam-3889	32	40	the	the	DET
ejpam-3889	32	41	non	non	ADJ
ejpam-3889	32	42	-	-	ADJ
ejpam-3889	32	43	zero	zero	ADJ
ejpam-3889	32	44	homogeneous	homogeneous	ADJ
ejpam-3889	32	45	elements	element	NOUN
ejpam-3889	32	46	of	of	ADP
ejpam-3889	32	47	a	a	DET
ejpam-3889	32	48	verifying	verifying	NOUN
ejpam-3889	32	49	the	the	DET
ejpam-3889	32	50	left	left	ADJ
ejpam-3889	32	51	ore	ore	NOUN
ejpam-3889	32	52	conditions	condition	NOUN
ejpam-3889	32	53	and	and	CCONJ
ejpam-3889	32	54	m	m	NOUN
ejpam-3889	32	55	=	=	PROPN
ejpam-3889	32	56	⊕	⊕	PROPN
ejpam-3889	32	57	n∈z	n∈z	VERB
ejpam-3889	32	58	mn	mn	PROPN
ejpam-3889	32	59	a	a	DET
ejpam-3889	32	60	graded	grade	VERB
ejpam-3889	32	61	left	leave	VERB
ejpam-3889	32	62	a	a	DET
ejpam-3889	32	63	-	-	PUNCT
ejpam-3889	32	64	module	module	NOUN
ejpam-3889	32	65	,	,	PUNCT
ejpam-3889	32	66	then	then	ADV
ejpam-3889	32	67	s−1(m	s−1(m	PROPN
ejpam-3889	32	68	)	)	PUNCT
ejpam-3889	33	1	=	=	PROPN
ejpam-3889	33	2	⊕	⊕	PROPN
ejpam-3889	33	3	i∈z	i∈z	PROPN
ejpam-3889	33	4	(	(	PUNCT
ejpam-3889	33	5	s−1m)i	s−1m)i	NOUN
ejpam-3889	33	6	is	be	AUX
ejpam-3889	33	7	hopfian(respectively	hopfian(respectively	ADV
ejpam-3889	33	8	cohopfian	cohopfian	ADJ
ejpam-3889	33	9	)	)	PUNCT
ejpam-3889	33	10	if	if	SCONJ
ejpam-3889	33	11	,	,	PUNCT
ejpam-3889	33	12	and	and	CCONJ
ejpam-3889	33	13	only	only	ADV
ejpam-3889	33	14	,	,	PUNCT
ejpam-3889	33	15	if	if	SCONJ
ejpam-3889	33	16	(	(	PUNCT
ejpam-3889	33	17	s−1m)i	s−1m)i	NOUN
ejpam-3889	33	18	is	be	AUX
ejpam-3889	33	19	a	a	DET
ejpam-3889	33	20	hopfian(respectiveley	hopfian(respectiveley	PROPN
ejpam-3889	33	21	cohopfien	cohopfien	NOUN
ejpam-3889	33	22	)	)	PUNCT
ejpam-3889	33	23	group	group	NOUN
ejpam-3889	33	24	;	;	PUNCT
ejpam-3889	33	25	(	(	PUNCT
ejpam-3889	33	26	ii	ii	NOUN
ejpam-3889	33	27	)	)	PUNCT
ejpam-3889	33	28	let	let	VERB
ejpam-3889	33	29	a	a	PRON
ejpam-3889	33	30	be	be	AUX
ejpam-3889	33	31	a	a	DET
ejpam-3889	33	32	graded	grade	VERB
ejpam-3889	33	33	ring	ring	NOUN
ejpam-3889	33	34	,	,	PUNCT
ejpam-3889	33	35	s	s	VERB
ejpam-3889	33	36	a	a	DET
ejpam-3889	33	37	saturated	saturate	VERB
ejpam-3889	33	38	multiplicative	multiplicative	ADJ
ejpam-3889	33	39	part	part	NOUN
ejpam-3889	33	40	formed	form	VERB
ejpam-3889	33	41	by	by	ADP
ejpam-3889	33	42	the	the	DET
ejpam-3889	33	43	non	non	ADJ
ejpam-3889	33	44	-	-	ADJ
ejpam-3889	33	45	zero	zero	ADJ
ejpam-3889	33	46	homogeneous	homogeneous	ADJ
ejpam-3889	33	47	elements	element	NOUN
ejpam-3889	33	48	of	of	ADP
ejpam-3889	33	49	a	a	DET
ejpam-3889	33	50	verifying	verifying	NOUN
ejpam-3889	33	51	the	the	DET
ejpam-3889	33	52	left	left	ADJ
ejpam-3889	33	53	ore	ore	NOUN
ejpam-3889	33	54	conditions	condition	NOUN
ejpam-3889	33	55	,	,	PUNCT
ejpam-3889	33	56	m	m	VERB
ejpam-3889	33	57	a	a	DET
ejpam-3889	33	58	graded	grade	VERB
ejpam-3889	33	59	left	leave	VERB
ejpam-3889	33	60	a	a	DET
ejpam-3889	33	61	-	-	PUNCT
ejpam-3889	33	62	module	module	NOUN
ejpam-3889	33	63	.	.	PUNCT
ejpam-3889	34	1	then	then	ADV
ejpam-3889	34	2	,	,	PUNCT
ejpam-3889	34	3	if	if	SCONJ
ejpam-3889	34	4	s−1(m	s−1(m	PROPN
ejpam-3889	34	5	)	)	PUNCT
ejpam-3889	34	6	is	be	AUX
ejpam-3889	34	7	a	a	DET
ejpam-3889	34	8	hopfian	hopfian	ADJ
ejpam-3889	34	9	left	leave	VERB
ejpam-3889	34	10	graded	grade	VERB
ejpam-3889	34	11	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	34	12	,	,	PUNCT
ejpam-3889	34	13	implies	imply	VERB
ejpam-3889	34	14	that	that	SCONJ
ejpam-3889	34	15	m	m	PROPN
ejpam-3889	34	16	is	be	AUX
ejpam-3889	34	17	a	a	DET
ejpam-3889	34	18	hopfian	hopfian	ADJ
ejpam-3889	34	19	left	leave	VERB
ejpam-3889	34	20	graded	grade	VERB
ejpam-3889	34	21	a	a	DET
ejpam-3889	34	22	-	-	PUNCT
ejpam-3889	34	23	module	module	NOUN
ejpam-3889	34	24	;	;	PUNCT
ejpam-3889	34	25	(	(	PUNCT
ejpam-3889	34	26	iii	iii	X
ejpam-3889	34	27	)	)	PUNCT
ejpam-3889	34	28	if	if	SCONJ
ejpam-3889	34	29	s−1(m	s−1(m	PROPN
ejpam-3889	34	30	)	)	PUNCT
ejpam-3889	34	31	is	be	AUX
ejpam-3889	34	32	a	a	DET
ejpam-3889	34	33	left	left	NOUN
ejpam-3889	34	34	graded	grade	VERB
ejpam-3889	34	35	hopfian	hopfian	PROPN
ejpam-3889	34	36	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	34	37	,	,	PUNCT
ejpam-3889	34	38	then	then	ADV
ejpam-3889	34	39	mn	mn	PROPN
ejpam-3889	34	40	for	for	ADP
ejpam-3889	34	41	all	all	DET
ejpam-3889	34	42	n	n	PRON
ejpam-3889	34	43	∈	∈	PROPN
ejpam-3889	34	44	z	z	NOUN
ejpam-3889	34	45	is	be	AUX
ejpam-3889	34	46	a	a	DET
ejpam-3889	34	47	hopfian	hopfian	ADJ
ejpam-3889	34	48	group	group	NOUN
ejpam-3889	34	49	;	;	PUNCT
ejpam-3889	34	50	(	(	PUNCT
ejpam-3889	34	51	iv	iv	X
ejpam-3889	34	52	)	)	PUNCT
ejpam-3889	34	53	let	let	VERB
ejpam-3889	34	54	a	a	PRON
ejpam-3889	34	55	be	be	AUX
ejpam-3889	34	56	a	a	DET
ejpam-3889	34	57	graded	grade	VERB
ejpam-3889	34	58	ring	ring	NOUN
ejpam-3889	34	59	,	,	PUNCT
ejpam-3889	34	60	s	s	VERB
ejpam-3889	34	61	a	a	DET
ejpam-3889	34	62	saturated	saturate	VERB
ejpam-3889	34	63	multiplicative	multiplicative	ADJ
ejpam-3889	34	64	part	part	NOUN
ejpam-3889	34	65	formed	form	VERB
ejpam-3889	34	66	by	by	ADP
ejpam-3889	34	67	the	the	DET
ejpam-3889	34	68	non	non	ADJ
ejpam-3889	34	69	-	-	ADJ
ejpam-3889	34	70	zero	zero	ADJ
ejpam-3889	34	71	homogeneous	homogeneous	ADJ
ejpam-3889	34	72	elements	element	NOUN
ejpam-3889	34	73	of	of	ADP
ejpam-3889	34	74	a	a	DET
ejpam-3889	34	75	verifying	verifying	NOUN
ejpam-3889	34	76	the	the	DET
ejpam-3889	34	77	left	left	ADJ
ejpam-3889	34	78	ore	ore	NOUN
ejpam-3889	34	79	conditions	condition	NOUN
ejpam-3889	34	80	,	,	PUNCT
ejpam-3889	34	81	m	m	AUX
ejpam-3889	34	82	left	leave	VERB
ejpam-3889	34	83	graded	grade	VERB
ejpam-3889	34	84	a	a	DET
ejpam-3889	34	85	-	-	PUNCT
ejpam-3889	34	86	module	module	NOUN
ejpam-3889	34	87	.	.	PUNCT
ejpam-3889	35	1	then	then	ADV
ejpam-3889	35	2	,	,	PUNCT
ejpam-3889	35	3	if	if	SCONJ
ejpam-3889	35	4	m	m	NOUN
ejpam-3889	35	5	is	be	AUX
ejpam-3889	35	6	a	a	DET
ejpam-3889	35	7	cohopfian	cohopfian	ADJ
ejpam-3889	35	8	and	and	CCONJ
ejpam-3889	35	9	completely	completely	ADV
ejpam-3889	35	10	invariant	invariant	ADJ
ejpam-3889	35	11	submodule	submodule	NOUN
ejpam-3889	35	12	of	of	ADP
ejpam-3889	35	13	left	leave	VERB
ejpam-3889	35	14	a	a	DET
ejpam-3889	35	15	-	-	PUNCT
ejpam-3889	35	16	module	module	NOUN
ejpam-3889	35	17	s−1(m	s−1(m	PROPN
ejpam-3889	35	18	)	)	PUNCT
ejpam-3889	35	19	,	,	PUNCT
ejpam-3889	35	20	implies	imply	VERB
ejpam-3889	35	21	that	that	SCONJ
ejpam-3889	35	22	,	,	PUNCT
ejpam-3889	35	23	s−1(m	s−1(m	PROPN
ejpam-3889	35	24	)	)	PUNCT
ejpam-3889	35	25	is	be	AUX
ejpam-3889	35	26	a	a	DET
ejpam-3889	35	27	cohopfian	cohopfian	ADJ
ejpam-3889	35	28	graded	grade	VERB
ejpam-3889	35	29	left	leave	VERB
ejpam-3889	35	30	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	35	31	;	;	PUNCT
ejpam-3889	35	32	(	(	PUNCT
ejpam-3889	35	33	v	v	NOUN
ejpam-3889	35	34	)	)	PUNCT
ejpam-3889	35	35	ifm	ifm	PROPN
ejpam-3889	35	36	is	be	AUX
ejpam-3889	35	37	a	a	DET
ejpam-3889	35	38	cohopfian	cohopfian	ADJ
ejpam-3889	35	39	and	and	CCONJ
ejpam-3889	35	40	completely	completely	ADV
ejpam-3889	35	41	invariant	invariant	ADJ
ejpam-3889	35	42	submodule	submodule	NOUN
ejpam-3889	35	43	of	of	ADP
ejpam-3889	35	44	the	the	DET
ejpam-3889	35	45	lefta	lefta	NOUN
ejpam-3889	35	46	-	-	PUNCT
ejpam-3889	35	47	module	module	NOUN
ejpam-3889	35	48	s−1(m	s−1(m	PROPN
ejpam-3889	35	49	)	)	PUNCT
ejpam-3889	35	50	,	,	PUNCT
ejpam-3889	35	51	then	then	ADV
ejpam-3889	35	52	(	(	PUNCT
ejpam-3889	35	53	s−1m)i	s−1m)i	NOUN
ejpam-3889	35	54	is	be	AUX
ejpam-3889	35	55	a	a	DET
ejpam-3889	35	56	cohopfian	cohopfian	ADJ
ejpam-3889	35	57	group	group	NOUN
ejpam-3889	35	58	;	;	PUNCT
ejpam-3889	35	59	(	(	PUNCT
ejpam-3889	35	60	vi	vi	X
ejpam-3889	35	61	)	)	PUNCT
ejpam-3889	35	62	let	let	VERB
ejpam-3889	35	63	m	m	PRON
ejpam-3889	35	64	be	be	AUX
ejpam-3889	35	65	a	a	DET
ejpam-3889	35	66	noetherian	noetherian	ADJ
ejpam-3889	35	67	quasi	quasi	ADJ
ejpam-3889	35	68	-	-	ADJ
ejpam-3889	35	69	injective	injective	ADJ
ejpam-3889	35	70	graded	grade	VERB
ejpam-3889	35	71	left	leave	VERB
ejpam-3889	35	72	a	a	DET
ejpam-3889	35	73	-	-	PUNCT
ejpam-3889	35	74	module	module	NOUN
ejpam-3889	35	75	,	,	PUNCT
ejpam-3889	35	76	n	n	PRON
ejpam-3889	35	77	be	be	VERB
ejpam-3889	35	78	an	an	DET
ejpam-3889	35	79	essential	essential	ADJ
ejpam-3889	35	80	and	and	CCONJ
ejpam-3889	35	81	completely	completely	ADV
ejpam-3889	35	82	invariant	invariant	ADJ
ejpam-3889	35	83	graded	grade	VERB
ejpam-3889	35	84	submodule	submodule	NOUN
ejpam-3889	35	85	of	of	ADP
ejpam-3889	35	86	m	m	PROPN
ejpam-3889	35	87	and	and	CCONJ
ejpam-3889	35	88	s	s	VERB
ejpam-3889	35	89	a	a	DET
ejpam-3889	35	90	saturated	saturate	VERB
ejpam-3889	35	91	multiplicative	multiplicative	ADJ
ejpam-3889	35	92	part	part	NOUN
ejpam-3889	35	93	formed	form	VERB
ejpam-3889	35	94	by	by	ADP
ejpam-3889	35	95	the	the	DET
ejpam-3889	35	96	non	non	ADJ
ejpam-3889	35	97	-	-	ADJ
ejpam-3889	35	98	zero	zero	ADJ
ejpam-3889	35	99	homogeneous	homogeneous	ADJ
ejpam-3889	35	100	elements	element	NOUN
ejpam-3889	35	101	of	of	ADP
ejpam-3889	35	102	a	a	DET
ejpam-3889	35	103	verifying	verifying	NOUN
ejpam-3889	35	104	the	the	DET
ejpam-3889	35	105	left	left	ADJ
ejpam-3889	35	106	ore	ore	NOUN
ejpam-3889	35	107	conditions	condition	NOUN
ejpam-3889	35	108	.	.	PUNCT
ejpam-3889	36	1	then	then	ADV
ejpam-3889	36	2	,	,	PUNCT
ejpam-3889	36	3	the	the	DET
ejpam-3889	36	4	graded	grade	VERB
ejpam-3889	36	5	left	leave	VERB
ejpam-3889	36	6	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	36	7	s−1(m	s−1(m	PROPN
ejpam-3889	36	8	)	)	PUNCT
ejpam-3889	36	9	is	be	AUX
ejpam-3889	36	10	cohopfian	cohopfian	ADJ
ejpam-3889	36	11	if	if	SCONJ
ejpam-3889	36	12	,	,	PUNCT
ejpam-3889	36	13	and	and	CCONJ
ejpam-3889	36	14	only	only	ADV
ejpam-3889	36	15	,	,	PUNCT
ejpam-3889	36	16	if	if	SCONJ
ejpam-3889	36	17	s−1(n	s−1(n	NOUN
ejpam-3889	36	18	)	)	PUNCT
ejpam-3889	36	19	is	be	AUX
ejpam-3889	36	20	cohopfian	cohopfian	ADJ
ejpam-3889	36	21	graded	grade	VERB
ejpam-3889	36	22	left	leave	VERB
ejpam-3889	36	23	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	36	24	;	;	PUNCT
ejpam-3889	36	25	(	(	PUNCT
ejpam-3889	36	26	vii	vii	PROPN
ejpam-3889	36	27	)	)	PUNCT
ejpam-3889	36	28	let	let	VERB
ejpam-3889	36	29	m	m	PRON
ejpam-3889	36	30	be	be	AUX
ejpam-3889	36	31	a	a	DET
ejpam-3889	36	32	graded	grade	VERB
ejpam-3889	36	33	quasi	quasi	NOUN
ejpam-3889	36	34	-	-	NOUN
ejpam-3889	36	35	projective	projective	ADJ
ejpam-3889	36	36	left	leave	VERB
ejpam-3889	36	37	a	a	DET
ejpam-3889	36	38	-	-	PUNCT
ejpam-3889	36	39	module	module	NOUN
ejpam-3889	36	40	,	,	PUNCT
ejpam-3889	36	41	n	n	CCONJ
ejpam-3889	36	42	a	a	PRON
ejpam-3889	36	43	superfluous	superfluous	ADJ
ejpam-3889	36	44	and	and	CCONJ
ejpam-3889	36	45	completely	completely	ADV
ejpam-3889	36	46	invariant	invariant	ADJ
ejpam-3889	36	47	graded	grade	VERB
ejpam-3889	36	48	sub	sub	NOUN
ejpam-3889	36	49	-	-	NOUN
ejpam-3889	36	50	module	module	NOUN
ejpam-3889	36	51	of	of	ADP
ejpam-3889	36	52	m	m	PRON
ejpam-3889	36	53	,	,	PUNCT
ejpam-3889	36	54	s	s	VERB
ejpam-3889	36	55	a	a	DET
ejpam-3889	36	56	saturated	saturate	VERB
ejpam-3889	36	57	multiplicative	multiplicative	ADJ
ejpam-3889	36	58	part	part	NOUN
ejpam-3889	36	59	formed	form	VERB
ejpam-3889	36	60	by	by	ADP
ejpam-3889	36	61	the	the	DET
ejpam-3889	36	62	non	non	ADJ
ejpam-3889	36	63	-	-	ADJ
ejpam-3889	36	64	zero	zero	ADJ
ejpam-3889	36	65	homogeneous	homogeneous	ADJ
ejpam-3889	36	66	elements	element	NOUN
ejpam-3889	36	67	of	of	ADP
ejpam-3889	36	68	a	a	DET
ejpam-3889	36	69	verifying	verifying	NOUN
ejpam-3889	36	70	the	the	DET
ejpam-3889	36	71	left	left	ADJ
ejpam-3889	36	72	ore	ore	NOUN
ejpam-3889	36	73	conditions	condition	NOUN
ejpam-3889	36	74	.	.	PUNCT
ejpam-3889	37	1	then	then	ADV
ejpam-3889	37	2	the	the	DET
ejpam-3889	37	3	left	left	ADJ
ejpam-3889	37	4	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	37	5	s−1(n	s−1(n	NOUN
ejpam-3889	37	6	)	)	PUNCT
ejpam-3889	37	7	is	be	AUX
ejpam-3889	37	8	hopfian	hopfian	ADJ
ejpam-3889	37	9	if	if	SCONJ
ejpam-3889	37	10	,	,	PUNCT
ejpam-3889	37	11	and	and	CCONJ
ejpam-3889	37	12	only	only	ADV
ejpam-3889	37	13	if	if	SCONJ
ejpam-3889	37	14	,	,	PUNCT
ejpam-3889	37	15	s−1(m	s−1(m	PROPN
ejpam-3889	37	16	/	/	SYM
ejpam-3889	37	17	n	n	CCONJ
ejpam-3889	37	18	)	)	PUNCT
ejpam-3889	37	19	is	be	AUX
ejpam-3889	37	20	hopfian	hopfian	ADJ
ejpam-3889	37	21	.	.	PUNCT
ejpam-3889	38	1	in	in	ADP
ejpam-3889	38	2	the	the	DET
ejpam-3889	38	3	section	section	NOUN
ejpam-3889	38	4	4	4	NUM
ejpam-3889	38	5	,	,	PUNCT
ejpam-3889	38	6	we	we	PRON
ejpam-3889	38	7	study	study	VERB
ejpam-3889	38	8	the	the	DET
ejpam-3889	38	9	localization	localization	NOUN
ejpam-3889	38	10	of	of	ADP
ejpam-3889	38	11	hopfian	hopfian	ADJ
ejpam-3889	38	12	and	and	CCONJ
ejpam-3889	38	13	cohopfian	cohopfian	ADJ
ejpam-3889	38	14	objects	object	NOUN
ejpam-3889	38	15	in	in	ADP
ejpam-3889	38	16	the	the	DET
ejpam-3889	38	17	category	category	NOUN
ejpam-3889	38	18	comp	comp	NOUN
ejpam-3889	38	19	(	(	PUNCT
ejpam-3889	38	20	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	38	21	)	)	PUNCT
ejpam-3889	38	22	)	)	PUNCT
ejpam-3889	38	23	and	and	CCONJ
ejpam-3889	38	24	we	we	PRON
ejpam-3889	38	25	show	show	VERB
ejpam-3889	38	26	the	the	DET
ejpam-3889	38	27	following	follow	VERB
ejpam-3889	38	28	results	result	NOUN
ejpam-3889	38	29	:	:	PUNCT
ejpam-3889	38	30	(	(	PUNCT
ejpam-3889	38	31	i	i	NOUN
ejpam-3889	38	32	)	)	PUNCT
ejpam-3889	38	33	let	let	VERB
ejpam-3889	38	34	a	a	PRON
ejpam-3889	38	35	be	be	AUX
ejpam-3889	38	36	a	a	DET
ejpam-3889	38	37	graded	grade	VERB
ejpam-3889	38	38	ring	ring	NOUN
ejpam-3889	38	39	,	,	PUNCT
ejpam-3889	38	40	s	s	VERB
ejpam-3889	38	41	a	a	DET
ejpam-3889	38	42	saturated	saturate	VERB
ejpam-3889	38	43	multiplicative	multiplicative	ADJ
ejpam-3889	38	44	part	part	NOUN
ejpam-3889	38	45	formed	form	VERB
ejpam-3889	38	46	by	by	ADP
ejpam-3889	38	47	the	the	DET
ejpam-3889	38	48	non	non	ADJ
ejpam-3889	38	49	-	-	ADJ
ejpam-3889	38	50	zero	zero	ADJ
ejpam-3889	38	51	homogeneous	homogeneous	ADJ
ejpam-3889	38	52	elements	element	NOUN
ejpam-3889	38	53	of	of	ADP
ejpam-3889	38	54	a	a	DET
ejpam-3889	38	55	verifying	verifying	NOUN
ejpam-3889	38	56	the	the	DET
ejpam-3889	38	57	left	left	ADJ
ejpam-3889	38	58	ore	ore	NOUN
ejpam-3889	38	59	conditions	condition	NOUN
ejpam-3889	38	60	,	,	PUNCT
ejpam-3889	38	61	m	m	VERB
ejpam-3889	38	62	a	a	DET
ejpam-3889	38	63	graded	grade	VERB
ejpam-3889	38	64	left	leave	VERB
ejpam-3889	38	65	a	a	DET
ejpam-3889	38	66	-	-	PUNCT
ejpam-3889	38	67	module	module	NOUN
ejpam-3889	38	68	and	and	CCONJ
ejpam-3889	38	69	m∗	m∗	VERB
ejpam-3889	38	70	the	the	DET
ejpam-3889	38	71	complex	complex	ADJ
ejpam-3889	38	72	sequence	sequence	NOUN
ejpam-3889	38	73	of	of	ADP
ejpam-3889	38	74	morphisms	morphism	NOUN
ejpam-3889	38	75	of	of	ADP
ejpam-3889	38	76	graded	grade	VERB
ejpam-3889	38	77	left	leave	VERB
ejpam-3889	38	78	a	a	DET
ejpam-3889	38	79	-	-	PUNCT
ejpam-3889	38	80	modules	module	NOUN
ejpam-3889	38	81	associated	associate	VERB
ejpam-3889	38	82	with	with	ADP
ejpam-3889	38	83	m	m	PROPN
ejpam-3889	38	84	.	.	PUNCT
ejpam-3889	39	1	then	then	ADV
ejpam-3889	39	2	,	,	PUNCT
ejpam-3889	39	3	if	if	SCONJ
ejpam-3889	39	4	the	the	DET
ejpam-3889	39	5	complexe	complexe	PROPN
ejpam-3889	39	6	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	39	7	)	)	PUNCT
ejpam-3889	39	8	of	of	ADP
ejpam-3889	39	9	morphisms	morphism	NOUN
ejpam-3889	39	10	of	of	ADP
ejpam-3889	39	11	graded	grade	VERB
ejpam-3889	39	12	left	leave	VERB
ejpam-3889	39	13	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	39	14	is	be	AUX
ejpam-3889	39	15	hopfian	hopfian	ADJ
ejpam-3889	39	16	,	,	PUNCT
ejpam-3889	39	17	implies	imply	VERB
ejpam-3889	39	18	that	that	SCONJ
ejpam-3889	39	19	m∗	m∗	PROPN
ejpam-3889	39	20	is	be	AUX
ejpam-3889	39	21	hopfian	hopfian	ADJ
ejpam-3889	39	22	;	;	PUNCT
ejpam-3889	39	23	(	(	PUNCT
ejpam-3889	39	24	ii	ii	NOUN
ejpam-3889	39	25	)	)	PUNCT
ejpam-3889	39	26	let	let	VERB
ejpam-3889	39	27	a	a	PRON
ejpam-3889	39	28	be	be	AUX
ejpam-3889	39	29	a	a	DET
ejpam-3889	39	30	graded	grade	VERB
ejpam-3889	39	31	ring	ring	NOUN
ejpam-3889	39	32	,	,	PUNCT
ejpam-3889	39	33	s	s	VERB
ejpam-3889	39	34	a	a	DET
ejpam-3889	39	35	saturated	saturate	VERB
ejpam-3889	39	36	multiplicative	multiplicative	ADJ
ejpam-3889	39	37	part	part	NOUN
ejpam-3889	39	38	formed	form	VERB
ejpam-3889	39	39	by	by	ADP
ejpam-3889	39	40	the	the	DET
ejpam-3889	39	41	non	non	ADJ
ejpam-3889	39	42	-	-	ADJ
ejpam-3889	39	43	zero	zero	ADJ
ejpam-3889	39	44	homogeneous	homogeneous	ADJ
ejpam-3889	39	45	elements	element	NOUN
ejpam-3889	39	46	of	of	ADP
ejpam-3889	39	47	a	a	DET
ejpam-3889	39	48	verifying	verifying	NOUN
ejpam-3889	39	49	the	the	DET
ejpam-3889	39	50	left	left	ADJ
ejpam-3889	39	51	ore	ore	NOUN
ejpam-3889	39	52	conditions	condition	NOUN
ejpam-3889	39	53	,	,	PUNCT
ejpam-3889	39	54	m	m	VERB
ejpam-3889	39	55	a	a	DET
ejpam-3889	39	56	graded	grade	VERB
ejpam-3889	39	57	left	leave	VERB
ejpam-3889	39	58	amodule	amodule	NOUN
ejpam-3889	39	59	.	.	PUNCT
ejpam-3889	40	1	if	if	SCONJ
ejpam-3889	40	2	m∗	m∗	VERB
ejpam-3889	40	3	the	the	DET
ejpam-3889	40	4	complex	complex	ADJ
ejpam-3889	40	5	sequence	sequence	NOUN
ejpam-3889	40	6	associated	associate	VERB
ejpam-3889	40	7	with	with	ADP
ejpam-3889	40	8	m	m	PRON
ejpam-3889	40	9	,	,	PUNCT
ejpam-3889	40	10	is	be	AUX
ejpam-3889	40	11	cohopfian	cohopfian	ADJ
ejpam-3889	40	12	and	and	CCONJ
ejpam-3889	40	13	completely	completely	ADV
ejpam-3889	40	14	invariant	invariant	ADJ
ejpam-3889	40	15	,	,	PUNCT
ejpam-3889	40	16	then	then	ADV
ejpam-3889	40	17	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	40	18	)	)	PUNCT
ejpam-3889	40	19	,	,	PUNCT
ejpam-3889	40	20	the	the	DET
ejpam-3889	40	21	complex	complex	ADJ
ejpam-3889	40	22	sequence	sequence	NOUN
ejpam-3889	40	23	associated	associate	VERB
ejpam-3889	40	24	with	with	ADP
ejpam-3889	40	25	s−1(m	s−1(m	PROPN
ejpam-3889	40	26	)	)	PUNCT
ejpam-3889	40	27	is	be	AUX
ejpam-3889	40	28	cohopfian	cohopfian	ADJ
ejpam-3889	40	29	;	;	PUNCT
ejpam-3889	40	30	s.	s.	PROPN
ejpam-3889	40	31	a.	a.	PROPN
ejpam-3889	40	32	balde	balde	PROPN
ejpam-3889	40	33	,	,	PUNCT
ejpam-3889	40	34	m.	m.	PROPN
ejpam-3889	40	35	b.	b.	PROPN
ejpam-3889	40	36	maaouia	maaouia	PROPN
ejpam-3889	40	37	,	,	PUNCT
ejpam-3889	40	38	a.	a.	NOUN
ejpam-3889	40	39	o.	o.	NOUN
ejpam-3889	40	40	chbih	chbih	PROPN
ejpam-3889	40	41	/	/	SYM
ejpam-3889	40	42	eur	eur	PROPN
ejpam-3889	40	43	.	.	PUNCT
ejpam-3889	41	1	j.	j.	PROPN
ejpam-3889	41	2	pure	pure	PROPN
ejpam-3889	41	3	appl	appl	PROPN
ejpam-3889	41	4	.	.	PROPN
ejpam-3889	41	5	math	math	PROPN
ejpam-3889	41	6	,	,	PUNCT
ejpam-3889	41	7	14	14	NUM
ejpam-3889	41	8	(	(	PUNCT
ejpam-3889	41	9	2	2	NUM
ejpam-3889	41	10	)	)	PUNCT
ejpam-3889	41	11	(	(	PUNCT
ejpam-3889	41	12	2021	2021	NUM
ejpam-3889	41	13	)	)	PUNCT
ejpam-3889	41	14	,	,	PUNCT
ejpam-3889	41	15	404	404	NUM
ejpam-3889	41	16	-	-	SYM
ejpam-3889	41	17	422	422	NUM
ejpam-3889	41	18	407	407	NUM
ejpam-3889	41	19	(	(	PUNCT
ejpam-3889	41	20	iii	iii	NOUN
ejpam-3889	41	21	)	)	PUNCT
ejpam-3889	41	22	let	let	VERB
ejpam-3889	41	23	m	m	PRON
ejpam-3889	41	24	be	be	AUX
ejpam-3889	41	25	a	a	DET
ejpam-3889	41	26	noetherian	noetherian	ADJ
ejpam-3889	41	27	graded	grade	VERB
ejpam-3889	41	28	left	leave	VERB
ejpam-3889	41	29	a	a	DET
ejpam-3889	41	30	-	-	PUNCT
ejpam-3889	41	31	module	module	NOUN
ejpam-3889	41	32	,	,	PUNCT
ejpam-3889	41	33	s	s	VERB
ejpam-3889	41	34	a	a	DET
ejpam-3889	41	35	saturated	saturate	VERB
ejpam-3889	41	36	multiplicative	multiplicative	ADJ
ejpam-3889	41	37	part	part	NOUN
ejpam-3889	41	38	formed	form	VERB
ejpam-3889	41	39	by	by	ADP
ejpam-3889	41	40	the	the	DET
ejpam-3889	41	41	non	non	ADJ
ejpam-3889	41	42	-	-	ADJ
ejpam-3889	41	43	zero	zero	ADJ
ejpam-3889	41	44	homogeneous	homogeneous	ADJ
ejpam-3889	41	45	elements	element	NOUN
ejpam-3889	41	46	of	of	ADP
ejpam-3889	41	47	a	a	DET
ejpam-3889	41	48	verifying	verifying	NOUN
ejpam-3889	41	49	the	the	DET
ejpam-3889	41	50	left	left	ADJ
ejpam-3889	41	51	ore	ore	NOUN
ejpam-3889	41	52	conditions	condition	NOUN
ejpam-3889	41	53	,	,	PUNCT
ejpam-3889	41	54	n	n	CCONJ
ejpam-3889	41	55	a	a	DET
ejpam-3889	41	56	submodule	submodule	NOUN
ejpam-3889	41	57	of	of	ADP
ejpam-3889	41	58	m	m	PROPN
ejpam-3889	41	59	,	,	PUNCT
ejpam-3889	41	60	m∗	m∗	PROPN
ejpam-3889	41	61	is	be	AUX
ejpam-3889	41	62	a	a	DET
ejpam-3889	41	63	noetherian	noetherian	ADJ
ejpam-3889	41	64	quasi	quasi	ADJ
ejpam-3889	41	65	-	-	ADJ
ejpam-3889	41	66	injective	injective	ADJ
ejpam-3889	41	67	complex	complex	ADJ
ejpam-3889	41	68	sequence	sequence	NOUN
ejpam-3889	41	69	associated	associate	VERB
ejpam-3889	41	70	with	with	ADP
ejpam-3889	41	71	m	m	NOUN
ejpam-3889	41	72	and	and	CCONJ
ejpam-3889	41	73	n∗	n∗	PROPN
ejpam-3889	41	74	is	be	AUX
ejpam-3889	41	75	an	an	DET
ejpam-3889	41	76	essential	essential	ADJ
ejpam-3889	41	77	and	and	CCONJ
ejpam-3889	41	78	completely	completely	ADV
ejpam-3889	41	79	invariant	invariant	ADJ
ejpam-3889	41	80	complex	complex	ADJ
ejpam-3889	41	81	sub	sub	NOUN
ejpam-3889	41	82	-	-	NOUN
ejpam-3889	41	83	sequence	sequence	NOUN
ejpam-3889	41	84	of	of	ADP
ejpam-3889	41	85	m∗.	m∗.	PROPN
ejpam-3889	41	86	then	then	ADV
ejpam-3889	41	87	,	,	PUNCT
ejpam-3889	41	88	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	41	89	)	)	PUNCT
ejpam-3889	41	90	the	the	DET
ejpam-3889	41	91	complex	complex	ADJ
ejpam-3889	41	92	sequence	sequence	NOUN
ejpam-3889	41	93	of	of	ADP
ejpam-3889	41	94	morphisms	morphism	NOUN
ejpam-3889	41	95	of	of	ADP
ejpam-3889	41	96	left	left	ADJ
ejpam-3889	41	97	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	41	98	is	be	AUX
ejpam-3889	41	99	cohopfian	cohopfian	ADJ
ejpam-3889	41	100	if	if	SCONJ
ejpam-3889	41	101	,	,	PUNCT
ejpam-3889	41	102	and	and	CCONJ
ejpam-3889	41	103	only	only	ADV
ejpam-3889	41	104	,	,	PUNCT
ejpam-3889	41	105	if	if	SCONJ
ejpam-3889	41	106	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	41	107	)	)	PUNCT
ejpam-3889	41	108	is	be	AUX
ejpam-3889	41	109	cohopfian	cohopfian	ADJ
ejpam-3889	41	110	;	;	PUNCT
ejpam-3889	41	111	(	(	PUNCT
ejpam-3889	41	112	iv	iv	X
ejpam-3889	41	113	)	)	PUNCT
ejpam-3889	41	114	let	let	VERB
ejpam-3889	41	115	m	m	PRON
ejpam-3889	41	116	be	be	AUX
ejpam-3889	41	117	a	a	DET
ejpam-3889	41	118	graded	grade	VERB
ejpam-3889	41	119	left	leave	VERB
ejpam-3889	41	120	a	a	DET
ejpam-3889	41	121	-	-	PUNCT
ejpam-3889	41	122	module	module	NOUN
ejpam-3889	41	123	and	and	CCONJ
ejpam-3889	41	124	s	s	VERB
ejpam-3889	41	125	a	a	DET
ejpam-3889	41	126	saturated	saturate	VERB
ejpam-3889	41	127	multiplicative	multiplicative	ADJ
ejpam-3889	41	128	part	part	NOUN
ejpam-3889	41	129	formed	form	VERB
ejpam-3889	41	130	by	by	ADP
ejpam-3889	41	131	the	the	DET
ejpam-3889	41	132	non	non	ADJ
ejpam-3889	41	133	-	-	ADJ
ejpam-3889	41	134	zero	zero	ADJ
ejpam-3889	41	135	homogeneous	homogeneous	ADJ
ejpam-3889	41	136	elements	element	NOUN
ejpam-3889	41	137	of	of	ADP
ejpam-3889	41	138	a	a	DET
ejpam-3889	41	139	verifying	verifying	NOUN
ejpam-3889	41	140	the	the	DET
ejpam-3889	41	141	left	left	ADJ
ejpam-3889	41	142	ore	ore	NOUN
ejpam-3889	41	143	conditions	condition	NOUN
ejpam-3889	41	144	.	.	PUNCT
ejpam-3889	42	1	if	if	SCONJ
ejpam-3889	42	2	m∗	m∗	PROPN
ejpam-3889	42	3	is	be	AUX
ejpam-3889	42	4	a	a	DET
ejpam-3889	42	5	hopfian	hopfian	ADJ
ejpam-3889	42	6	,	,	PUNCT
ejpam-3889	42	7	noetherian	noetherian	ADJ
ejpam-3889	42	8	and	and	CCONJ
ejpam-3889	42	9	quasi	quasi	ADJ
ejpam-3889	42	10	-	-	ADJ
ejpam-3889	42	11	injective	injective	ADJ
ejpam-3889	42	12	complex	complex	ADJ
ejpam-3889	42	13	sequence	sequence	NOUN
ejpam-3889	42	14	associated	associate	VERB
ejpam-3889	42	15	with	with	ADP
ejpam-3889	42	16	m	m	PROPN
ejpam-3889	42	17	,	,	PUNCT
ejpam-3889	42	18	then	then	ADV
ejpam-3889	42	19	the	the	DET
ejpam-3889	42	20	complex	complex	ADJ
ejpam-3889	42	21	sequence	sequence	NOUN
ejpam-3889	42	22	of	of	ADP
ejpam-3889	42	23	morphisms	morphism	NOUN
ejpam-3889	42	24	of	of	ADP
ejpam-3889	42	25	left	left	ADJ
ejpam-3889	42	26	s−1a	s−1a	NOUN
ejpam-3889	42	27	-	-	PUNCT
ejpam-3889	42	28	modules	module	NOUN
ejpam-3889	42	29	s−1m∗	s−1m∗	NOUN
ejpam-3889	42	30	has	have	VERB
ejpam-3889	42	31	the	the	DET
ejpam-3889	42	32	following	follow	VERB
ejpam-3889	42	33	property	property	NOUN
ejpam-3889	42	34	:	:	PUNCT
ejpam-3889	42	35	�	�	VERB
ejpam-3889	42	36	any	any	DET
ejpam-3889	42	37	epimorphism	epimorphism	NOUN
ejpam-3889	42	38	of	of	ADP
ejpam-3889	42	39	sub	sub	ADJ
ejpam-3889	42	40	-	-	ADJ
ejpam-3889	42	41	complex	complex	ADJ
ejpam-3889	42	42	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	42	43	)	)	PUNCT
ejpam-3889	42	44	of	of	ADP
ejpam-3889	42	45	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	42	46	)	)	PUNCT
ejpam-3889	42	47	is	be	AUX
ejpam-3889	42	48	an	an	DET
ejpam-3889	42	49	isomorphism	isomorphism	NOUN
ejpam-3889	42	50	�	�	PROPN
ejpam-3889	42	51	;	;	PUNCT
ejpam-3889	42	52	(	(	PUNCT
ejpam-3889	42	53	v	v	NOUN
ejpam-3889	42	54	)	)	PUNCT
ejpam-3889	42	55	let	let	VERB
ejpam-3889	42	56	m	m	PRON
ejpam-3889	42	57	be	be	AUX
ejpam-3889	42	58	a	a	DET
ejpam-3889	42	59	graded	grade	VERB
ejpam-3889	42	60	left	leave	VERB
ejpam-3889	42	61	a	a	DET
ejpam-3889	42	62	-	-	PUNCT
ejpam-3889	42	63	module	module	NOUN
ejpam-3889	42	64	,	,	PUNCT
ejpam-3889	42	65	n	n	CCONJ
ejpam-3889	42	66	a	a	DET
ejpam-3889	42	67	graded	grade	VERB
ejpam-3889	42	68	submodule	submodule	NOUN
ejpam-3889	42	69	of	of	ADP
ejpam-3889	42	70	m	m	PROPN
ejpam-3889	42	71	,	,	PUNCT
ejpam-3889	42	72	s	s	VERB
ejpam-3889	42	73	a	a	DET
ejpam-3889	42	74	saturated	saturate	VERB
ejpam-3889	42	75	multiplicative	multiplicative	ADJ
ejpam-3889	42	76	part	part	NOUN
ejpam-3889	42	77	formed	form	VERB
ejpam-3889	42	78	by	by	ADP
ejpam-3889	42	79	the	the	DET
ejpam-3889	42	80	non	non	ADJ
ejpam-3889	42	81	-	-	ADJ
ejpam-3889	42	82	zero	zero	ADJ
ejpam-3889	42	83	homogeneous	homogeneous	ADJ
ejpam-3889	42	84	elements	element	NOUN
ejpam-3889	42	85	of	of	ADP
ejpam-3889	42	86	a	a	DET
ejpam-3889	42	87	verifying	verifying	NOUN
ejpam-3889	42	88	the	the	DET
ejpam-3889	42	89	left	left	ADJ
ejpam-3889	42	90	ore	ore	NOUN
ejpam-3889	42	91	conditions	condition	NOUN
ejpam-3889	42	92	.	.	PUNCT
ejpam-3889	43	1	m∗	m∗	VERB
ejpam-3889	43	2	the	the	DET
ejpam-3889	43	3	quasi	quasi	ADJ
ejpam-3889	43	4	-	-	ADJ
ejpam-3889	43	5	projective	projective	ADJ
ejpam-3889	43	6	complex	complex	ADJ
ejpam-3889	43	7	sequence	sequence	NOUN
ejpam-3889	43	8	associated	associate	VERB
ejpam-3889	43	9	with	with	ADP
ejpam-3889	43	10	m	m	PROPN
ejpam-3889	43	11	and	and	CCONJ
ejpam-3889	43	12	n∗	n∗	VERB
ejpam-3889	43	13	a	a	DET
ejpam-3889	43	14	superfluous	superfluous	ADJ
ejpam-3889	43	15	and	and	CCONJ
ejpam-3889	43	16	completely	completely	ADV
ejpam-3889	43	17	invariant	invariant	ADJ
ejpam-3889	43	18	complex	complex	ADJ
ejpam-3889	43	19	sub	sub	NOUN
ejpam-3889	43	20	-	-	NOUN
ejpam-3889	43	21	sequence	sequence	NOUN
ejpam-3889	43	22	of	of	ADP
ejpam-3889	43	23	m∗.	m∗.	PROPN
ejpam-3889	43	24	then	then	ADV
ejpam-3889	43	25	the	the	DET
ejpam-3889	43	26	complex	complex	ADJ
ejpam-3889	43	27	morphism	morphism	NOUN
ejpam-3889	43	28	sequence	sequence	NOUN
ejpam-3889	43	29	of	of	ADP
ejpam-3889	43	30	left	left	ADJ
ejpam-3889	43	31	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	43	32	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	43	33	)	)	PUNCT
ejpam-3889	43	34	is	be	AUX
ejpam-3889	43	35	hopfian	hopfian	ADJ
ejpam-3889	43	36	if	if	SCONJ
ejpam-3889	43	37	,	,	PUNCT
ejpam-3889	43	38	and	and	CCONJ
ejpam-3889	43	39	only	only	ADV
ejpam-3889	43	40	if	if	SCONJ
ejpam-3889	43	41	,	,	PUNCT
ejpam-3889	43	42	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	43	43	)	)	PUNCT
ejpam-3889	43	44	the	the	DET
ejpam-3889	43	45	complex	complex	ADJ
ejpam-3889	43	46	sequence	sequence	NOUN
ejpam-3889	43	47	associated	associate	VERB
ejpam-3889	43	48	with	with	ADP
ejpam-3889	43	49	s−1(m	s−1(m	PROPN
ejpam-3889	43	50	/	/	SYM
ejpam-3889	43	51	n	n	CCONJ
ejpam-3889	43	52	)	)	PUNCT
ejpam-3889	43	53	is	be	AUX
ejpam-3889	43	54	hopfian	hopfian	ADJ
ejpam-3889	43	55	.	.	PUNCT
ejpam-3889	44	1	2	2	X
ejpam-3889	44	2	.	.	NUM
ejpam-3889	44	3	preliminaries	preliminary	NOUN
ejpam-3889	44	4	definition	definition	NOUN
ejpam-3889	44	5	1	1	X
ejpam-3889	44	6	.	.	PUNCT
ejpam-3889	45	1	let	let	VERB
ejpam-3889	45	2	a	a	PRON
ejpam-3889	45	3	be	be	AUX
ejpam-3889	45	4	a	a	DET
ejpam-3889	45	5	ring	ring	NOUN
ejpam-3889	45	6	and	and	CCONJ
ejpam-3889	45	7	{	{	PUNCT
ejpam-3889	45	8	an}n∈z	an}n∈z	VERB
ejpam-3889	45	9	a	a	DET
ejpam-3889	45	10	family	family	NOUN
ejpam-3889	45	11	of	of	ADP
ejpam-3889	45	12	sub	sub	NOUN
ejpam-3889	45	13	-	-	NOUN
ejpam-3889	45	14	group	group	NOUN
ejpam-3889	45	15	of	of	ADP
ejpam-3889	45	16	a.	a.	NOUN
ejpam-3889	46	1	if	if	SCONJ
ejpam-3889	46	2	(	(	PUNCT
ejpam-3889	46	3	i	i	NOUN
ejpam-3889	46	4	)	)	PUNCT
ejpam-3889	46	5	a	a	DET
ejpam-3889	46	6	=	=	PROPN
ejpam-3889	46	7	⊕	⊕	PROPN
ejpam-3889	46	8	n∈z	n∈z	VERB
ejpam-3889	46	9	an	an	PRON
ejpam-3889	46	10	;	;	PUNCT
ejpam-3889	46	11	(	(	PUNCT
ejpam-3889	46	12	ii	ii	NOUN
ejpam-3889	46	13	)	)	PUNCT
ejpam-3889	46	14	an	an	DET
ejpam-3889	46	15	·	·	PUNCT
ejpam-3889	46	16	am	am	NOUN
ejpam-3889	46	17	⊂	⊂	PROPN
ejpam-3889	46	18	an+m	an+m	PROPN
ejpam-3889	46	19	,	,	PUNCT
ejpam-3889	46	20	∀	∀	X
ejpam-3889	46	21	n	n	CCONJ
ejpam-3889	46	22	,	,	PUNCT
ejpam-3889	46	23	m	m	VERB
ejpam-3889	46	24	∈	∈	PROPN
ejpam-3889	46	25	z.	z.	PROPN
ejpam-3889	47	1	then	then	ADV
ejpam-3889	47	2	we	we	PRON
ejpam-3889	47	3	say	say	VERB
ejpam-3889	47	4	that	that	SCONJ
ejpam-3889	47	5	a	a	PRON
ejpam-3889	47	6	is	be	AUX
ejpam-3889	47	7	a	a	DET
ejpam-3889	47	8	graded	grade	VERB
ejpam-3889	47	9	ring	ring	NOUN
ejpam-3889	47	10	.	.	PUNCT
ejpam-3889	48	1	else	else	ADV
ejpam-3889	48	2	,	,	PUNCT
ejpam-3889	48	3	if	if	SCONJ
ejpam-3889	48	4	an	an	PRON
ejpam-3889	48	5	=	=	SYM
ejpam-3889	48	6	0,∀n	0,∀n	NOUN
ejpam-3889	48	7	<	<	X
ejpam-3889	48	8	0	0	NUM
ejpam-3889	48	9	.	.	PUNCT
ejpam-3889	49	1	then	then	ADV
ejpam-3889	49	2	a	a	PRON
ejpam-3889	49	3	is	be	AUX
ejpam-3889	49	4	called	call	VERB
ejpam-3889	49	5	positively	positively	ADV
ejpam-3889	49	6	graded	grade	VERB
ejpam-3889	49	7	ring	ring	NOUN
ejpam-3889	49	8	.	.	PUNCT
ejpam-3889	50	1	definition	definition	NOUN
ejpam-3889	50	2	2	2	NUM
ejpam-3889	50	3	.	.	PUNCT
ejpam-3889	51	1	let	let	VERB
ejpam-3889	51	2	a	a	DET
ejpam-3889	51	3	a	a	DET
ejpam-3889	51	4	graded	grade	VERB
ejpam-3889	51	5	ring	ring	NOUN
ejpam-3889	51	6	,	,	PUNCT
ejpam-3889	51	7	x	x	PRON
ejpam-3889	51	8	be	be	AUX
ejpam-3889	51	9	a	a	DET
ejpam-3889	51	10	non	non	ADJ
ejpam-3889	51	11	-	-	ADJ
ejpam-3889	51	12	zero	zero	NUM
ejpam-3889	51	13	element	element	NOUN
ejpam-3889	51	14	of	of	ADP
ejpam-3889	51	15	a	a	PRON
ejpam-3889	51	16	,	,	PUNCT
ejpam-3889	51	17	we	we	PRON
ejpam-3889	51	18	say	say	VERB
ejpam-3889	51	19	that	that	SCONJ
ejpam-3889	51	20	x	x	PRON
ejpam-3889	51	21	is	be	AUX
ejpam-3889	51	22	homogeneous	homogeneous	ADJ
ejpam-3889	51	23	of	of	ADP
ejpam-3889	51	24	degree	degree	NOUN
ejpam-3889	51	25	n	n	CCONJ
ejpam-3889	51	26	,	,	PUNCT
ejpam-3889	51	27	if	if	SCONJ
ejpam-3889	51	28	there	there	PRON
ejpam-3889	51	29	exists	exist	VERB
ejpam-3889	51	30	n	n	PRON
ejpam-3889	51	31	such	such	ADJ
ejpam-3889	51	32	that	that	SCONJ
ejpam-3889	51	33	x	x	SYM
ejpam-3889	51	34	∈	∈	PROPN
ejpam-3889	51	35	an	an	PRON
ejpam-3889	52	1	and	and	CCONJ
ejpam-3889	52	2	we	we	PRON
ejpam-3889	52	3	note	note	VERB
ejpam-3889	52	4	deg(x	deg(x	ADV
ejpam-3889	52	5	)	)	PUNCT
ejpam-3889	52	6	=	=	VERB
ejpam-3889	52	7	n.	n.	NOUN
ejpam-3889	52	8	such	such	ADJ
ejpam-3889	52	9	that	that	SCONJ
ejpam-3889	52	10	x	x	SYM
ejpam-3889	52	11	∈	∈	PROPN
ejpam-3889	52	12	an	an	PRON
ejpam-3889	53	1	and	and	CCONJ
ejpam-3889	53	2	we	we	PRON
ejpam-3889	53	3	note	note	VERB
ejpam-3889	53	4	that	that	SCONJ
ejpam-3889	53	5	deg(x	deg(x	ADV
ejpam-3889	53	6	)	)	PUNCT
ejpam-3889	53	7	=	=	SYM
ejpam-3889	53	8	n.	n.	NOUN
ejpam-3889	53	9	definition	definition	NOUN
ejpam-3889	53	10	3	3	X
ejpam-3889	53	11	.	.	PUNCT
ejpam-3889	54	1	let	let	VERB
ejpam-3889	54	2	a	a	DET
ejpam-3889	54	3	=	=	SYM
ejpam-3889	54	4	⊕	⊕	PROPN
ejpam-3889	54	5	n∈z	n∈z	VERB
ejpam-3889	54	6	an	an	DET
ejpam-3889	54	7	be	be	AUX
ejpam-3889	54	8	a	a	DET
ejpam-3889	54	9	graded	grade	VERB
ejpam-3889	54	10	ring	ring	NOUN
ejpam-3889	54	11	and	and	CCONJ
ejpam-3889	54	12	m	m	AUX
ejpam-3889	54	13	be	be	AUX
ejpam-3889	54	14	a	a	DET
ejpam-3889	54	15	left	left	ADJ
ejpam-3889	54	16	a−module	a−module	ADP
ejpam-3889	54	17	,	,	PUNCT
ejpam-3889	54	18	then	then	ADV
ejpam-3889	54	19	m	m	VERB
ejpam-3889	54	20	is	be	AUX
ejpam-3889	54	21	called	call	VERB
ejpam-3889	54	22	a	a	DET
ejpam-3889	54	23	graded	grade	VERB
ejpam-3889	54	24	left	leave	VERB
ejpam-3889	54	25	a−module	a−module	ADP
ejpam-3889	54	26	if	if	SCONJ
ejpam-3889	54	27	there	there	PRON
ejpam-3889	54	28	exists	exist	VERB
ejpam-3889	54	29	a	a	DET
ejpam-3889	54	30	sequence	sequence	NOUN
ejpam-3889	54	31	(	(	PUNCT
ejpam-3889	54	32	mn)n∈z	mn)n∈z	NUM
ejpam-3889	54	33	of	of	ADP
ejpam-3889	54	34	sub	sub	NOUN
ejpam-3889	54	35	-	-	NOUN
ejpam-3889	54	36	group	group	NOUN
ejpam-3889	54	37	of	of	ADP
ejpam-3889	54	38	m	m	PROPN
ejpam-3889	55	1	such	such	ADJ
ejpam-3889	55	2	that	that	SCONJ
ejpam-3889	55	3	(	(	PUNCT
ejpam-3889	55	4	i	i	NOUN
ejpam-3889	55	5	)	)	PUNCT
ejpam-3889	55	6	m	m	VERB
ejpam-3889	55	7	=	=	PROPN
ejpam-3889	55	8	⊕	⊕	PROPN
ejpam-3889	55	9	n∈z	n∈z	PROPN
ejpam-3889	55	10	mn	mn	PROPN
ejpam-3889	55	11	;	;	PUNCT
ejpam-3889	55	12	s.	s.	PROPN
ejpam-3889	55	13	a.	a.	PROPN
ejpam-3889	55	14	balde	balde	PROPN
ejpam-3889	55	15	,	,	PUNCT
ejpam-3889	55	16	m.	m.	PROPN
ejpam-3889	55	17	b.	b.	PROPN
ejpam-3889	55	18	maaouia	maaouia	PROPN
ejpam-3889	55	19	,	,	PUNCT
ejpam-3889	55	20	a.	a.	NOUN
ejpam-3889	55	21	o.	o.	NOUN
ejpam-3889	55	22	chbih	chbih	PROPN
ejpam-3889	55	23	/	/	SYM
ejpam-3889	55	24	eur	eur	PROPN
ejpam-3889	55	25	.	.	PUNCT
ejpam-3889	56	1	j.	j.	PROPN
ejpam-3889	56	2	pure	pure	PROPN
ejpam-3889	56	3	appl	appl	PROPN
ejpam-3889	56	4	.	.	PROPN
ejpam-3889	56	5	math	math	PROPN
ejpam-3889	56	6	,	,	PUNCT
ejpam-3889	56	7	14	14	NUM
ejpam-3889	56	8	(	(	PUNCT
ejpam-3889	56	9	2	2	NUM
ejpam-3889	56	10	)	)	PUNCT
ejpam-3889	56	11	(	(	PUNCT
ejpam-3889	56	12	2021	2021	NUM
ejpam-3889	56	13	)	)	PUNCT
ejpam-3889	56	14	,	,	PUNCT
ejpam-3889	56	15	404	404	NUM
ejpam-3889	56	16	-	-	SYM
ejpam-3889	56	17	422	422	NUM
ejpam-3889	56	18	408	408	NUM
ejpam-3889	56	19	(	(	PUNCT
ejpam-3889	56	20	ii	ii	NOUN
ejpam-3889	56	21	)	)	PUNCT
ejpam-3889	56	22	an	an	DET
ejpam-3889	56	23	·	·	PUNCT
ejpam-3889	56	24	md	md	X
ejpam-3889	56	25	⊂mn+d	⊂mn+d	PROPN
ejpam-3889	56	26	,	,	PUNCT
ejpam-3889	56	27	∀	∀	X
ejpam-3889	56	28	n	n	CCONJ
ejpam-3889	56	29	,	,	PUNCT
ejpam-3889	56	30	d	d	PROPN
ejpam-3889	56	31	∈	∈	PROPN
ejpam-3889	56	32	z.	z.	PROPN
ejpam-3889	56	33	definition	definition	NOUN
ejpam-3889	56	34	4	4	X
ejpam-3889	56	35	.	.	PUNCT
ejpam-3889	57	1	let	let	VERB
ejpam-3889	57	2	a	a	DET
ejpam-3889	57	3	=	=	SYM
ejpam-3889	57	4	⊕	⊕	PROPN
ejpam-3889	57	5	n∈z	n∈z	VERB
ejpam-3889	57	6	an	an	DET
ejpam-3889	57	7	be	be	AUX
ejpam-3889	57	8	a	a	DET
ejpam-3889	57	9	graded	grade	VERB
ejpam-3889	57	10	ring	ring	NOUN
ejpam-3889	57	11	,	,	PUNCT
ejpam-3889	57	12	m	m	VERB
ejpam-3889	57	13	=	=	ADJ
ejpam-3889	57	14	⊕	⊕	PROPN
ejpam-3889	58	1	n∈z	n∈z	ADJ
ejpam-3889	58	2	mn	mn	PROPN
ejpam-3889	58	3	be	be	AUX
ejpam-3889	58	4	a	a	DET
ejpam-3889	58	5	graded	grade	VERB
ejpam-3889	58	6	left	leave	VERB
ejpam-3889	58	7	a−module	a−module	NOUN
ejpam-3889	58	8	and	and	CCONJ
ejpam-3889	58	9	n	n	PRON
ejpam-3889	58	10	is	be	AUX
ejpam-3889	58	11	a	a	DET
ejpam-3889	58	12	sub	sub	NOUN
ejpam-3889	58	13	-	-	NOUN
ejpam-3889	58	14	module	module	NOUN
ejpam-3889	58	15	of	of	ADP
ejpam-3889	58	16	m	m	PROPN
ejpam-3889	58	17	,	,	PUNCT
ejpam-3889	58	18	then	then	ADV
ejpam-3889	58	19	n	n	CCONJ
ejpam-3889	58	20	is	be	AUX
ejpam-3889	58	21	called	call	VERB
ejpam-3889	58	22	a	a	DET
ejpam-3889	58	23	graded	grade	VERB
ejpam-3889	58	24	sub	sub	NOUN
ejpam-3889	58	25	-	-	NOUN
ejpam-3889	58	26	module	module	NOUN
ejpam-3889	58	27	of	of	ADP
ejpam-3889	58	28	m	m	PRON
ejpam-3889	58	29	,	,	PUNCT
ejpam-3889	58	30	if	if	SCONJ
ejpam-3889	58	31	∀x	∀x	NOUN
ejpam-3889	58	32	=	=	PUNCT
ejpam-3889	58	33	∑	∑	PUNCT
ejpam-3889	58	34	n∈z	n∈z	PRON
ejpam-3889	58	35	xn	xn	PROPN
ejpam-3889	58	36	∈	∈	PROPN
ejpam-3889	58	37	n	n	CCONJ
ejpam-3889	58	38	,	,	PUNCT
ejpam-3889	58	39	with	with	ADP
ejpam-3889	58	40	xn	xn	PROPN
ejpam-3889	58	41	∈mn	∈mn	PROPN
ejpam-3889	58	42	,	,	PUNCT
ejpam-3889	58	43	then	then	ADV
ejpam-3889	58	44	xn	xn	PROPN
ejpam-3889	58	45	∈	∈	PROPN
ejpam-3889	58	46	n	n	PRON
ejpam-3889	58	47	,	,	PUNCT
ejpam-3889	58	48	∀n	∀n	PROPN
ejpam-3889	58	49	∈	∈	PROPN
ejpam-3889	58	50	z.	z.	PROPN
ejpam-3889	58	51	definition	definition	NOUN
ejpam-3889	58	52	5	5	NUM
ejpam-3889	58	53	.	.	PUNCT
ejpam-3889	59	1	let	let	VERB
ejpam-3889	59	2	a	a	DET
ejpam-3889	59	3	=	=	PROPN
ejpam-3889	59	4	⊕	⊕	PROPN
ejpam-3889	59	5	n∈zan	n∈zan	PROPN
ejpam-3889	59	6	be	be	AUX
ejpam-3889	59	7	a	a	DET
ejpam-3889	59	8	graded	grade	VERB
ejpam-3889	59	9	ring	ring	NOUN
ejpam-3889	59	10	,	,	PUNCT
ejpam-3889	59	11	m	m	VERB
ejpam-3889	59	12	=	=	PROPN
ejpam-3889	59	13	⊕	⊕	PROPN
ejpam-3889	59	14	n∈z	n∈z	PROPN
ejpam-3889	59	15	mn	mn	PROPN
ejpam-3889	59	16	,	,	PUNCT
ejpam-3889	59	17	n	n	PROPN
ejpam-3889	59	18	=	=	PROPN
ejpam-3889	59	19	⊕	⊕	PROPN
ejpam-3889	59	20	n∈z	n∈z	ADJ
ejpam-3889	59	21	nn	nn	PROPN
ejpam-3889	59	22	are	be	AUX
ejpam-3889	59	23	two	two	NUM
ejpam-3889	59	24	graded	grade	VERB
ejpam-3889	59	25	left	leave	VERB
ejpam-3889	59	26	a−modules	a−module	NOUN
ejpam-3889	59	27	and	and	CCONJ
ejpam-3889	59	28	f	f	X
ejpam-3889	59	29	:	:	PUNCT
ejpam-3889	59	30	m	m	VERB
ejpam-3889	59	31	−→	−→	ADJ
ejpam-3889	60	1	n	n	NOUN
ejpam-3889	60	2	is	be	AUX
ejpam-3889	60	3	a	a	DET
ejpam-3889	60	4	morphism	morphism	NOUN
ejpam-3889	60	5	of	of	ADP
ejpam-3889	60	6	left	left	ADJ
ejpam-3889	60	7	a−modules	a−module	NOUN
ejpam-3889	60	8	,	,	PUNCT
ejpam-3889	60	9	then	then	ADV
ejpam-3889	60	10	f	f	PROPN
ejpam-3889	60	11	is	be	AUX
ejpam-3889	60	12	called	call	VERB
ejpam-3889	60	13	a	a	DET
ejpam-3889	60	14	graded	grade	VERB
ejpam-3889	60	15	morphism	morphism	NOUN
ejpam-3889	60	16	if	if	SCONJ
ejpam-3889	60	17	for	for	ADP
ejpam-3889	60	18	any	any	DET
ejpam-3889	60	19	ms	ms	PROPN
ejpam-3889	60	20	∈ms	∈ms	PROPN
ejpam-3889	60	21	then	then	ADV
ejpam-3889	60	22	f(ms	f(ms	ADV
ejpam-3889	60	23	)	)	PUNCT
ejpam-3889	60	24	∈	∈	PROPN
ejpam-3889	60	25	ns	ns	PROPN
ejpam-3889	60	26	.	.	PUNCT
ejpam-3889	60	27	theorem	theorem	NOUN
ejpam-3889	60	28	1	1	NUM
ejpam-3889	60	29	.	.	PUNCT
ejpam-3889	61	1	let	let	VERB
ejpam-3889	61	2	a	a	DET
ejpam-3889	61	3	be	be	AUX
ejpam-3889	61	4	a	a	DET
ejpam-3889	61	5	graded	grade	VERB
ejpam-3889	61	6	ring	ring	NOUN
ejpam-3889	61	7	,	,	PUNCT
ejpam-3889	61	8	the	the	DET
ejpam-3889	61	9	category	category	NOUN
ejpam-3889	61	10	of	of	ADP
ejpam-3889	61	11	graded	grade	VERB
ejpam-3889	61	12	left	leave	VERB
ejpam-3889	61	13	a−module	a−module	NOUN
ejpam-3889	61	14	is	be	AUX
ejpam-3889	61	15	the	the	DET
ejpam-3889	61	16	category	category	NOUN
ejpam-3889	61	17	denoted	denote	VERB
ejpam-3889	61	18	by	by	ADP
ejpam-3889	61	19	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	61	20	)	)	PUNCT
ejpam-3889	61	21	whose	whose	DET
ejpam-3889	61	22	(	(	PUNCT
ejpam-3889	61	23	i	i	NOUN
ejpam-3889	61	24	)	)	PUNCT
ejpam-3889	61	25	the	the	DET
ejpam-3889	61	26	objects	object	NOUN
ejpam-3889	61	27	are	be	AUX
ejpam-3889	61	28	the	the	DET
ejpam-3889	61	29	graded	grade	VERB
ejpam-3889	61	30	left	leave	VERB
ejpam-3889	61	31	a−modules	a−module	NOUN
ejpam-3889	61	32	;	;	PUNCT
ejpam-3889	61	33	(	(	PUNCT
ejpam-3889	61	34	ii	ii	NOUN
ejpam-3889	61	35	)	)	PUNCT
ejpam-3889	61	36	the	the	DET
ejpam-3889	61	37	morphisms	morphism	NOUN
ejpam-3889	61	38	are	be	AUX
ejpam-3889	61	39	the	the	DET
ejpam-3889	61	40	graded	grade	VERB
ejpam-3889	61	41	morphisms	morphism	NOUN
ejpam-3889	61	42	.	.	PUNCT
ejpam-3889	62	1	proof	proof	NOUN
ejpam-3889	62	2	.	.	PUNCT
ejpam-3889	63	1	see	see	VERB
ejpam-3889	63	2	[	[	X
ejpam-3889	63	3	23	23	NUM
ejpam-3889	63	4	]	]	PUNCT
ejpam-3889	63	5	theorem	theorem	NOUN
ejpam-3889	63	6	2	2	NUM
ejpam-3889	63	7	.	.	PUNCT
ejpam-3889	63	8	let	let	VERB
ejpam-3889	63	9	a	a	PRON
ejpam-3889	63	10	be	be	AUX
ejpam-3889	63	11	a	a	DET
ejpam-3889	63	12	graded	grade	VERB
ejpam-3889	63	13	ring	ring	NOUN
ejpam-3889	63	14	,	,	PUNCT
ejpam-3889	63	15	m	m	VERB
ejpam-3889	63	16	a	a	DET
ejpam-3889	63	17	graded	grade	VERB
ejpam-3889	63	18	left	leave	VERB
ejpam-3889	63	19	a	a	DET
ejpam-3889	63	20	-	-	PUNCT
ejpam-3889	63	21	module	module	NOUN
ejpam-3889	63	22	and	and	CCONJ
ejpam-3889	63	23	s	s	NOUN
ejpam-3889	63	24	s	s	X
ejpam-3889	63	25	a	a	DET
ejpam-3889	63	26	saturated	saturate	VERB
ejpam-3889	63	27	multiplicative	multiplicative	ADJ
ejpam-3889	63	28	part	part	NOUN
ejpam-3889	63	29	formed	form	VERB
ejpam-3889	63	30	by	by	ADP
ejpam-3889	63	31	the	the	DET
ejpam-3889	63	32	non	non	ADJ
ejpam-3889	63	33	-	-	ADJ
ejpam-3889	63	34	zero	zero	ADJ
ejpam-3889	63	35	homogeneous	homogeneous	ADJ
ejpam-3889	63	36	elements	element	NOUN
ejpam-3889	63	37	of	of	ADP
ejpam-3889	63	38	a	a	DET
ejpam-3889	63	39	verifying	verifying	NOUN
ejpam-3889	63	40	the	the	DET
ejpam-3889	63	41	left	left	ADJ
ejpam-3889	63	42	ore	ore	NOUN
ejpam-3889	63	43	conditions	condition	NOUN
ejpam-3889	63	44	,	,	PUNCT
ejpam-3889	63	45	then	then	ADV
ejpam-3889	63	46	:	:	PUNCT
ejpam-3889	63	47	(	(	PUNCT
ejpam-3889	63	48	i	i	NOUN
ejpam-3889	63	49	)	)	PUNCT
ejpam-3889	63	50	s−1a	s−1a	PROPN
ejpam-3889	63	51	=	=	PROPN
ejpam-3889	63	52	⊕	⊕	PROPN
ejpam-3889	63	53	i∈z	i∈z	PROPN
ejpam-3889	63	54	(	(	PUNCT
ejpam-3889	63	55	s−1a)i	s−1a)i	ADJ
ejpam-3889	63	56	is	be	AUX
ejpam-3889	63	57	a	a	DET
ejpam-3889	63	58	graded	grade	VERB
ejpam-3889	63	59	ring	ring	NOUN
ejpam-3889	63	60	,	,	PUNCT
ejpam-3889	63	61	where	where	SCONJ
ejpam-3889	63	62	(	(	PUNCT
ejpam-3889	63	63	s−1a)i	s−1a)i	NOUN
ejpam-3889	63	64	=	=	NOUN
ejpam-3889	63	65	{	{	PUNCT
ejpam-3889	63	66	as	as	ADP
ejpam-3889	63	67	∈	∈	PROPN
ejpam-3889	63	68	s	s	PART
ejpam-3889	63	69	−1a,∃k	−1a,∃k	NOUN
ejpam-3889	63	70	,	,	PUNCT
ejpam-3889	63	71	a	a	DET
ejpam-3889	63	72	∈	∈	PROPN
ejpam-3889	63	73	ak	ak	NOUN
ejpam-3889	63	74	and	and	CCONJ
ejpam-3889	63	75	deg(s	deg(s	PROPN
ejpam-3889	63	76	)	)	PUNCT
ejpam-3889	64	1	=	=	SYM
ejpam-3889	65	1	k	k	X
ejpam-3889	65	2	−	−	PUNCT
ejpam-3889	66	1	i	i	NOUN
ejpam-3889	66	2	}	}	PUNCT
ejpam-3889	66	3	.	.	PUNCT
ejpam-3889	67	1	(	(	PUNCT
ejpam-3889	67	2	ii	ii	X
ejpam-3889	67	3	)	)	PUNCT
ejpam-3889	67	4	s−1	s−1	PROPN
ejpam-3889	67	5	m	m	NOUN
ejpam-3889	67	6	=	=	PROPN
ejpam-3889	67	7	⊕	⊕	PROPN
ejpam-3889	67	8	i∈z	i∈z	PROPN
ejpam-3889	67	9	(	(	PUNCT
ejpam-3889	67	10	s−1m)i	s−1m)i	NOUN
ejpam-3889	67	11	is	be	AUX
ejpam-3889	67	12	a	a	DET
ejpam-3889	67	13	graded	grade	VERB
ejpam-3889	67	14	left	leave	VERB
ejpam-3889	67	15	s−1a	s−1a	NOUN
ejpam-3889	67	16	-	-	PUNCT
ejpam-3889	67	17	module	module	NOUN
ejpam-3889	67	18	,	,	PUNCT
ejpam-3889	67	19	where	where	SCONJ
ejpam-3889	67	20	(	(	PUNCT
ejpam-3889	67	21	s−1m)i	s−1m)i	NOUN
ejpam-3889	67	22	=	=	X
ejpam-3889	67	23	{	{	PUNCT
ejpam-3889	67	24	ms	ms	PROPN
ejpam-3889	67	25	∈	∈	PROPN
ejpam-3889	67	26	s	s	PART
ejpam-3889	67	27	−1m,∃p	−1m,∃p	NOUN
ejpam-3889	67	28	,	,	PUNCT
ejpam-3889	67	29	m	m	VERB
ejpam-3889	67	30	∈mp	∈mp	NOUN
ejpam-3889	67	31	and	and	CCONJ
ejpam-3889	67	32	deg(s	deg(s	NUM
ejpam-3889	67	33	)	)	PUNCT
ejpam-3889	67	34	=	=	PUNCT
ejpam-3889	68	1	p−	p−	X
ejpam-3889	68	2	i	i	NOUN
ejpam-3889	68	3	}	}	PUNCT
ejpam-3889	68	4	.	.	PUNCT
ejpam-3889	69	1	proof	proof	NOUN
ejpam-3889	69	2	.	.	PUNCT
ejpam-3889	70	1	see	see	VERB
ejpam-3889	70	2	[	[	X
ejpam-3889	70	3	1	1	X
ejpam-3889	70	4	]	]	PUNCT
ejpam-3889	70	5	proposition	proposition	NOUN
ejpam-3889	70	6	1	1	NUM
ejpam-3889	70	7	.	.	PUNCT
ejpam-3889	71	1	let	let	VERB
ejpam-3889	71	2	a	a	DET
ejpam-3889	71	3	=	=	SYM
ejpam-3889	71	4	⊕	⊕	PROPN
ejpam-3889	71	5	n∈n	n∈n	VERB
ejpam-3889	71	6	an	an	DET
ejpam-3889	71	7	be	be	AUX
ejpam-3889	71	8	a	a	DET
ejpam-3889	71	9	graded	grade	VERB
ejpam-3889	71	10	ring	ring	NOUN
ejpam-3889	71	11	,	,	PUNCT
ejpam-3889	71	12	m	m	VERB
ejpam-3889	71	13	=	=	ADJ
ejpam-3889	71	14	⊕	⊕	PROPN
ejpam-3889	71	15	n∈z	n∈z	VERB
ejpam-3889	71	16	mn	mn	PROPN
ejpam-3889	71	17	and	and	CCONJ
ejpam-3889	71	18	n	n	PROPN
ejpam-3889	71	19	=	=	PROPN
ejpam-3889	71	20	⊕	⊕	PROPN
ejpam-3889	71	21	n∈z	n∈z	ADJ
ejpam-3889	71	22	nn	nn	PROPN
ejpam-3889	71	23	are	be	AUX
ejpam-3889	71	24	two	two	NUM
ejpam-3889	71	25	graded	grade	VERB
ejpam-3889	71	26	left	leave	VERB
ejpam-3889	71	27	a	a	DET
ejpam-3889	71	28	-	-	PUNCT
ejpam-3889	71	29	modules	module	NOUN
ejpam-3889	71	30	,	,	PUNCT
ejpam-3889	71	31	f	f	X
ejpam-3889	71	32	:	:	PUNCT
ejpam-3889	71	33	m	m	VERB
ejpam-3889	71	34	−→	−→	ADJ
ejpam-3889	71	35	n	n	NOUN
ejpam-3889	71	36	is	be	AUX
ejpam-3889	71	37	graded	grade	VERB
ejpam-3889	71	38	morphism	morphism	NOUN
ejpam-3889	71	39	and	and	CCONJ
ejpam-3889	71	40	s	s	VERB
ejpam-3889	71	41	the	the	DET
ejpam-3889	71	42	set	set	NOUN
ejpam-3889	71	43	of	of	ADP
ejpam-3889	71	44	non	non	ADJ
ejpam-3889	71	45	-	-	ADJ
ejpam-3889	71	46	zero	zero	ADJ
ejpam-3889	71	47	homogeneous	homogeneous	ADJ
ejpam-3889	71	48	elements	element	NOUN
ejpam-3889	71	49	of	of	ADP
ejpam-3889	71	50	a	a	PRON
ejpam-3889	71	51	,	,	PUNCT
ejpam-3889	71	52	then	then	ADV
ejpam-3889	71	53	we	we	PRON
ejpam-3889	71	54	have	have	VERB
ejpam-3889	71	55	:	:	PUNCT
ejpam-3889	71	56	(	(	PUNCT
ejpam-3889	71	57	i	i	NOUN
ejpam-3889	71	58	)	)	PUNCT
ejpam-3889	71	59	the	the	DET
ejpam-3889	71	60	following	follow	VERB
ejpam-3889	71	61	complex	complex	ADJ
ejpam-3889	71	62	sequences	sequence	NOUN
ejpam-3889	71	63	:	:	PUNCT
ejpam-3889	71	64	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	71	65	)	)	PUNCT
ejpam-3889	71	66	:	:	PUNCT
ejpam-3889	71	67	·	·	PUNCT
ejpam-3889	71	68	·	·	PUNCT
ejpam-3889	71	69	·	·	PUNCT
ejpam-3889	72	1	−→	−→	NOUN
ejpam-3889	72	2	s−1(m(n+1	s−1(m(n+1	PROPN
ejpam-3889	72	3	)	)	PUNCT
ejpam-3889	72	4	)	)	PUNCT
ejpam-3889	72	5	s−1(dn+1)−→	s−1(dn+1)−→	ADP
ejpam-3889	72	6	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	72	7	)	)	PUNCT
ejpam-3889	72	8	)	)	PUNCT
ejpam-3889	73	1	s−1(dn)−→	s−1(dn)−→	PROPN
ejpam-3889	73	2	s−1(m(n−1	s−1(m(n−1	PROPN
ejpam-3889	73	3	)	)	PUNCT
ejpam-3889	73	4	)	)	PUNCT
ejpam-3889	74	1	−→	−→	NOUN
ejpam-3889	74	2	·	·	PUNCT
ejpam-3889	74	3	·	·	PUNCT
ejpam-3889	74	4	·	·	PUNCT
ejpam-3889	74	5	s.	s.	PROPN
ejpam-3889	74	6	a.	a.	PROPN
ejpam-3889	74	7	balde	balde	PROPN
ejpam-3889	74	8	,	,	PUNCT
ejpam-3889	74	9	m.	m.	PROPN
ejpam-3889	74	10	b.	b.	PROPN
ejpam-3889	74	11	maaouia	maaouia	PROPN
ejpam-3889	74	12	,	,	PUNCT
ejpam-3889	74	13	a.	a.	NOUN
ejpam-3889	74	14	o.	o.	NOUN
ejpam-3889	74	15	chbih	chbih	PROPN
ejpam-3889	74	16	/	/	SYM
ejpam-3889	74	17	eur	eur	PROPN
ejpam-3889	74	18	.	.	PUNCT
ejpam-3889	75	1	j.	j.	PROPN
ejpam-3889	75	2	pure	pure	PROPN
ejpam-3889	75	3	appl	appl	PROPN
ejpam-3889	75	4	.	.	PROPN
ejpam-3889	75	5	math	math	PROPN
ejpam-3889	75	6	,	,	PUNCT
ejpam-3889	75	7	14	14	NUM
ejpam-3889	75	8	(	(	PUNCT
ejpam-3889	75	9	2	2	NUM
ejpam-3889	75	10	)	)	PUNCT
ejpam-3889	75	11	(	(	PUNCT
ejpam-3889	75	12	2021	2021	NUM
ejpam-3889	75	13	)	)	PUNCT
ejpam-3889	75	14	,	,	PUNCT
ejpam-3889	75	15	404	404	NUM
ejpam-3889	75	16	-	-	SYM
ejpam-3889	75	17	422	422	NUM
ejpam-3889	75	18	409	409	NUM
ejpam-3889	75	19	(	(	PUNCT
ejpam-3889	75	20	ii	ii	NOUN
ejpam-3889	75	21	)	)	PUNCT
ejpam-3889	75	22	the	the	DET
ejpam-3889	75	23	following	follow	VERB
ejpam-3889	75	24	chain	chain	NOUN
ejpam-3889	75	25	complexes	complex	NOUN
ejpam-3889	75	26	:	:	PUNCT
ejpam-3889	75	27	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	75	28	)	)	PUNCT
ejpam-3889	75	29	:	:	PUNCT
ejpam-3889	75	30	·	·	PUNCT
ejpam-3889	75	31	·	·	PUNCT
ejpam-3889	75	32	·	·	PUNCT
ejpam-3889	76	1	//	//	PUNCT
ejpam-3889	77	1	s−1(f∗	s−1(f∗	NOUN
ejpam-3889	77	2	)	)	PUNCT
ejpam-3889	77	3	�	�	PROPN
ejpam-3889	77	4	�	�	PROPN
ejpam-3889	77	5	//	//	NUM
ejpam-3889	77	6	s−1(m(n+	s−1(m(n+	NOUN
ejpam-3889	77	7	1	1	NUM
ejpam-3889	77	8	)	)	PUNCT
ejpam-3889	77	9	)	)	PUNCT
ejpam-3889	77	10	s−1(dn+1)//	s−1(dn+1)//	PROPN
ejpam-3889	77	11	s−1(f(n+1	s−1(f(n+1	PROPN
ejpam-3889	77	12	)	)	PUNCT
ejpam-3889	77	13	)	)	PUNCT
ejpam-3889	77	14	�	�	PROPN
ejpam-3889	77	15	�	�	PROPN
ejpam-3889	77	16	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	77	17	)	)	PUNCT
ejpam-3889	77	18	)	)	PUNCT
ejpam-3889	78	1	s−1(dn)//	s−1(dn)//	PROPN
ejpam-3889	78	2	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	78	3	)	)	PUNCT
ejpam-3889	78	4	)	)	PUNCT
ejpam-3889	78	5	�	�	PROPN
ejpam-3889	78	6	�	�	PROPN
ejpam-3889	78	7	s−1(m(n−	s−1(m(n−	VERB
ejpam-3889	78	8	1	1	NUM
ejpam-3889	78	9	)	)	PUNCT
ejpam-3889	78	10	)	)	PUNCT
ejpam-3889	79	1	//	//	X
ejpam-3889	79	2	s−1(f(n−1	s−1(f(n−1	PROPN
ejpam-3889	79	3	)	)	PUNCT
ejpam-3889	79	4	)	)	PUNCT
ejpam-3889	79	5	�	�	PROPN
ejpam-3889	79	6	�	�	PROPN
ejpam-3889	79	7	·	·	PUNCT
ejpam-3889	79	8	·	·	PUNCT
ejpam-3889	79	9	·	·	PUNCT
ejpam-3889	80	1	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	80	2	)	)	PUNCT
ejpam-3889	80	3	:	:	PUNCT
ejpam-3889	80	4	·	·	PUNCT
ejpam-3889	80	5	·	·	PUNCT
ejpam-3889	80	6	·	·	PUNCT
ejpam-3889	80	7	//	//	NUM
ejpam-3889	80	8	s−1(n(n+	s−1(n(n+	NOUN
ejpam-3889	80	9	1	1	NUM
ejpam-3889	80	10	)	)	PUNCT
ejpam-3889	80	11	)	)	PUNCT
ejpam-3889	80	12	s−1(d′n+1)//	s−1(d′n+1)//	ADP
ejpam-3889	80	13	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	80	14	)	)	PUNCT
ejpam-3889	80	15	)	)	PUNCT
ejpam-3889	80	16	s−1(d′n)//	s−1(d′n)//	NOUN
ejpam-3889	80	17	s−1(n(n−	s−1(n(n−	VERB
ejpam-3889	80	18	1	1	NUM
ejpam-3889	80	19	)	)	PUNCT
ejpam-3889	80	20	)	)	PUNCT
ejpam-3889	80	21	//	//	X
ejpam-3889	80	22	·	·	PUNCT
ejpam-3889	80	23	·	·	PUNCT
ejpam-3889	80	24	·	·	PUNCT
ejpam-3889	81	1	proof	proof	NOUN
ejpam-3889	81	2	.	.	PUNCT
ejpam-3889	82	1	see	see	VERB
ejpam-3889	82	2	[	[	X
ejpam-3889	82	3	3	3	NUM
ejpam-3889	82	4	]	]	X
ejpam-3889	82	5	theorem	theorem	NOUN
ejpam-3889	82	6	3	3	X
ejpam-3889	82	7	.	.	PUNCT
ejpam-3889	83	1	let	let	VERB
ejpam-3889	83	2	m∗	m∗	NOUN
ejpam-3889	83	3	:	:	PUNCT
ejpam-3889	83	4	.	.	PUNCT
ejpam-3889	83	5	.	.	PUNCT
ejpam-3889	84	1	.m(n	.m(n	NOUN
ejpam-3889	85	1	+	+	CCONJ
ejpam-3889	85	2	1	1	X
ejpam-3889	85	3	)	)	PUNCT
ejpam-3889	85	4	dn+1−−−→	dn+1−−−→	PROPN
ejpam-3889	85	5	m(n	m(n	PROPN
ejpam-3889	85	6	)	)	PUNCT
ejpam-3889	85	7	dn−→	dn−→	PROPN
ejpam-3889	86	1	m(n	m(n	PROPN
ejpam-3889	86	2	−	−	PROPN
ejpam-3889	86	3	1	1	X
ejpam-3889	86	4	)	)	PUNCT
ejpam-3889	86	5	−→	−→	NOUN
ejpam-3889	86	6	.	.	PUNCT
ejpam-3889	86	7	.	.	PUNCT
ejpam-3889	86	8	.	.	PUNCT
ejpam-3889	87	1	be	be	AUX
ejpam-3889	87	2	an	an	DET
ejpam-3889	87	3	object	object	NOUN
ejpam-3889	87	4	of	of	ADP
ejpam-3889	87	5	comp	comp	NOUN
ejpam-3889	87	6	(	(	PUNCT
ejpam-3889	87	7	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	87	8	)	)	PUNCT
ejpam-3889	87	9	)	)	PUNCT
ejpam-3889	87	10	.	.	PUNCT
ejpam-3889	88	1	m∗	m∗	PROPN
ejpam-3889	88	2	is	be	AUX
ejpam-3889	88	3	quasi	quasi	ADJ
ejpam-3889	88	4	-	-	ADJ
ejpam-3889	88	5	injective	injective	ADJ
ejpam-3889	88	6	in	in	ADP
ejpam-3889	88	7	comp	comp	NOUN
ejpam-3889	88	8	(	(	PUNCT
ejpam-3889	88	9	gr(a−mod	gr(a−mod	NOUN
ejpam-3889	88	10	)	)	PUNCT
ejpam-3889	88	11	)	)	PUNCT
ejpam-3889	89	1	if	if	SCONJ
ejpam-3889	89	2	,	,	PUNCT
ejpam-3889	89	3	and	and	CCONJ
ejpam-3889	89	4	only	only	ADV
ejpam-3889	89	5	,	,	PUNCT
ejpam-3889	89	6	if	if	SCONJ
ejpam-3889	89	7	for	for	ADP
ejpam-3889	89	8	all	all	DET
ejpam-3889	89	9	n	n	PRON
ejpam-3889	89	10	∈	∈	PROPN
ejpam-3889	89	11	z	z	PROPN
ejpam-3889	89	12	,	,	PUNCT
ejpam-3889	89	13	m(n	m(n	PROPN
ejpam-3889	89	14	)	)	PUNCT
ejpam-3889	89	15	is	be	AUX
ejpam-3889	89	16	a	a	DET
ejpam-3889	89	17	quasi	quasi	NOUN
ejpam-3889	89	18	-	-	ADJ
ejpam-3889	89	19	injective	injective	ADJ
ejpam-3889	89	20	in	in	ADP
ejpam-3889	89	21	comp	comp	NOUN
ejpam-3889	89	22	(	(	PUNCT
ejpam-3889	89	23	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	89	24	)	)	PUNCT
ejpam-3889	89	25	)	)	PUNCT
ejpam-3889	89	26	.	.	PUNCT
ejpam-3889	90	1	proof	proof	NOUN
ejpam-3889	90	2	.	.	PUNCT
ejpam-3889	91	1	see	see	VERB
ejpam-3889	91	2	[	[	X
ejpam-3889	91	3	23	23	NUM
ejpam-3889	91	4	]	]	PUNCT
ejpam-3889	91	5	theorem	theorem	NOUN
ejpam-3889	91	6	4	4	NUM
ejpam-3889	91	7	.	.	PUNCT
ejpam-3889	92	1	let	let	VERB
ejpam-3889	92	2	m∗	m∗	NOUN
ejpam-3889	92	3	:	:	PUNCT
ejpam-3889	92	4	.	.	PUNCT
ejpam-3889	92	5	.	.	PUNCT
ejpam-3889	93	1	.m(n	.m(n	NOUN
ejpam-3889	94	1	+	+	CCONJ
ejpam-3889	94	2	1	1	X
ejpam-3889	94	3	)	)	PUNCT
ejpam-3889	94	4	dn+1−−−→	dn+1−−−→	PROPN
ejpam-3889	94	5	m(n	m(n	PROPN
ejpam-3889	94	6	)	)	PUNCT
ejpam-3889	94	7	dn−→	dn−→	PROPN
ejpam-3889	95	1	m(n	m(n	PROPN
ejpam-3889	95	2	−	−	PROPN
ejpam-3889	95	3	1	1	X
ejpam-3889	95	4	)	)	PUNCT
ejpam-3889	95	5	−→	−→	NOUN
ejpam-3889	95	6	.	.	PUNCT
ejpam-3889	95	7	.	.	PUNCT
ejpam-3889	95	8	.	.	PUNCT
ejpam-3889	96	1	be	be	AUX
ejpam-3889	96	2	an	an	DET
ejpam-3889	96	3	object	object	NOUN
ejpam-3889	96	4	of	of	ADP
ejpam-3889	96	5	comp	comp	NOUN
ejpam-3889	96	6	(	(	PUNCT
ejpam-3889	96	7	gr(a−mod	gr(a−mod	NOUN
ejpam-3889	96	8	)	)	PUNCT
ejpam-3889	96	9	)	)	PUNCT
ejpam-3889	96	10	.	.	PUNCT
ejpam-3889	97	1	m∗	m∗	PROPN
ejpam-3889	97	2	is	be	AUX
ejpam-3889	97	3	quasi	quasi	ADJ
ejpam-3889	97	4	-	-	NOUN
ejpam-3889	97	5	projective	projective	ADJ
ejpam-3889	97	6	if	if	SCONJ
ejpam-3889	97	7	,	,	PUNCT
ejpam-3889	97	8	and	and	CCONJ
ejpam-3889	97	9	only	only	ADV
ejpam-3889	97	10	,	,	PUNCT
ejpam-3889	97	11	if	if	SCONJ
ejpam-3889	97	12	for	for	SCONJ
ejpam-3889	97	13	all	all	DET
ejpam-3889	97	14	n	n	PRON
ejpam-3889	97	15	∈	∈	PROPN
ejpam-3889	97	16	z	z	PROPN
ejpam-3889	97	17	,	,	PUNCT
ejpam-3889	97	18	m(n	m(n	PROPN
ejpam-3889	97	19	)	)	PUNCT
ejpam-3889	97	20	is	be	AUX
ejpam-3889	97	21	a	a	DET
ejpam-3889	97	22	quasi	quasi	NOUN
ejpam-3889	97	23	-	-	ADJ
ejpam-3889	97	24	projective	projective	ADJ
ejpam-3889	97	25	left	leave	VERB
ejpam-3889	97	26	a	a	DET
ejpam-3889	97	27	-	-	PUNCT
ejpam-3889	97	28	modules	module	NOUN
ejpam-3889	97	29	.	.	PUNCT
ejpam-3889	98	1	proof	proof	NOUN
ejpam-3889	98	2	.	.	PUNCT
ejpam-3889	99	1	see	see	VERB
ejpam-3889	100	1	[	[	X
ejpam-3889	100	2	23	23	NUM
ejpam-3889	100	3	]	]	SYM
ejpam-3889	100	4	3	3	X
ejpam-3889	100	5	.	.	X
ejpam-3889	100	6	localization	localization	NOUN
ejpam-3889	100	7	of	of	ADP
ejpam-3889	100	8	hopfian	hopfian	ADJ
ejpam-3889	100	9	and	and	CCONJ
ejpam-3889	100	10	cohopfian	cohopfian	ADJ
ejpam-3889	100	11	objects	object	NOUN
ejpam-3889	100	12	in	in	ADP
ejpam-3889	100	13	the	the	DET
ejpam-3889	100	14	category	category	NOUN
ejpam-3889	100	15	of	of	ADP
ejpam-3889	100	16	a−mod	a−mod	NOUN
ejpam-3889	100	17	definition	definition	NOUN
ejpam-3889	100	18	6	6	NUM
ejpam-3889	100	19	.	.	PUNCT
ejpam-3889	101	1	let	let	VERB
ejpam-3889	101	2	m	m	PRON
ejpam-3889	101	3	a	a	DET
ejpam-3889	101	4	left	left	ADJ
ejpam-3889	101	5	a	a	DET
ejpam-3889	101	6	-	-	PUNCT
ejpam-3889	101	7	module	module	NOUN
ejpam-3889	101	8	.	.	PUNCT
ejpam-3889	102	1	then	then	ADV
ejpam-3889	102	2	m	m	NOUN
ejpam-3889	102	3	is	be	AUX
ejpam-3889	102	4	said	say	VERB
ejpam-3889	102	5	hopfian	hopfian	ADJ
ejpam-3889	102	6	(	(	PUNCT
ejpam-3889	102	7	respectively	respectively	ADV
ejpam-3889	102	8	cohopfian	cohopfian	ADJ
ejpam-3889	102	9	)	)	PUNCT
ejpam-3889	102	10	,	,	PUNCT
ejpam-3889	102	11	if	if	SCONJ
ejpam-3889	102	12	any	any	DET
ejpam-3889	102	13	epimorphism	epimorphism	NOUN
ejpam-3889	102	14	(	(	PUNCT
ejpam-3889	102	15	respectively	respectively	ADV
ejpam-3889	102	16	monomorphism	monomorphism	NOUN
ejpam-3889	102	17	)	)	PUNCT
ejpam-3889	102	18	of	of	ADP
ejpam-3889	102	19	m	m	PROPN
ejpam-3889	102	20	is	be	AUX
ejpam-3889	102	21	an	an	DET
ejpam-3889	102	22	automorphism	automorphism	NOUN
ejpam-3889	102	23	of	of	ADP
ejpam-3889	102	24	m	m	PROPN
ejpam-3889	102	25	.	.	PUNCT
ejpam-3889	103	1	theorem	theorem	ADJ
ejpam-3889	103	2	5	5	NUM
ejpam-3889	103	3	.	.	PUNCT
ejpam-3889	104	1	let	let	VERB
ejpam-3889	104	2	a	a	PRON
ejpam-3889	104	3	be	be	AUX
ejpam-3889	104	4	a	a	DET
ejpam-3889	104	5	ring	ring	NOUN
ejpam-3889	104	6	,	,	PUNCT
ejpam-3889	104	7	s	s	VERB
ejpam-3889	104	8	a	a	DET
ejpam-3889	104	9	saturated	saturate	VERB
ejpam-3889	104	10	multiplicative	multiplicative	ADJ
ejpam-3889	104	11	part	part	NOUN
ejpam-3889	104	12	of	of	ADP
ejpam-3889	104	13	a	a	DET
ejpam-3889	104	14	verifying	verifying	NOUN
ejpam-3889	104	15	the	the	DET
ejpam-3889	104	16	left	left	ADJ
ejpam-3889	104	17	ore	ore	NOUN
ejpam-3889	104	18	conditions	condition	NOUN
ejpam-3889	104	19	,	,	PUNCT
ejpam-3889	104	20	m	m	VERB
ejpam-3889	104	21	a	a	DET
ejpam-3889	104	22	left	left	ADJ
ejpam-3889	104	23	a	a	DET
ejpam-3889	104	24	-	-	PUNCT
ejpam-3889	104	25	module	module	NOUN
ejpam-3889	104	26	.	.	PUNCT
ejpam-3889	105	1	if	if	SCONJ
ejpam-3889	105	2	s−1(m	s−1(m	PROPN
ejpam-3889	105	3	)	)	PUNCT
ejpam-3889	105	4	is	be	AUX
ejpam-3889	105	5	a	a	DET
ejpam-3889	105	6	hopfian	hopfian	NOUN
ejpam-3889	105	7	left	leave	VERB
ejpam-3889	105	8	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	105	9	,	,	PUNCT
ejpam-3889	105	10	then	then	ADV
ejpam-3889	105	11	m	m	VERB
ejpam-3889	105	12	is	be	AUX
ejpam-3889	105	13	a	a	DET
ejpam-3889	105	14	hopfian	hopfian	NOUN
ejpam-3889	105	15	left	leave	VERB
ejpam-3889	105	16	a	a	DET
ejpam-3889	105	17	-	-	PUNCT
ejpam-3889	105	18	module	module	NOUN
ejpam-3889	105	19	.	.	PUNCT
ejpam-3889	106	1	proof	proof	NOUN
ejpam-3889	106	2	.	.	PUNCT
ejpam-3889	107	1	let	let	VERB
ejpam-3889	107	2	f	f	NOUN
ejpam-3889	107	3	:	:	PUNCT
ejpam-3889	107	4	m	m	VERB
ejpam-3889	107	5	−→m	−→m	VERB
ejpam-3889	107	6	an	an	DET
ejpam-3889	107	7	epimorphism	epimorphism	NOUN
ejpam-3889	107	8	of	of	ADP
ejpam-3889	107	9	left	leave	VERB
ejpam-3889	107	10	a	a	DET
ejpam-3889	107	11	-	-	PUNCT
ejpam-3889	107	12	module	module	NOUN
ejpam-3889	107	13	.	.	PUNCT
ejpam-3889	108	1	then	then	ADV
ejpam-3889	108	2	s−1(f	s−1(f	PROPN
ejpam-3889	108	3	)	)	PUNCT
ejpam-3889	108	4	:	:	PUNCT
ejpam-3889	109	1	s−1(m	s−1(m	PROPN
ejpam-3889	109	2	)	)	PUNCT
ejpam-3889	109	3	−→	−→	NOUN
ejpam-3889	109	4	s−1(m	s−1(m	PROPN
ejpam-3889	109	5	)	)	PUNCT
ejpam-3889	109	6	define	define	VERB
ejpam-3889	109	7	by	by	ADP
ejpam-3889	109	8	[	[	X
ejpam-3889	109	9	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	109	10	)	)	PUNCT
ejpam-3889	109	11	=	=	SYM
ejpam-3889	109	12	f(m	f(m	PROPN
ejpam-3889	109	13	)	)	PUNCT
ejpam-3889	109	14	s	s	VERB
ejpam-3889	109	15	is	be	AUX
ejpam-3889	109	16	an	an	DET
ejpam-3889	109	17	endomorphisme	endomorphisme	NOUN
ejpam-3889	109	18	of	of	ADP
ejpam-3889	109	19	s−1(a)-mod	s−1(a)-mod	NOUN
ejpam-3889	109	20	.	.	PUNCT
ejpam-3889	110	1	let	let	VERB
ejpam-3889	110	2	m′	m′	NOUN
ejpam-3889	110	3	s	s	PART
ejpam-3889	110	4	∈	∈	NOUN
ejpam-3889	110	5	s	s	PART
ejpam-3889	110	6	−1(m	−1(m	ADJ
ejpam-3889	110	7	)	)	PUNCT
ejpam-3889	110	8	,	,	PUNCT
ejpam-3889	110	9	as	as	SCONJ
ejpam-3889	110	10	f	f	PROPN
ejpam-3889	110	11	is	be	AUX
ejpam-3889	110	12	an	an	DET
ejpam-3889	110	13	epimorphism	epimorphism	NOUN
ejpam-3889	110	14	,	,	PUNCT
ejpam-3889	110	15	then	then	ADV
ejpam-3889	110	16	there	there	PRON
ejpam-3889	110	17	exists	exist	VERB
ejpam-3889	110	18	m	m	VERB
ejpam-3889	110	19	∈m	∈m	NOUN
ejpam-3889	110	20	,	,	PUNCT
ejpam-3889	110	21	f(m	f(m	PROPN
ejpam-3889	110	22	)	)	PUNCT
ejpam-3889	110	23	=	=	SYM
ejpam-3889	110	24	m′.	m′.	PROPN
ejpam-3889	111	1	so	so	SCONJ
ejpam-3889	112	1	[	[	X
ejpam-3889	112	2	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	112	3	)	)	PUNCT
ejpam-3889	112	4	=	=	SYM
ejpam-3889	112	5	f(m	f(m	PROPN
ejpam-3889	112	6	)	)	PUNCT
ejpam-3889	112	7	s	s	PART
ejpam-3889	112	8	=	=	PUNCT
ejpam-3889	112	9	m′	m′	X
ejpam-3889	112	10	s	s	PART
ejpam-3889	112	11	,	,	PUNCT
ejpam-3889	112	12	thus	thus	ADV
ejpam-3889	112	13	s−1(f	s−1(f	PROPN
ejpam-3889	112	14	)	)	PUNCT
ejpam-3889	112	15	is	be	AUX
ejpam-3889	112	16	an	an	DET
ejpam-3889	112	17	epimorphism	epimorphism	NOUN
ejpam-3889	112	18	,	,	PUNCT
ejpam-3889	112	19	since	since	SCONJ
ejpam-3889	112	20	s−1(m	s−1(m	PROPN
ejpam-3889	112	21	)	)	PUNCT
ejpam-3889	112	22	is	be	AUX
ejpam-3889	112	23	hopfian	hopfian	ADJ
ejpam-3889	112	24	,	,	PUNCT
ejpam-3889	112	25	so	so	ADV
ejpam-3889	112	26	s−1(f	s−1(f	PROPN
ejpam-3889	112	27	)	)	PUNCT
ejpam-3889	112	28	is	be	AUX
ejpam-3889	112	29	an	an	DET
ejpam-3889	112	30	automorphism	automorphism	NOUN
ejpam-3889	112	31	of	of	ADP
ejpam-3889	112	32	s−1	s−1	PROPN
ejpam-3889	112	33	m	m	NOUN
ejpam-3889	112	34	.	.	PUNCT
ejpam-3889	113	1	s.	s.	PROPN
ejpam-3889	113	2	a.	a.	PROPN
ejpam-3889	113	3	balde	balde	PROPN
ejpam-3889	113	4	,	,	PUNCT
ejpam-3889	113	5	m.	m.	PROPN
ejpam-3889	113	6	b.	b.	PROPN
ejpam-3889	113	7	maaouia	maaouia	PROPN
ejpam-3889	113	8	,	,	PUNCT
ejpam-3889	113	9	a.	a.	NOUN
ejpam-3889	113	10	o.	o.	NOUN
ejpam-3889	113	11	chbih	chbih	PROPN
ejpam-3889	113	12	/	/	SYM
ejpam-3889	113	13	eur	eur	PROPN
ejpam-3889	113	14	.	.	PUNCT
ejpam-3889	114	1	j.	j.	PROPN
ejpam-3889	114	2	pure	pure	PROPN
ejpam-3889	114	3	appl	appl	PROPN
ejpam-3889	114	4	.	.	PROPN
ejpam-3889	114	5	math	math	PROPN
ejpam-3889	114	6	,	,	PUNCT
ejpam-3889	114	7	14	14	NUM
ejpam-3889	114	8	(	(	PUNCT
ejpam-3889	114	9	2	2	NUM
ejpam-3889	114	10	)	)	PUNCT
ejpam-3889	114	11	(	(	PUNCT
ejpam-3889	114	12	2021	2021	NUM
ejpam-3889	114	13	)	)	PUNCT
ejpam-3889	114	14	,	,	PUNCT
ejpam-3889	114	15	404	404	NUM
ejpam-3889	114	16	-	-	SYM
ejpam-3889	114	17	422	422	NUM
ejpam-3889	114	18	410	410	NUM
ejpam-3889	114	19	let	let	VERB
ejpam-3889	114	20	m1	m1	PROPN
ejpam-3889	114	21	and	and	CCONJ
ejpam-3889	114	22	m2	m2	PROPN
ejpam-3889	114	23	∈m	∈m	NOUN
ejpam-3889	114	24	such	such	ADJ
ejpam-3889	114	25	that	that	DET
ejpam-3889	114	26	f(m1	f(m1	NOUN
ejpam-3889	114	27	)	)	PUNCT
ejpam-3889	114	28	=	=	SYM
ejpam-3889	114	29	f(m2	f(m2	NOUN
ejpam-3889	114	30	)	)	PUNCT
ejpam-3889	115	1	=	=	NOUN
ejpam-3889	115	2	⇒	⇒	NOUN
ejpam-3889	115	3	f(m1	f(m1	NOUN
ejpam-3889	115	4	)	)	PUNCT
ejpam-3889	116	1	1	1	NUM
ejpam-3889	116	2	=	=	SYM
ejpam-3889	116	3	m2	m2	PROPN
ejpam-3889	116	4	1	1	NUM
ejpam-3889	116	5	=	=	NOUN
ejpam-3889	116	6	⇒	⇒	NOUN
ejpam-3889	116	7	[	[	X
ejpam-3889	116	8	s−1(f)](m1	s−1(f)](m1	NOUN
ejpam-3889	116	9	)	)	PUNCT
ejpam-3889	116	10	=	=	PUNCT
ejpam-3889	117	1	[	[	X
ejpam-3889	117	2	s−1(f)](m2	s−1(f)](m2	NOUN
ejpam-3889	117	3	)	)	PUNCT
ejpam-3889	117	4	=	=	NOUN
ejpam-3889	117	5	⇒	⇒	NOUN
ejpam-3889	117	6	m1	m1	PROPN
ejpam-3889	117	7	=	=	SYM
ejpam-3889	117	8	m2	m2	PROPN
ejpam-3889	117	9	thus	thus	ADV
ejpam-3889	117	10	f	f	PROPN
ejpam-3889	117	11	is	be	AUX
ejpam-3889	117	12	an	an	DET
ejpam-3889	117	13	automorphism	automorphism	NOUN
ejpam-3889	117	14	of	of	ADP
ejpam-3889	117	15	m	m	PRON
ejpam-3889	117	16	,	,	PUNCT
ejpam-3889	117	17	so	so	ADV
ejpam-3889	117	18	m	m	VERB
ejpam-3889	117	19	is	be	AUX
ejpam-3889	117	20	hopfian	hopfian	PROPN
ejpam-3889	117	21	.	.	PUNCT
ejpam-3889	118	1	theorem	theorem	VERB
ejpam-3889	118	2	6	6	NUM
ejpam-3889	118	3	.	.	PUNCT
ejpam-3889	119	1	let	let	VERB
ejpam-3889	119	2	a	a	PRON
ejpam-3889	119	3	be	be	AUX
ejpam-3889	119	4	a	a	DET
ejpam-3889	119	5	ring	ring	NOUN
ejpam-3889	119	6	,	,	PUNCT
ejpam-3889	119	7	s	s	VERB
ejpam-3889	119	8	a	a	DET
ejpam-3889	119	9	saturated	saturate	VERB
ejpam-3889	119	10	multiplicative	multiplicative	ADJ
ejpam-3889	119	11	part	part	NOUN
ejpam-3889	119	12	of	of	ADP
ejpam-3889	119	13	a	a	DET
ejpam-3889	119	14	verifying	verifying	NOUN
ejpam-3889	119	15	the	the	DET
ejpam-3889	119	16	left	left	ADJ
ejpam-3889	119	17	ore	ore	NOUN
ejpam-3889	119	18	conditions	condition	NOUN
ejpam-3889	119	19	,	,	PUNCT
ejpam-3889	119	20	m	m	VERB
ejpam-3889	119	21	a	a	DET
ejpam-3889	119	22	left	left	ADJ
ejpam-3889	119	23	a	a	DET
ejpam-3889	119	24	-	-	PUNCT
ejpam-3889	119	25	module	module	NOUN
ejpam-3889	119	26	.	.	PUNCT
ejpam-3889	120	1	if	if	SCONJ
ejpam-3889	120	2	m	m	NOUN
ejpam-3889	120	3	is	be	AUX
ejpam-3889	120	4	a	a	DET
ejpam-3889	120	5	cohopfian	cohopfian	ADJ
ejpam-3889	120	6	and	and	CCONJ
ejpam-3889	120	7	completely	completely	ADV
ejpam-3889	120	8	invariant	invariant	ADJ
ejpam-3889	120	9	submodule	submodule	NOUN
ejpam-3889	120	10	of	of	ADP
ejpam-3889	120	11	left	leave	VERB
ejpam-3889	120	12	a	a	DET
ejpam-3889	120	13	-	-	PUNCT
ejpam-3889	120	14	module	module	NOUN
ejpam-3889	120	15	s−1(m	s−1(m	PROPN
ejpam-3889	120	16	)	)	PUNCT
ejpam-3889	120	17	,	,	PUNCT
ejpam-3889	120	18	then	then	ADV
ejpam-3889	120	19	s−1	s−1	PROPN
ejpam-3889	120	20	m	m	NOUN
ejpam-3889	120	21	is	be	AUX
ejpam-3889	120	22	a	a	DET
ejpam-3889	120	23	cohopfian	cohopfian	ADJ
ejpam-3889	120	24	left	leave	VERB
ejpam-3889	120	25	s−1(a)(respectively	s−1(a)(respectively	ADV
ejpam-3889	120	26	a)-module	a)-module	NOUN
ejpam-3889	120	27	.	.	PUNCT
ejpam-3889	121	1	proof	proof	NOUN
ejpam-3889	121	2	.	.	PUNCT
ejpam-3889	122	1	let	let	VERB
ejpam-3889	122	2	g	g	NOUN
ejpam-3889	122	3	:	:	PUNCT
ejpam-3889	122	4	s−1(m	s−1(m	PROPN
ejpam-3889	122	5	)	)	PUNCT
ejpam-3889	122	6	−→	−→	NOUN
ejpam-3889	122	7	s−1(m	s−1(m	PROPN
ejpam-3889	122	8	)	)	PUNCT
ejpam-3889	122	9	be	be	AUX
ejpam-3889	122	10	a	a	DET
ejpam-3889	122	11	s−1a	s−1a	NOUN
ejpam-3889	122	12	-	-	NOUN
ejpam-3889	122	13	morphism	morphism	NOUN
ejpam-3889	122	14	,	,	PUNCT
ejpam-3889	122	15	we	we	PRON
ejpam-3889	122	16	remark	remark	VERB
ejpam-3889	122	17	also	also	ADV
ejpam-3889	122	18	that	that	SCONJ
ejpam-3889	122	19	g	g	PROPN
ejpam-3889	122	20	is	be	AUX
ejpam-3889	122	21	a	a	DET
ejpam-3889	122	22	a	a	DET
ejpam-3889	122	23	-	-	PUNCT
ejpam-3889	122	24	morphism	morphism	NOUN
ejpam-3889	122	25	.	.	PUNCT
ejpam-3889	123	1	as	as	SCONJ
ejpam-3889	123	2	m	m	PROPN
ejpam-3889	123	3	is	be	AUX
ejpam-3889	123	4	a	a	DET
ejpam-3889	123	5	completely	completely	ADV
ejpam-3889	123	6	invariant	invariant	ADJ
ejpam-3889	123	7	submodule	submodule	NOUN
ejpam-3889	123	8	of	of	ADP
ejpam-3889	123	9	left	leave	VERB
ejpam-3889	123	10	a	a	DET
ejpam-3889	123	11	-	-	PUNCT
ejpam-3889	123	12	module	module	NOUN
ejpam-3889	123	13	s−1(m	s−1(m	PROPN
ejpam-3889	123	14	)	)	PUNCT
ejpam-3889	123	15	,	,	PUNCT
ejpam-3889	123	16	so	so	ADV
ejpam-3889	123	17	g(m	g(m	ADJ
ejpam-3889	123	18	)	)	PUNCT
ejpam-3889	123	19	⊂m	⊂m	PROPN
ejpam-3889	123	20	.	.	PUNCT
ejpam-3889	124	1	suppose	suppose	VERB
ejpam-3889	124	2	that	that	SCONJ
ejpam-3889	124	3	g	g	PROPN
ejpam-3889	124	4	is	be	AUX
ejpam-3889	124	5	a	a	DET
ejpam-3889	124	6	monomorphism	monomorphism	NOUN
ejpam-3889	124	7	.	.	PUNCT
ejpam-3889	125	1	thus	thus	ADV
ejpam-3889	125	2	the	the	DET
ejpam-3889	125	3	induce	induce	ADJ
ejpam-3889	125	4	morphism	morphism	NOUN
ejpam-3889	125	5	gind	gind	NOUN
ejpam-3889	125	6	:	:	PUNCT
ejpam-3889	125	7	m	m	AUX
ejpam-3889	125	8	−→	−→	ADJ
ejpam-3889	125	9	m	m	VERB
ejpam-3889	125	10	is	be	AUX
ejpam-3889	125	11	a	a	DET
ejpam-3889	125	12	monomorphisme	monomorphisme	NOUN
ejpam-3889	125	13	of	of	ADP
ejpam-3889	125	14	m	m	PRON
ejpam-3889	125	15	,	,	PUNCT
ejpam-3889	125	16	since	since	SCONJ
ejpam-3889	125	17	m	m	PROPN
ejpam-3889	125	18	is	be	AUX
ejpam-3889	125	19	cohopfian	cohopfian	ADJ
ejpam-3889	125	20	,	,	PUNCT
ejpam-3889	125	21	then	then	ADV
ejpam-3889	125	22	gind	gind	NOUN
ejpam-3889	125	23	is	be	AUX
ejpam-3889	125	24	an	an	DET
ejpam-3889	125	25	automorphism	automorphism	NOUN
ejpam-3889	125	26	of	of	ADP
ejpam-3889	125	27	m	m	PROPN
ejpam-3889	125	28	.	.	PUNCT
ejpam-3889	126	1	let	let	VERB
ejpam-3889	126	2	’s	’s	NOUN
ejpam-3889	126	3	consider	consider	VERB
ejpam-3889	126	4	s−1(gind	s−1(gind	NOUN
ejpam-3889	126	5	)	)	PUNCT
ejpam-3889	126	6	:	:	PUNCT
ejpam-3889	127	1	s−1(m	s−1(m	PROPN
ejpam-3889	127	2	)	)	PUNCT
ejpam-3889	127	3	−→	−→	NOUN
ejpam-3889	127	4	s−1(m	s−1(m	PROPN
ejpam-3889	127	5	)	)	PUNCT
ejpam-3889	127	6	m	m	PROPN
ejpam-3889	127	7	s	s	PROPN
ejpam-3889	127	8	7−→	7−→	PROPN
ejpam-3889	127	9	gind(m	gind(m	NOUN
ejpam-3889	127	10	)	)	PUNCT
ejpam-3889	127	11	s	s	PART
ejpam-3889	127	12	so	so	ADV
ejpam-3889	127	13	gind(m	gind(m	NOUN
ejpam-3889	127	14	)	)	PUNCT
ejpam-3889	127	15	s	s	PART
ejpam-3889	127	16	=	=	SYM
ejpam-3889	127	17	1	1	NUM
ejpam-3889	127	18	s	s	NOUN
ejpam-3889	127	19	.	.	PUNCT
ejpam-3889	128	1	g(m	g(m	PROPN
ejpam-3889	128	2	)	)	PUNCT
ejpam-3889	128	3	1	1	NUM
ejpam-3889	128	4	.	.	PUNCT
ejpam-3889	129	1	or	or	CCONJ
ejpam-3889	129	2	g	g	PROPN
ejpam-3889	129	3	is	be	AUX
ejpam-3889	129	4	a	a	DET
ejpam-3889	129	5	s−1(a)-morphisme	s−1(a)-morphisme	NOUN
ejpam-3889	129	6	,	,	PUNCT
ejpam-3889	129	7	so	so	ADV
ejpam-3889	129	8	1	1	NUM
ejpam-3889	129	9	s	s	NOUN
ejpam-3889	129	10	.	.	PUNCT
ejpam-3889	130	1	gind(m	gind(m	NOUN
ejpam-3889	130	2	)	)	PUNCT
ejpam-3889	130	3	1	1	NUM
ejpam-3889	130	4	=	=	SYM
ejpam-3889	130	5	gind	gind	NOUN
ejpam-3889	130	6	(	(	PUNCT
ejpam-3889	130	7	1	1	NUM
ejpam-3889	130	8	s	s	NOUN
ejpam-3889	130	9	m	m	NOUN
ejpam-3889	130	10	1	1	NUM
ejpam-3889	130	11	)	)	PUNCT
ejpam-3889	130	12	=	=	SYM
ejpam-3889	130	13	gind	gind	NOUN
ejpam-3889	130	14	(	(	PUNCT
ejpam-3889	130	15	m	m	PROPN
ejpam-3889	130	16	s	s	PART
ejpam-3889	130	17	)	)	PUNCT
ejpam-3889	131	1	=	=	NOUN
ejpam-3889	131	2	⇒	⇒	NOUN
ejpam-3889	131	3	s−1(gind	s−1(gind	NOUN
ejpam-3889	131	4	)	)	PUNCT
ejpam-3889	131	5	=	=	PUNCT
ejpam-3889	132	1	g	g	PROPN
ejpam-3889	132	2	g	g	PROPN
ejpam-3889	132	3	is	be	AUX
ejpam-3889	132	4	a	a	DET
ejpam-3889	132	5	monomorphism	monomorphism	NOUN
ejpam-3889	132	6	,	,	PUNCT
ejpam-3889	132	7	then	then	ADV
ejpam-3889	132	8	gind	gind	NOUN
ejpam-3889	132	9	is	be	AUX
ejpam-3889	132	10	a	a	DET
ejpam-3889	132	11	monomorphism	monomorphism	NOUN
ejpam-3889	132	12	.	.	PUNCT
ejpam-3889	133	1	as	as	SCONJ
ejpam-3889	133	2	m	m	PROPN
ejpam-3889	133	3	is	be	AUX
ejpam-3889	133	4	cohopfian	cohopfian	ADJ
ejpam-3889	133	5	,	,	PUNCT
ejpam-3889	133	6	then	then	ADV
ejpam-3889	133	7	gind	gind	NOUN
ejpam-3889	133	8	is	be	AUX
ejpam-3889	133	9	an	an	DET
ejpam-3889	133	10	automorphism	automorphism	NOUN
ejpam-3889	133	11	of	of	ADP
ejpam-3889	133	12	m	m	PROPN
ejpam-3889	133	13	.	.	PUNCT
ejpam-3889	134	1	let	let	VERB
ejpam-3889	134	2	m′	m′	NOUN
ejpam-3889	134	3	s	s	PART
ejpam-3889	134	4	∈	∈	NOUN
ejpam-3889	134	5	s	s	PART
ejpam-3889	134	6	−1(m	−1(m	ADJ
ejpam-3889	134	7	)	)	PUNCT
ejpam-3889	134	8	=	=	VERB
ejpam-3889	134	9	⇒	⇒	VERB
ejpam-3889	134	10	m′	m′	NOUN
ejpam-3889	134	11	∈m	∈m	NOUN
ejpam-3889	134	12	,	,	PUNCT
ejpam-3889	134	13	so	so	CCONJ
ejpam-3889	134	14	there	there	PRON
ejpam-3889	134	15	exists	exist	VERB
ejpam-3889	134	16	m	m	VERB
ejpam-3889	134	17	∈m	∈m	NOUN
ejpam-3889	134	18	:	:	PUNCT
ejpam-3889	134	19	gind(m	gind(m	NOUN
ejpam-3889	134	20	)	)	PUNCT
ejpam-3889	134	21	=	=	PUNCT
ejpam-3889	134	22	m′	m′	NOUN
ejpam-3889	135	1	thus	thus	ADV
ejpam-3889	135	2	m	m	PROPN
ejpam-3889	135	3	s	s	NOUN
ejpam-3889	135	4	∈	∈	PROPN
ejpam-3889	135	5	s	s	PART
ejpam-3889	135	6	−1	−1	NOUN
ejpam-3889	135	7	m	m	PRON
ejpam-3889	135	8	,	,	PUNCT
ejpam-3889	135	9	[	[	X
ejpam-3889	135	10	s−1(gind	s−1(gind	NOUN
ejpam-3889	135	11	)	)	PUNCT
ejpam-3889	135	12	]	]	PUNCT
ejpam-3889	135	13	(	(	PUNCT
ejpam-3889	135	14	m	m	PROPN
ejpam-3889	135	15	s	s	PART
ejpam-3889	135	16	)	)	PUNCT
ejpam-3889	135	17	=	=	SYM
ejpam-3889	135	18	g(m	g(m	VERB
ejpam-3889	135	19	)	)	PUNCT
ejpam-3889	135	20	s	s	PART
ejpam-3889	135	21	=	=	PUNCT
ejpam-3889	135	22	m′	m′	NOUN
ejpam-3889	135	23	s	s	PART
ejpam-3889	135	24	hence	hence	ADV
ejpam-3889	135	25	(	(	PUNCT
ejpam-3889	135	26	g(ms	g(ms	PROPN
ejpam-3889	135	27	)	)	PUNCT
ejpam-3889	135	28	)	)	PUNCT
ejpam-3889	136	1	=	=	PUNCT
ejpam-3889	136	2	m′	m′	NOUN
ejpam-3889	136	3	s	s	PART
ejpam-3889	136	4	.	.	PUNCT
ejpam-3889	137	1	thus	thus	ADV
ejpam-3889	137	2	s−1	s−1	PROPN
ejpam-3889	137	3	m	m	NOUN
ejpam-3889	137	4	is	be	AUX
ejpam-3889	137	5	a	a	DET
ejpam-3889	137	6	cohopfian	cohopfian	ADJ
ejpam-3889	137	7	left	leave	VERB
ejpam-3889	137	8	s−1a	s−1a	NOUN
ejpam-3889	137	9	-	-	PUNCT
ejpam-3889	137	10	module	module	NOUN
ejpam-3889	137	11	.	.	PUNCT
ejpam-3889	138	1	let	let	VERB
ejpam-3889	138	2	’s	’s	PRON
ejpam-3889	138	3	prove	prove	VERB
ejpam-3889	138	4	now	now	ADV
ejpam-3889	138	5	that	that	SCONJ
ejpam-3889	138	6	s−1(m	s−1(m	PROPN
ejpam-3889	138	7	)	)	PUNCT
ejpam-3889	138	8	is	be	AUX
ejpam-3889	138	9	a	a	DET
ejpam-3889	138	10	cohopfian	cohopfian	ADJ
ejpam-3889	138	11	left	leave	VERB
ejpam-3889	138	12	a	a	DET
ejpam-3889	138	13	-	-	PUNCT
ejpam-3889	138	14	module	module	NOUN
ejpam-3889	138	15	let	let	VERB
ejpam-3889	138	16	f	f	NOUN
ejpam-3889	138	17	:	:	PUNCT
ejpam-3889	138	18	s−1(m	s−1(m	PROPN
ejpam-3889	138	19	)	)	PUNCT
ejpam-3889	138	20	−→	−→	NOUN
ejpam-3889	138	21	s−1(m	s−1(m	PROPN
ejpam-3889	138	22	)	)	PUNCT
ejpam-3889	138	23	be	be	AUX
ejpam-3889	138	24	a	a	DET
ejpam-3889	138	25	monomorphism	monomorphism	NOUN
ejpam-3889	138	26	of	of	ADP
ejpam-3889	138	27	lefta	lefta	NOUN
ejpam-3889	138	28	-	-	PUNCT
ejpam-3889	138	29	module	module	NOUN
ejpam-3889	138	30	,	,	PUNCT
ejpam-3889	138	31	then	then	ADV
ejpam-3889	138	32	f	f	X
ejpam-3889	138	33	:	:	PUNCT
ejpam-3889	138	34	s−1(m	s−1(m	PROPN
ejpam-3889	138	35	)	)	PUNCT
ejpam-3889	138	36	−→	−→	NOUN
ejpam-3889	138	37	s−1(m	s−1(m	PROPN
ejpam-3889	138	38	)	)	PUNCT
ejpam-3889	138	39	is	be	AUX
ejpam-3889	138	40	a	a	DET
ejpam-3889	138	41	monomorphism	monomorphism	NOUN
ejpam-3889	138	42	,	,	PUNCT
ejpam-3889	138	43	indeed	indeed	ADV
ejpam-3889	138	44	,	,	PUNCT
ejpam-3889	138	45	let	let	VERB
ejpam-3889	138	46	m	m	PRON
ejpam-3889	138	47	s	s	NOUN
ejpam-3889	138	48	and	and	CCONJ
ejpam-3889	138	49	m′	m′	NOUN
ejpam-3889	138	50	s′	s′	VERB
ejpam-3889	138	51	∈	∈	PROPN
ejpam-3889	138	52	s	s	PART
ejpam-3889	138	53	−1	−1	NOUN
ejpam-3889	138	54	m	m	VERB
ejpam-3889	138	55	:	:	PUNCT
ejpam-3889	139	1	[	[	X
ejpam-3889	139	2	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	139	3	)	)	PUNCT
ejpam-3889	139	4	=	=	PUNCT
ejpam-3889	140	1	[	[	X
ejpam-3889	140	2	s−1(f)](m	s−1(f)](m	X
ejpam-3889	140	3	′	′	NOUN
ejpam-3889	140	4	s′	s′	NOUN
ejpam-3889	140	5	)	)	PUNCT
ejpam-3889	141	1	=	=	SYM
ejpam-3889	141	2	⇒	⇒	NOUN
ejpam-3889	141	3	f(m	f(m	PROPN
ejpam-3889	141	4	)	)	PUNCT
ejpam-3889	141	5	s	s	PART
ejpam-3889	141	6	=	=	SYM
ejpam-3889	141	7	f(m′	f(m′	NOUN
ejpam-3889	141	8	)	)	PUNCT
ejpam-3889	141	9	s′	s′	PUNCT
ejpam-3889	141	10	=	=	NOUN
ejpam-3889	141	11	⇒	⇒	NOUN
ejpam-3889	141	12	∃x	∃x	NOUN
ejpam-3889	141	13	,	,	PUNCT
ejpam-3889	141	14	y	y	PROPN
ejpam-3889	141	15	∈	∈	PROPN
ejpam-3889	141	16	s	s	VERB
ejpam-3889	141	17	such	such	ADJ
ejpam-3889	141	18	that	that	PRON
ejpam-3889	141	19	:	:	PUNCT
ejpam-3889	141	20	{	{	PUNCT
ejpam-3889	141	21	x.f(m	x.f(m	X
ejpam-3889	141	22	)	)	PUNCT
ejpam-3889	141	23	=	=	SYM
ejpam-3889	141	24	y.f(m′	y.f(m′	PROPN
ejpam-3889	141	25	)	)	PUNCT
ejpam-3889	141	26	xs	xs	PROPN
ejpam-3889	142	1	=	=	SYM
ejpam-3889	142	2	ys′	ys′	PUNCT
ejpam-3889	143	1	=	=	PRON
ejpam-3889	143	2	⇒	⇒	NOUN
ejpam-3889	143	3	{	{	PUNCT
ejpam-3889	143	4	f(x.m	f(x.m	ADJ
ejpam-3889	143	5	)	)	PUNCT
ejpam-3889	143	6	=	=	SYM
ejpam-3889	143	7	f(y.m′	f(y.m′	PROPN
ejpam-3889	143	8	)	)	PUNCT
ejpam-3889	143	9	xs	xs	PROPN
ejpam-3889	143	10	=	=	SYM
ejpam-3889	143	11	ys′	ys′	PUNCT
ejpam-3889	144	1	=	=	PRON
ejpam-3889	144	2	⇒	⇒	NOUN
ejpam-3889	144	3	{	{	PUNCT
ejpam-3889	144	4	x.m	x.m	PROPN
ejpam-3889	144	5	=	=	PUNCT
ejpam-3889	144	6	y.m′	y.m′	PROPN
ejpam-3889	144	7	xs	xs	PROPN
ejpam-3889	144	8	=	=	SYM
ejpam-3889	144	9	ys′	ys′	PUNCT
ejpam-3889	145	1	=	=	PRON
ejpam-3889	145	2	⇒	⇒	VERB
ejpam-3889	145	3	m	m	VERB
ejpam-3889	145	4	s	s	NOUN
ejpam-3889	145	5	=	=	PUNCT
ejpam-3889	145	6	m′	m′	NOUN
ejpam-3889	145	7	s	s	PART
ejpam-3889	145	8	=	=	NOUN
ejpam-3889	145	9	⇒	⇒	PROPN
ejpam-3889	145	10	s−1(f	s−1(f	PROPN
ejpam-3889	145	11	)	)	PUNCT
ejpam-3889	145	12	is	be	AUX
ejpam-3889	145	13	a	a	DET
ejpam-3889	145	14	monomorphisme	monomorphisme	NOUN
ejpam-3889	145	15	,	,	PUNCT
ejpam-3889	145	16	or	or	CCONJ
ejpam-3889	145	17	s−1(m	s−1(m	PROPN
ejpam-3889	145	18	)	)	PUNCT
ejpam-3889	145	19	is	be	AUX
ejpam-3889	145	20	cohopfian	cohopfian	ADJ
ejpam-3889	145	21	,	,	PUNCT
ejpam-3889	145	22	then	then	ADV
ejpam-3889	145	23	s−1(f	s−1(f	PROPN
ejpam-3889	145	24	)	)	PUNCT
ejpam-3889	145	25	is	be	AUX
ejpam-3889	145	26	an	an	DET
ejpam-3889	145	27	automorphism	automorphism	NOUN
ejpam-3889	145	28	.	.	PUNCT
ejpam-3889	146	1	let	let	VERB
ejpam-3889	146	2	m′	m′	NOUN
ejpam-3889	146	3	s′	s′	ADJ
ejpam-3889	146	4	∈	∈	PROPN
ejpam-3889	146	5	s−1(m	s−1(m	PROPN
ejpam-3889	146	6	)	)	PUNCT
ejpam-3889	146	7	,	,	PUNCT
ejpam-3889	146	8	since	since	SCONJ
ejpam-3889	146	9	s−1(f	s−1(f	PROPN
ejpam-3889	146	10	)	)	PUNCT
ejpam-3889	146	11	is	be	AUX
ejpam-3889	146	12	an	an	DET
ejpam-3889	146	13	automorphism	automorphism	NOUN
ejpam-3889	146	14	,	,	PUNCT
ejpam-3889	146	15	then	then	ADV
ejpam-3889	146	16	there	there	PRON
ejpam-3889	146	17	exists	exist	VERB
ejpam-3889	146	18	m	m	PROPN
ejpam-3889	146	19	s	s	NOUN
ejpam-3889	146	20	∈	∈	NOUN
ejpam-3889	146	21	s−1	s−1	PROPN
ejpam-3889	146	22	m	m	NOUN
ejpam-3889	146	23	such	such	ADJ
ejpam-3889	146	24	that	that	SCONJ
ejpam-3889	146	25	[	[	X
ejpam-3889	146	26	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	146	27	)	)	PUNCT
ejpam-3889	146	28	=	=	SYM
ejpam-3889	146	29	m′	m′	NOUN
ejpam-3889	146	30	s′	s′	VERB
ejpam-3889	146	31	=	=	NOUN
ejpam-3889	146	32	⇒	⇒	NOUN
ejpam-3889	146	33	f(m	f(m	PROPN
ejpam-3889	146	34	)	)	PUNCT
ejpam-3889	146	35	s	s	PART
ejpam-3889	146	36	=	=	PUNCT
ejpam-3889	146	37	m′	m′	NOUN
ejpam-3889	146	38	s′	s′	VERB
ejpam-3889	146	39	=	=	NOUN
ejpam-3889	146	40	⇒	⇒	NOUN
ejpam-3889	146	41	s	s	PART
ejpam-3889	146	42	s	s	NOUN
ejpam-3889	146	43	.m	.m	NOUN
ejpam-3889	146	44	s	s	PART
ejpam-3889	146	45	=	=	NOUN
ejpam-3889	146	46	m′	m′	NOUN
ejpam-3889	146	47	s′	s′	VERB
ejpam-3889	146	48	=	=	NOUN
ejpam-3889	146	49	⇒	⇒	VERB
ejpam-3889	146	50	s.f(m	s.f(m	NOUN
ejpam-3889	146	51	s	s	PART
ejpam-3889	146	52	)	)	PUNCT
ejpam-3889	146	53	s	s	PART
ejpam-3889	146	54	=	=	VERB
ejpam-3889	146	55	m′	m′	NOUN
ejpam-3889	146	56	s′	s′	VERB
ejpam-3889	146	57	=	=	NOUN
ejpam-3889	146	58	⇒	⇒	NOUN
ejpam-3889	146	59	f(m	f(m	PROPN
ejpam-3889	146	60	)	)	PUNCT
ejpam-3889	146	61	s	s	PART
ejpam-3889	146	62	=	=	NOUN
ejpam-3889	146	63	m′	m′	NOUN
ejpam-3889	146	64	s′	s′	NUM
ejpam-3889	146	65	,	,	PUNCT
ejpam-3889	146	66	thus	thus	ADV
ejpam-3889	146	67	there	there	PRON
ejpam-3889	146	68	exists	exist	VERB
ejpam-3889	146	69	m	m	PROPN
ejpam-3889	146	70	s	s	NOUN
ejpam-3889	146	71	∈	∈	NOUN
ejpam-3889	146	72	s−1	s−1	PROPN
ejpam-3889	146	73	m	m	NOUN
ejpam-3889	146	74	such	such	ADJ
ejpam-3889	146	75	that	that	SCONJ
ejpam-3889	146	76	f(ms	f(ms	ADV
ejpam-3889	146	77	)	)	PUNCT
ejpam-3889	146	78	=	=	SYM
ejpam-3889	146	79	m′	m′	NOUN
ejpam-3889	146	80	s′	s′	VERB
ejpam-3889	146	81	=	=	NOUN
ejpam-3889	146	82	⇒	⇒	NOUN
ejpam-3889	146	83	f	f	X
ejpam-3889	146	84	is	be	AUX
ejpam-3889	146	85	an	an	DET
ejpam-3889	146	86	epimorphism	epimorphism	NOUN
ejpam-3889	146	87	,	,	PUNCT
ejpam-3889	146	88	hence	hence	ADV
ejpam-3889	146	89	s−1(m	s−1(m	PROPN
ejpam-3889	146	90	)	)	PUNCT
ejpam-3889	146	91	is	be	AUX
ejpam-3889	146	92	cohopfian	cohopfian	ADJ
ejpam-3889	146	93	.	.	PUNCT
ejpam-3889	147	1	s.	s.	PROPN
ejpam-3889	147	2	a.	a.	PROPN
ejpam-3889	147	3	balde	balde	PROPN
ejpam-3889	147	4	,	,	PUNCT
ejpam-3889	147	5	m.	m.	PROPN
ejpam-3889	147	6	b.	b.	PROPN
ejpam-3889	147	7	maaouia	maaouia	PROPN
ejpam-3889	147	8	,	,	PUNCT
ejpam-3889	147	9	a.	a.	NOUN
ejpam-3889	147	10	o.	o.	NOUN
ejpam-3889	147	11	chbih	chbih	PROPN
ejpam-3889	147	12	/	/	SYM
ejpam-3889	147	13	eur	eur	PROPN
ejpam-3889	147	14	.	.	PUNCT
ejpam-3889	148	1	j.	j.	PROPN
ejpam-3889	148	2	pure	pure	PROPN
ejpam-3889	148	3	appl	appl	PROPN
ejpam-3889	148	4	.	.	PROPN
ejpam-3889	148	5	math	math	PROPN
ejpam-3889	148	6	,	,	PUNCT
ejpam-3889	148	7	14	14	NUM
ejpam-3889	148	8	(	(	PUNCT
ejpam-3889	148	9	2	2	NUM
ejpam-3889	148	10	)	)	PUNCT
ejpam-3889	148	11	(	(	PUNCT
ejpam-3889	148	12	2021	2021	NUM
ejpam-3889	148	13	)	)	PUNCT
ejpam-3889	148	14	,	,	PUNCT
ejpam-3889	148	15	404	404	NUM
ejpam-3889	148	16	-	-	SYM
ejpam-3889	148	17	422	422	NUM
ejpam-3889	148	18	411	411	NUM
ejpam-3889	148	19	theorem	theorem	NOUN
ejpam-3889	148	20	7	7	NUM
ejpam-3889	148	21	.	.	PUNCT
ejpam-3889	149	1	let	let	VERB
ejpam-3889	149	2	m	m	PRON
ejpam-3889	149	3	be	be	AUX
ejpam-3889	149	4	a	a	DET
ejpam-3889	149	5	noetherian	noetherian	ADJ
ejpam-3889	149	6	quasi	quasi	ADJ
ejpam-3889	149	7	-	-	ADJ
ejpam-3889	149	8	injective	injective	ADJ
ejpam-3889	149	9	left	leave	VERB
ejpam-3889	149	10	a	a	DET
ejpam-3889	149	11	-	-	PUNCT
ejpam-3889	149	12	module	module	NOUN
ejpam-3889	149	13	,	,	PUNCT
ejpam-3889	149	14	n	n	PRON
ejpam-3889	149	15	be	be	VERB
ejpam-3889	149	16	an	an	DET
ejpam-3889	149	17	essential	essential	ADJ
ejpam-3889	149	18	and	and	CCONJ
ejpam-3889	149	19	completely	completely	ADV
ejpam-3889	149	20	invariant	invariant	ADJ
ejpam-3889	149	21	submodule	submodule	NOUN
ejpam-3889	149	22	of	of	ADP
ejpam-3889	149	23	m	m	PROPN
ejpam-3889	149	24	and	and	CCONJ
ejpam-3889	149	25	s	s	VERB
ejpam-3889	149	26	a	a	DET
ejpam-3889	149	27	saturated	saturate	VERB
ejpam-3889	149	28	multiplicative	multiplicative	NOUN
ejpam-3889	149	29	of	of	ADP
ejpam-3889	149	30	a	a	DET
ejpam-3889	149	31	verifying	verifying	NOUN
ejpam-3889	149	32	the	the	DET
ejpam-3889	149	33	left	left	ADJ
ejpam-3889	149	34	ore	ore	NOUN
ejpam-3889	149	35	conditions	condition	NOUN
ejpam-3889	149	36	.	.	PUNCT
ejpam-3889	150	1	then	then	ADV
ejpam-3889	150	2	,	,	PUNCT
ejpam-3889	150	3	the	the	DET
ejpam-3889	150	4	left	leave	VERB
ejpam-3889	150	5	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	150	6	s−1(n	s−1(n	NOUN
ejpam-3889	150	7	)	)	PUNCT
ejpam-3889	150	8	is	be	AUX
ejpam-3889	150	9	cohopfian	cohopfian	ADJ
ejpam-3889	150	10	if	if	SCONJ
ejpam-3889	150	11	,	,	PUNCT
ejpam-3889	150	12	and	and	CCONJ
ejpam-3889	150	13	only	only	ADV
ejpam-3889	150	14	,	,	PUNCT
ejpam-3889	150	15	if	if	SCONJ
ejpam-3889	150	16	s−1(m	s−1(m	PROPN
ejpam-3889	150	17	)	)	PUNCT
ejpam-3889	150	18	is	be	AUX
ejpam-3889	150	19	cohopfian	cohopfian	ADJ
ejpam-3889	150	20	left	leave	VERB
ejpam-3889	150	21	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	150	22	.	.	PUNCT
ejpam-3889	151	1	proof	proof	NOUN
ejpam-3889	151	2	.	.	PUNCT
ejpam-3889	152	1	suppose	suppose	VERB
ejpam-3889	152	2	that	that	SCONJ
ejpam-3889	152	3	s−1(m	s−1(m	PROPN
ejpam-3889	152	4	)	)	PUNCT
ejpam-3889	152	5	is	be	AUX
ejpam-3889	152	6	cohopfian	cohopfian	ADJ
ejpam-3889	152	7	and	and	CCONJ
ejpam-3889	152	8	left	leave	VERB
ejpam-3889	152	9	s−1(f	s−1(f	PROPN
ejpam-3889	152	10	)	)	PUNCT
ejpam-3889	152	11	:	:	PUNCT
ejpam-3889	152	12	s−1(n	s−1(n	PROPN
ejpam-3889	152	13	)	)	PUNCT
ejpam-3889	152	14	−→	−→	NOUN
ejpam-3889	152	15	s−1(n	s−1(n	PROPN
ejpam-3889	152	16	)	)	PUNCT
ejpam-3889	152	17	be	be	VERB
ejpam-3889	152	18	a	a	DET
ejpam-3889	152	19	monomorphism	monomorphism	NOUN
ejpam-3889	152	20	.	.	PUNCT
ejpam-3889	153	1	as	as	ADP
ejpam-3889	153	2	s−1(m	s−1(m	PROPN
ejpam-3889	153	3	)	)	PUNCT
ejpam-3889	153	4	is	be	AUX
ejpam-3889	153	5	quasi	quasi	ADJ
ejpam-3889	153	6	-	-	ADJ
ejpam-3889	153	7	injective	injective	ADJ
ejpam-3889	153	8	because	because	SCONJ
ejpam-3889	153	9	m	m	NOUN
ejpam-3889	153	10	is	be	AUX
ejpam-3889	153	11	noetherian	noetherian	ADJ
ejpam-3889	153	12	.	.	PUNCT
ejpam-3889	154	1	then	then	ADV
ejpam-3889	154	2	,	,	PUNCT
ejpam-3889	154	3	there	there	PRON
ejpam-3889	154	4	exists	exist	VERB
ejpam-3889	154	5	s−1(g	s−1(g	PROPN
ejpam-3889	154	6	)	)	PUNCT
ejpam-3889	154	7	∈	∈	PROPN
ejpam-3889	154	8	end(s−1(m	end(s−1(m	PROPN
ejpam-3889	154	9	)	)	PUNCT
ejpam-3889	154	10	)	)	PUNCT
ejpam-3889	154	11	such	such	ADJ
ejpam-3889	154	12	that	that	DET
ejpam-3889	154	13	s−1(g|s−1n	s−1(g|s−1n	NOUN
ejpam-3889	154	14	)	)	PUNCT
ejpam-3889	154	15	=	=	SYM
ejpam-3889	154	16	s−1(f	s−1(f	PROPN
ejpam-3889	154	17	)	)	PUNCT
ejpam-3889	154	18	.	.	PUNCT
ejpam-3889	155	1	s−1(g	s−1(g	PROPN
ejpam-3889	155	2	)	)	PUNCT
ejpam-3889	155	3	is	be	AUX
ejpam-3889	155	4	injective	injective	ADJ
ejpam-3889	155	5	since	since	SCONJ
ejpam-3889	155	6	s−1(n	s−1(n	PROPN
ejpam-3889	155	7	)	)	PUNCT
ejpam-3889	155	8	is	be	AUX
ejpam-3889	155	9	essential	essential	ADJ
ejpam-3889	155	10	in	in	ADP
ejpam-3889	155	11	s−1(m	s−1(m	PROPN
ejpam-3889	155	12	)	)	PUNCT
ejpam-3889	155	13	,	,	PUNCT
ejpam-3889	155	14	and	and	CCONJ
ejpam-3889	155	15	as	as	ADP
ejpam-3889	155	16	s−1(m	s−1(m	PROPN
ejpam-3889	155	17	)	)	PUNCT
ejpam-3889	155	18	is	be	AUX
ejpam-3889	155	19	cohopfian	cohopfian	ADJ
ejpam-3889	155	20	,	,	PUNCT
ejpam-3889	155	21	s−1(g	s−1(g	PROPN
ejpam-3889	155	22	)	)	PUNCT
ejpam-3889	155	23	is	be	AUX
ejpam-3889	155	24	invertible	invertible	ADJ
ejpam-3889	155	25	.	.	PUNCT
ejpam-3889	156	1	let	let	VERB
ejpam-3889	156	2	x	x	SYM
ejpam-3889	156	3	s	s	PART
ejpam-3889	156	4	∈	∈	PROPN
ejpam-3889	156	5	s	s	PART
ejpam-3889	156	6	−1(n	−1(n	NOUN
ejpam-3889	156	7	)	)	PUNCT
ejpam-3889	156	8	,	,	PUNCT
ejpam-3889	156	9	there	there	PRON
ejpam-3889	156	10	exists	exist	VERB
ejpam-3889	156	11	y	y	PROPN
ejpam-3889	156	12	t	t	PROPN
ejpam-3889	156	13	∈	∈	PROPN
ejpam-3889	156	14	s	s	PART
ejpam-3889	156	15	−1(m	−1(m	NOUN
ejpam-3889	156	16	)	)	PUNCT
ejpam-3889	156	17	such	such	ADJ
ejpam-3889	156	18	that	that	SCONJ
ejpam-3889	156	19	x	x	PRON
ejpam-3889	156	20	s	s	NOUN
ejpam-3889	156	21	=	=	X
ejpam-3889	157	1	[	[	X
ejpam-3889	157	2	s−1(g)](yt	s−1(g)](yt	PROPN
ejpam-3889	157	3	)	)	PUNCT
ejpam-3889	157	4	.	.	PUNCT
ejpam-3889	158	1	or	or	CCONJ
ejpam-3889	158	2	s−1(g−1	s−1(g−1	X
ejpam-3889	158	3	)	)	PUNCT
ejpam-3889	158	4	∈	∈	PROPN
ejpam-3889	158	5	end(s−1()n	end(s−1()n	NOUN
ejpam-3889	158	6	)	)	PUNCT
ejpam-3889	158	7	)	)	PUNCT
ejpam-3889	158	8	and	and	CCONJ
ejpam-3889	158	9	s−1(n	s−1(n	NOUN
ejpam-3889	158	10	)	)	PUNCT
ejpam-3889	158	11	is	be	AUX
ejpam-3889	158	12	completely	completely	ADV
ejpam-3889	158	13	invariant	invariant	ADJ
ejpam-3889	158	14	,	,	PUNCT
ejpam-3889	158	15	so	so	ADV
ejpam-3889	158	16	y	y	PROPN
ejpam-3889	158	17	t	t	PROPN
ejpam-3889	159	1	=	=	PUNCT
ejpam-3889	160	1	[	[	X
ejpam-3889	160	2	s−1(g−1)](xs	s−1(g−1)](xs	ADP
ejpam-3889	160	3	)	)	PUNCT
ejpam-3889	160	4	∈	∈	PROPN
ejpam-3889	160	5	s−1(n	s−1(n	PROPN
ejpam-3889	160	6	)	)	PUNCT
ejpam-3889	160	7	,	,	PUNCT
ejpam-3889	160	8	thus	thus	ADV
ejpam-3889	160	9	s−1(f	s−1(f	PROPN
ejpam-3889	160	10	)	)	PUNCT
ejpam-3889	160	11	is	be	AUX
ejpam-3889	160	12	an	an	DET
ejpam-3889	160	13	automorphisme	automorphisme	NOUN
ejpam-3889	160	14	,	,	PUNCT
ejpam-3889	160	15	consequentely	consequentely	ADV
ejpam-3889	160	16	s−1(n	s−1(n	PROPN
ejpam-3889	160	17	)	)	PUNCT
ejpam-3889	160	18	is	be	AUX
ejpam-3889	160	19	cohopfian	cohopfian	ADJ
ejpam-3889	160	20	.	.	PUNCT
ejpam-3889	161	1	reciprocally	reciprocally	PROPN
ejpam-3889	161	2	,	,	PUNCT
ejpam-3889	161	3	suppose	suppose	VERB
ejpam-3889	161	4	that	that	SCONJ
ejpam-3889	161	5	s−1(n	s−1(n	PROPN
ejpam-3889	161	6	)	)	PUNCT
ejpam-3889	161	7	is	be	AUX
ejpam-3889	161	8	cohopfian	cohopfian	ADJ
ejpam-3889	161	9	and	and	CCONJ
ejpam-3889	161	10	let	let	VERB
ejpam-3889	161	11	s−1(f	s−1(f	PROPN
ejpam-3889	161	12	)	)	PUNCT
ejpam-3889	161	13	:	:	PUNCT
ejpam-3889	162	1	s−1(m	s−1(m	PROPN
ejpam-3889	162	2	)	)	PUNCT
ejpam-3889	162	3	−→	−→	NOUN
ejpam-3889	162	4	s−1(m	s−1(m	PROPN
ejpam-3889	162	5	)	)	PUNCT
ejpam-3889	162	6	a	a	DET
ejpam-3889	162	7	monomorphism	monomorphism	NOUN
ejpam-3889	162	8	.	.	PUNCT
ejpam-3889	163	1	then	then	ADV
ejpam-3889	163	2	s−1(f|s−1(n	s−1(f|s−1(n	PROPN
ejpam-3889	163	3	)	)	PUNCT
ejpam-3889	163	4	)	)	PUNCT
ejpam-3889	163	5	is	be	AUX
ejpam-3889	163	6	a	a	DET
ejpam-3889	163	7	monomorphism	monomorphism	NOUN
ejpam-3889	163	8	of	of	ADP
ejpam-3889	163	9	s−1(n	s−1(n	PROPN
ejpam-3889	163	10	)	)	PUNCT
ejpam-3889	163	11	.	.	PUNCT
ejpam-3889	164	1	thus	thus	ADV
ejpam-3889	164	2	,	,	PUNCT
ejpam-3889	164	3	s−1(f	s−1(f	PROPN
ejpam-3889	164	4	)	)	PUNCT
ejpam-3889	164	5	∈	∈	PROPN
ejpam-3889	164	6	aut(s−1(n	aut(s−1(n	PROPN
ejpam-3889	164	7	)	)	PUNCT
ejpam-3889	164	8	)	)	PUNCT
ejpam-3889	164	9	,	,	PUNCT
ejpam-3889	164	10	hence	hence	ADV
ejpam-3889	164	11	[	[	X
ejpam-3889	164	12	s−1(f)](s−1(n	s−1(f)](s−1(n	NOUN
ejpam-3889	164	13	)	)	PUNCT
ejpam-3889	164	14	)	)	PUNCT
ejpam-3889	165	1	=	=	SYM
ejpam-3889	165	2	s−1(n	s−1(n	PROPN
ejpam-3889	165	3	)	)	PUNCT
ejpam-3889	165	4	.	.	PUNCT
ejpam-3889	166	1	as	as	ADP
ejpam-3889	166	2	s−1(m	s−1(m	PROPN
ejpam-3889	166	3	)	)	PUNCT
ejpam-3889	166	4	is	be	AUX
ejpam-3889	166	5	quasi	quasi	ADJ
ejpam-3889	166	6	-	-	ADJ
ejpam-3889	166	7	injective	injective	ADJ
ejpam-3889	166	8	,	,	PUNCT
ejpam-3889	166	9	then	then	ADV
ejpam-3889	166	10	there	there	PRON
ejpam-3889	166	11	exists	exist	VERB
ejpam-3889	166	12	s−1(l	s−1(l	PROPN
ejpam-3889	166	13	)	)	PUNCT
ejpam-3889	166	14	a	a	DET
ejpam-3889	166	15	submodule	submodule	NOUN
ejpam-3889	166	16	of	of	ADP
ejpam-3889	166	17	s−1(m	s−1(m	PROPN
ejpam-3889	166	18	)	)	PUNCT
ejpam-3889	166	19	such	such	ADJ
ejpam-3889	166	20	that	that	SCONJ
ejpam-3889	166	21	s−1(m	s−1(m	PROPN
ejpam-3889	166	22	)	)	PUNCT
ejpam-3889	167	1	=	=	PUNCT
ejpam-3889	168	1	[	[	X
ejpam-3889	168	2	s−1(f)](s−1(m	s−1(f)](s−1(m	NOUN
ejpam-3889	168	3	)	)	PUNCT
ejpam-3889	168	4	)	)	PUNCT
ejpam-3889	169	1	⊕	⊕	PROPN
ejpam-3889	169	2	s−1(l	s−1(l	PROPN
ejpam-3889	169	3	)	)	PUNCT
ejpam-3889	169	4	.	.	PUNCT
ejpam-3889	170	1	thus	thus	ADV
ejpam-3889	170	2	,	,	PUNCT
ejpam-3889	170	3	we	we	PRON
ejpam-3889	170	4	have	have	VERB
ejpam-3889	170	5	0	0	NUM
ejpam-3889	171	1	=	=	PUNCT
ejpam-3889	172	1	[	[	X
ejpam-3889	172	2	s−1(f)](s−1n	s−1(f)](s−1n	NOUN
ejpam-3889	172	3	)	)	PUNCT
ejpam-3889	172	4	∩	∩	PROPN
ejpam-3889	172	5	s−1(l	s−1(l	PROPN
ejpam-3889	172	6	)	)	PUNCT
ejpam-3889	172	7	=	=	SYM
ejpam-3889	172	8	s−1(n	s−1(n	PROPN
ejpam-3889	172	9	)	)	PUNCT
ejpam-3889	172	10	∩	∩	PROPN
ejpam-3889	172	11	s−1(l	s−1(l	PROPN
ejpam-3889	172	12	)	)	PUNCT
ejpam-3889	172	13	,	,	PUNCT
ejpam-3889	172	14	since	since	SCONJ
ejpam-3889	172	15	s−1(n	s−1(n	PROPN
ejpam-3889	172	16	)	)	PUNCT
ejpam-3889	172	17	is	be	AUX
ejpam-3889	172	18	essential	essential	ADJ
ejpam-3889	172	19	,	,	PUNCT
ejpam-3889	172	20	then	then	ADV
ejpam-3889	172	21	s−1(l	s−1(l	PROPN
ejpam-3889	172	22	)	)	PUNCT
ejpam-3889	173	1	=	=	SYM
ejpam-3889	173	2	0	0	NUM
ejpam-3889	173	3	,	,	PUNCT
ejpam-3889	173	4	hence	hence	ADV
ejpam-3889	173	5	s−1(m	s−1(m	PROPN
ejpam-3889	173	6	)	)	PUNCT
ejpam-3889	173	7	=	=	PUNCT
ejpam-3889	174	1	[	[	X
ejpam-3889	174	2	s−1(f)](s−1	s−1(f)](s−1	NOUN
ejpam-3889	174	3	m	m	NOUN
ejpam-3889	174	4	)	)	PUNCT
ejpam-3889	174	5	,	,	PUNCT
ejpam-3889	174	6	thus	thus	ADV
ejpam-3889	174	7	s−1(f	s−1(f	PROPN
ejpam-3889	174	8	)	)	PUNCT
ejpam-3889	174	9	is	be	AUX
ejpam-3889	174	10	an	an	DET
ejpam-3889	174	11	epimorphism	epimorphism	NOUN
ejpam-3889	174	12	,	,	PUNCT
ejpam-3889	174	13	so	so	ADV
ejpam-3889	174	14	s−1(m	s−1(m	PROPN
ejpam-3889	174	15	)	)	PUNCT
ejpam-3889	174	16	is	be	AUX
ejpam-3889	174	17	cohopfian	cohopfian	ADJ
ejpam-3889	174	18	.	.	PUNCT
ejpam-3889	175	1	theorem	theorem	NOUN
ejpam-3889	175	2	8	8	NUM
ejpam-3889	175	3	.	.	PUNCT
ejpam-3889	176	1	let	let	VERB
ejpam-3889	176	2	m	m	PRON
ejpam-3889	176	3	be	be	AUX
ejpam-3889	176	4	a	a	DET
ejpam-3889	176	5	quasi	quasi	NOUN
ejpam-3889	176	6	-	-	ADJ
ejpam-3889	176	7	projective	projective	ADJ
ejpam-3889	176	8	left	leave	VERB
ejpam-3889	176	9	a	a	DET
ejpam-3889	176	10	-	-	PUNCT
ejpam-3889	176	11	module	module	NOUN
ejpam-3889	176	12	,	,	PUNCT
ejpam-3889	176	13	n	n	CCONJ
ejpam-3889	176	14	an	an	DET
ejpam-3889	176	15	superfluous	superfluous	ADJ
ejpam-3889	176	16	and	and	CCONJ
ejpam-3889	176	17	completely	completely	ADV
ejpam-3889	176	18	invariant	invariant	ADJ
ejpam-3889	176	19	submodule	submodule	NOUN
ejpam-3889	176	20	of	of	ADP
ejpam-3889	176	21	m	m	PROPN
ejpam-3889	176	22	,	,	PUNCT
ejpam-3889	176	23	s	s	VERB
ejpam-3889	176	24	a	a	DET
ejpam-3889	176	25	saturated	saturate	VERB
ejpam-3889	176	26	multiplicative	multiplicative	ADJ
ejpam-3889	176	27	part	part	NOUN
ejpam-3889	176	28	of	of	ADP
ejpam-3889	176	29	a	a	DET
ejpam-3889	176	30	verifying	verifying	NOUN
ejpam-3889	176	31	the	the	DET
ejpam-3889	176	32	left	left	ADJ
ejpam-3889	176	33	ore	ore	NOUN
ejpam-3889	176	34	conditions	condition	NOUN
ejpam-3889	176	35	.	.	PUNCT
ejpam-3889	177	1	then	then	ADV
ejpam-3889	177	2	the	the	DET
ejpam-3889	177	3	left	left	ADJ
ejpam-3889	177	4	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	177	5	s−1(n	s−1(n	NOUN
ejpam-3889	177	6	)	)	PUNCT
ejpam-3889	177	7	is	be	AUX
ejpam-3889	177	8	hopfian	hopfian	ADJ
ejpam-3889	177	9	if	if	SCONJ
ejpam-3889	177	10	,	,	PUNCT
ejpam-3889	177	11	and	and	CCONJ
ejpam-3889	177	12	only	only	ADV
ejpam-3889	177	13	if	if	SCONJ
ejpam-3889	177	14	,	,	PUNCT
ejpam-3889	177	15	s−1(m	s−1(m	PROPN
ejpam-3889	177	16	/	/	SYM
ejpam-3889	177	17	n	n	CCONJ
ejpam-3889	177	18	)	)	PUNCT
ejpam-3889	177	19	is	be	AUX
ejpam-3889	177	20	hopfian	hopfian	ADJ
ejpam-3889	177	21	.	.	PUNCT
ejpam-3889	178	1	proof	proof	NOUN
ejpam-3889	178	2	.	.	PUNCT
ejpam-3889	179	1	suppose	suppose	VERB
ejpam-3889	179	2	that	that	SCONJ
ejpam-3889	179	3	s−1(m	s−1(m	PROPN
ejpam-3889	179	4	/	/	SYM
ejpam-3889	179	5	n	n	CCONJ
ejpam-3889	179	6	)	)	PUNCT
ejpam-3889	179	7	is	be	AUX
ejpam-3889	179	8	hopfian	hopfian	ADJ
ejpam-3889	179	9	and	and	CCONJ
ejpam-3889	179	10	let	let	VERB
ejpam-3889	179	11	s−1(f	s−1(f	PROPN
ejpam-3889	179	12	)	)	PUNCT
ejpam-3889	179	13	:	:	PUNCT
ejpam-3889	180	1	s−1(m	s−1(m	PROPN
ejpam-3889	180	2	)	)	PUNCT
ejpam-3889	180	3	−→	−→	NOUN
ejpam-3889	180	4	s−1(m	s−1(m	PROPN
ejpam-3889	180	5	)	)	PUNCT
ejpam-3889	180	6	an	an	DET
ejpam-3889	180	7	epimorphism	epimorphism	NOUN
ejpam-3889	180	8	.	.	PUNCT
ejpam-3889	181	1	as	as	SCONJ
ejpam-3889	181	2	s−1(n	s−1(n	PROPN
ejpam-3889	181	3	)	)	PUNCT
ejpam-3889	181	4	is	be	AUX
ejpam-3889	181	5	completely	completely	ADV
ejpam-3889	181	6	invariant	invariant	ADJ
ejpam-3889	181	7	,	,	PUNCT
ejpam-3889	181	8	then	then	ADV
ejpam-3889	181	9	[	[	X
ejpam-3889	181	10	s−1(f)](s−1(n	s−1(f)](s−1(n	NOUN
ejpam-3889	181	11	)	)	PUNCT
ejpam-3889	181	12	)	)	PUNCT
ejpam-3889	182	1	⊂	⊂	PROPN
ejpam-3889	182	2	s−1(n	s−1(n	PROPN
ejpam-3889	182	3	)	)	PUNCT
ejpam-3889	182	4	,	,	PUNCT
ejpam-3889	182	5	implies	imply	VERB
ejpam-3889	182	6	s−1(f	s−1(f	PROPN
ejpam-3889	182	7	)	)	PUNCT
ejpam-3889	182	8	induces	induce	VERB
ejpam-3889	182	9	an	an	DET
ejpam-3889	182	10	epimorphism	epimorphism	NOUN
ejpam-3889	182	11	s−1(f	s−1(f	PROPN
ejpam-3889	182	12	)	)	PUNCT
ejpam-3889	182	13	:	:	PUNCT
ejpam-3889	183	1	s−1(m	s−1(m	PROPN
ejpam-3889	183	2	/	/	SYM
ejpam-3889	183	3	n	n	CCONJ
ejpam-3889	183	4	)	)	PUNCT
ejpam-3889	183	5	−→	−→	NOUN
ejpam-3889	183	6	s−1(m	s−1(m	PROPN
ejpam-3889	183	7	/	/	SYM
ejpam-3889	183	8	n	n	CCONJ
ejpam-3889	183	9	)	)	PUNCT
ejpam-3889	183	10	,	,	PUNCT
ejpam-3889	183	11	since	since	SCONJ
ejpam-3889	183	12	s−1(m	s−1(m	PROPN
ejpam-3889	183	13	/	/	SYM
ejpam-3889	183	14	n	n	CCONJ
ejpam-3889	183	15	)	)	PUNCT
ejpam-3889	183	16	is	be	AUX
ejpam-3889	183	17	hopfian	hopfian	ADJ
ejpam-3889	183	18	,	,	PUNCT
ejpam-3889	183	19	then	then	ADV
ejpam-3889	183	20	s−1(f	s−1(f	PROPN
ejpam-3889	183	21	)	)	PUNCT
ejpam-3889	183	22	is	be	AUX
ejpam-3889	183	23	a	a	DET
ejpam-3889	183	24	graded	grade	VERB
ejpam-3889	183	25	automorphism	automorphism	NOUN
ejpam-3889	183	26	.	.	PUNCT
ejpam-3889	184	1	put	put	VERB
ejpam-3889	184	2	s−1(k	s−1(k	PROPN
ejpam-3889	184	3	)	)	PUNCT
ejpam-3889	184	4	=	=	SYM
ejpam-3889	184	5	ker(s−1(f	ker(s−1(f	PROPN
ejpam-3889	184	6	)	)	PUNCT
ejpam-3889	184	7	)	)	PUNCT
ejpam-3889	184	8	and	and	CCONJ
ejpam-3889	184	9	s−1(π	s−1(π	PROPN
ejpam-3889	184	10	)	)	PUNCT
ejpam-3889	184	11	:	:	PUNCT
ejpam-3889	185	1	s−1(m	s−1(m	PROPN
ejpam-3889	185	2	)	)	PUNCT
ejpam-3889	186	1	−→	−→	NOUN
ejpam-3889	186	2	s−1(m	s−1(m	PROPN
ejpam-3889	186	3	/	/	SYM
ejpam-3889	186	4	n	n	CCONJ
ejpam-3889	186	5	)	)	PUNCT
ejpam-3889	186	6	the	the	DET
ejpam-3889	186	7	canonical	canonical	ADJ
ejpam-3889	186	8	projection	projection	NOUN
ejpam-3889	186	9	,	,	PUNCT
ejpam-3889	186	10	we	we	PRON
ejpam-3889	186	11	have	have	VERB
ejpam-3889	186	12	:	:	PUNCT
ejpam-3889	186	13	s−1(f	s−1(f	PROPN
ejpam-3889	186	14	)	)	PUNCT
ejpam-3889	186	15	◦	◦	NOUN
ejpam-3889	187	1	[	[	X
ejpam-3889	187	2	s−1(π)](s−1(k	s−1(π)](s−1(k	NOUN
ejpam-3889	187	3	)	)	PUNCT
ejpam-3889	187	4	)	)	PUNCT
ejpam-3889	188	1	=	=	PUNCT
ejpam-3889	188	2	s−1(π	s−1(π	PROPN
ejpam-3889	188	3	◦	◦	NOUN
ejpam-3889	188	4	f(k	f(k	VERB
ejpam-3889	188	5	)	)	PUNCT
ejpam-3889	188	6	)	)	PUNCT
ejpam-3889	189	1	=	=	SYM
ejpam-3889	189	2	0	0	PUNCT
ejpam-3889	189	3	indeed	indeed	ADV
ejpam-3889	189	4	,	,	PUNCT
ejpam-3889	189	5	∀xs	∀xs	PROPN
ejpam-3889	189	6	∈	∈	PROPN
ejpam-3889	189	7	s	s	PART
ejpam-3889	189	8	−1(k	−1(k	NOUN
ejpam-3889	189	9	)	)	PUNCT
ejpam-3889	189	10	,	,	PUNCT
ejpam-3889	189	11	we	we	PRON
ejpam-3889	189	12	have	have	VERB
ejpam-3889	189	13	:	:	PUNCT
ejpam-3889	189	14	s−1[(f	s−1[(f	PART
ejpam-3889	189	15	◦	◦	VERB
ejpam-3889	189	16	π)](xs	π)](xs	X
ejpam-3889	189	17	)	)	PUNCT
ejpam-3889	189	18	=	=	PUNCT
ejpam-3889	190	1	[	[	X
ejpam-3889	190	2	s−1(π	s−1(π	NOUN
ejpam-3889	190	3	◦	◦	NOUN
ejpam-3889	190	4	f)](xs	f)](xs	ADP
ejpam-3889	190	5	)	)	PUNCT
ejpam-3889	190	6	,	,	PUNCT
ejpam-3889	190	7	so	so	CCONJ
ejpam-3889	190	8	[	[	X
ejpam-3889	190	9	s−1(π	s−1(π	NOUN
ejpam-3889	190	10	◦	◦	NOUN
ejpam-3889	190	11	f)](xs	f)](xs	ADP
ejpam-3889	190	12	)	)	PUNCT
ejpam-3889	190	13	=	=	PUNCT
ejpam-3889	191	1	[	[	X
ejpam-3889	191	2	s−1(π)](f(x)s	s−1(π)](f(x)s	NOUN
ejpam-3889	191	3	)	)	PUNCT
ejpam-3889	192	1	=	=	PUNCT
ejpam-3889	193	1	[	[	X
ejpam-3889	193	2	s−1(π)](0s	s−1(π)](0s	NOUN
ejpam-3889	193	3	)	)	PUNCT
ejpam-3889	193	4	=	=	SYM
ejpam-3889	193	5	0	0	NUM
ejpam-3889	193	6	,	,	PUNCT
ejpam-3889	193	7	thus	thus	ADV
ejpam-3889	193	8	[	[	X
ejpam-3889	193	9	s−1(f	s−1(f	PROPN
ejpam-3889	193	10	◦	◦	NOUN
ejpam-3889	193	11	π)](k	π)](k	PUNCT
ejpam-3889	193	12	)	)	PUNCT
ejpam-3889	193	13	=	=	SYM
ejpam-3889	194	1	0	0	X
ejpam-3889	194	2	.	.	PUNCT
ejpam-3889	195	1	we	we	PRON
ejpam-3889	195	2	have	have	VERB
ejpam-3889	195	3	:	:	PUNCT
ejpam-3889	196	1	[	[	X
ejpam-3889	196	2	s−1(f	s−1(f	PROPN
ejpam-3889	196	3	◦	◦	NOUN
ejpam-3889	196	4	π)](k	π)](k	PUNCT
ejpam-3889	196	5	)	)	PUNCT
ejpam-3889	196	6	=	=	PUNCT
ejpam-3889	197	1	[	[	X
ejpam-3889	197	2	s−1(π	s−1(π	NOUN
ejpam-3889	197	3	◦	◦	NOUN
ejpam-3889	197	4	f)](k	f)](k	NOUN
ejpam-3889	197	5	)	)	PUNCT
ejpam-3889	198	1	=	=	PUNCT
ejpam-3889	198	2	0	0	PUNCT
ejpam-3889	199	1	=	=	AUX
ejpam-3889	199	2	⇒	⇒	X
ejpam-3889	199	3	[	[	X
ejpam-3889	199	4	s−1(f)](π(k	s−1(f)](π(k	NUM
ejpam-3889	199	5	)	)	PUNCT
ejpam-3889	199	6	)	)	PUNCT
ejpam-3889	200	1	=	=	SYM
ejpam-3889	200	2	0	0	PUNCT
ejpam-3889	201	1	=	=	NOUN
ejpam-3889	201	2	⇒	⇒	NOUN
ejpam-3889	201	3	[	[	X
ejpam-3889	201	4	s−1(π)](k	s−1(π)](k	X
ejpam-3889	201	5	)	)	PUNCT
ejpam-3889	201	6	⊂	⊂	PROPN
ejpam-3889	201	7	s−1(n	s−1(n	PROPN
ejpam-3889	201	8	)	)	PUNCT
ejpam-3889	201	9	=	=	NOUN
ejpam-3889	201	10	⇒	⇒	NOUN
ejpam-3889	201	11	s−1(k	s−1(k	PROPN
ejpam-3889	201	12	)	)	PUNCT
ejpam-3889	201	13	⊂	⊂	PROPN
ejpam-3889	201	14	s−1(n	s−1(n	PROPN
ejpam-3889	201	15	)	)	PUNCT
ejpam-3889	201	16	since	since	SCONJ
ejpam-3889	201	17	m	m	PROPN
ejpam-3889	201	18	is	be	AUX
ejpam-3889	201	19	quasiprojective	quasiprojective	ADJ
ejpam-3889	201	20	=	=	NOUN
ejpam-3889	201	21	⇒	⇒	NOUN
ejpam-3889	201	22	s−1(m	s−1(m	PROPN
ejpam-3889	201	23	)	)	PUNCT
ejpam-3889	201	24	is	be	AUX
ejpam-3889	201	25	quasi	quasi	ADJ
ejpam-3889	201	26	-	-	ADJ
ejpam-3889	201	27	projective	projective	ADJ
ejpam-3889	201	28	,	,	PUNCT
ejpam-3889	201	29	there	there	PRON
ejpam-3889	201	30	exists	exist	VERB
ejpam-3889	201	31	an	an	DET
ejpam-3889	201	32	endomorphism	endomorphism	PROPN
ejpam-3889	201	33	s−1(s	s−1(s	PROPN
ejpam-3889	201	34	)	)	PUNCT
ejpam-3889	201	35	:	:	PUNCT
ejpam-3889	202	1	s−1(m	s−1(m	PROPN
ejpam-3889	202	2	)	)	PUNCT
ejpam-3889	202	3	−→	−→	NOUN
ejpam-3889	202	4	s−1(m	s−1(m	PROPN
ejpam-3889	202	5	)	)	PUNCT
ejpam-3889	202	6	such	such	ADJ
ejpam-3889	202	7	that	that	DET
ejpam-3889	202	8	s−1(f	s−1(f	PROPN
ejpam-3889	202	9	◦	◦	PROPN
ejpam-3889	202	10	s	s	PART
ejpam-3889	202	11	)	)	PUNCT
ejpam-3889	202	12	=	=	SYM
ejpam-3889	202	13	s−1(ids−1(m	s−1(ids−1(m	PROPN
ejpam-3889	202	14	)	)	PUNCT
ejpam-3889	202	15	)	)	PUNCT
ejpam-3889	202	16	,	,	PUNCT
ejpam-3889	202	17	this	this	PRON
ejpam-3889	202	18	implies	imply	VERB
ejpam-3889	202	19	s−1(m	s−1(m	PROPN
ejpam-3889	202	20	)	)	PUNCT
ejpam-3889	202	21	=	=	SYM
ejpam-3889	202	22	s−1(k	s−1(k	PROPN
ejpam-3889	202	23	⊕	⊕	PROPN
ejpam-3889	202	24	im(s	im(s	PROPN
ejpam-3889	202	25	)	)	PUNCT
ejpam-3889	202	26	)	)	PUNCT
ejpam-3889	202	27	,	,	PUNCT
ejpam-3889	202	28	or	or	CCONJ
ejpam-3889	202	29	k	k	X
ejpam-3889	202	30	=	=	PUNCT
ejpam-3889	202	31	n	n	PROPN
ejpam-3889	202	32	and	and	CCONJ
ejpam-3889	202	33	s−1(n	s−1(n	NOUN
ejpam-3889	202	34	)	)	PUNCT
ejpam-3889	202	35	is	be	AUX
ejpam-3889	202	36	superfluous	superfluous	ADJ
ejpam-3889	202	37	in	in	ADP
ejpam-3889	202	38	s−1(m	s−1(m	PROPN
ejpam-3889	202	39	)	)	PUNCT
ejpam-3889	202	40	,	,	PUNCT
ejpam-3889	202	41	then	then	ADV
ejpam-3889	202	42	s.	s.	PROPN
ejpam-3889	202	43	a.	a.	PROPN
ejpam-3889	202	44	balde	balde	PROPN
ejpam-3889	202	45	,	,	PUNCT
ejpam-3889	202	46	m.	m.	PROPN
ejpam-3889	202	47	b.	b.	PROPN
ejpam-3889	202	48	maaouia	maaouia	PROPN
ejpam-3889	202	49	,	,	PUNCT
ejpam-3889	202	50	a.	a.	NOUN
ejpam-3889	202	51	o.	o.	NOUN
ejpam-3889	202	52	chbih	chbih	PROPN
ejpam-3889	202	53	/	/	SYM
ejpam-3889	202	54	eur	eur	PROPN
ejpam-3889	202	55	.	.	PUNCT
ejpam-3889	203	1	j.	j.	PROPN
ejpam-3889	203	2	pure	pure	PROPN
ejpam-3889	203	3	appl	appl	PROPN
ejpam-3889	203	4	.	.	PROPN
ejpam-3889	203	5	math	math	PROPN
ejpam-3889	203	6	,	,	PUNCT
ejpam-3889	203	7	14	14	NUM
ejpam-3889	203	8	(	(	PUNCT
ejpam-3889	203	9	2	2	NUM
ejpam-3889	203	10	)	)	PUNCT
ejpam-3889	203	11	(	(	PUNCT
ejpam-3889	203	12	2021	2021	NUM
ejpam-3889	203	13	)	)	PUNCT
ejpam-3889	203	14	,	,	PUNCT
ejpam-3889	203	15	404	404	NUM
ejpam-3889	203	16	-	-	SYM
ejpam-3889	203	17	422	422	NUM
ejpam-3889	203	18	412	412	NUM
ejpam-3889	203	19	s−1(m	s−1(m	PROPN
ejpam-3889	203	20	)	)	PUNCT
ejpam-3889	203	21	=	=	SYM
ejpam-3889	203	22	s−1(im(s	s−1(im(s	NOUN
ejpam-3889	203	23	)	)	PUNCT
ejpam-3889	203	24	)	)	PUNCT
ejpam-3889	203	25	,	,	PUNCT
ejpam-3889	203	26	so	so	ADV
ejpam-3889	203	27	s−1(k	s−1(k	PROPN
ejpam-3889	203	28	)	)	PUNCT
ejpam-3889	204	1	=	=	SYM
ejpam-3889	204	2	ker[s−1(f	ker[s−1(f	PROPN
ejpam-3889	204	3	)	)	PUNCT
ejpam-3889	204	4	]	]	PUNCT
ejpam-3889	205	1	=	=	PUNCT
ejpam-3889	205	2	0	0	NUM
ejpam-3889	205	3	,	,	PUNCT
ejpam-3889	205	4	thus	thus	ADV
ejpam-3889	205	5	s−1(f	s−1(f	PROPN
ejpam-3889	205	6	)	)	PUNCT
ejpam-3889	205	7	is	be	AUX
ejpam-3889	205	8	a	a	DET
ejpam-3889	205	9	monomorphism	monomorphism	NOUN
ejpam-3889	205	10	,	,	PUNCT
ejpam-3889	205	11	finally	finally	ADV
ejpam-3889	205	12	,	,	PUNCT
ejpam-3889	205	13	s−1(f	s−1(f	PROPN
ejpam-3889	205	14	)	)	PUNCT
ejpam-3889	205	15	is	be	AUX
ejpam-3889	205	16	a	a	DET
ejpam-3889	205	17	monomorphism	monomorphism	NOUN
ejpam-3889	205	18	,	,	PUNCT
ejpam-3889	205	19	so	so	ADV
ejpam-3889	205	20	s−1(m	s−1(m	PROPN
ejpam-3889	205	21	)	)	PUNCT
ejpam-3889	205	22	is	be	AUX
ejpam-3889	205	23	a	a	DET
ejpam-3889	205	24	hopfian	hopfian	NOUN
ejpam-3889	205	25	left	leave	VERB
ejpam-3889	205	26	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	205	27	.	.	PUNCT
ejpam-3889	206	1	reciprocally	reciprocally	PROPN
ejpam-3889	206	2	,	,	PUNCT
ejpam-3889	206	3	let	let	VERB
ejpam-3889	206	4	s−1(m	s−1(m	PROPN
ejpam-3889	206	5	)	)	PUNCT
ejpam-3889	206	6	be	be	AUX
ejpam-3889	206	7	hopfian	hopfian	ADJ
ejpam-3889	206	8	,	,	PUNCT
ejpam-3889	206	9	show	show	VERB
ejpam-3889	206	10	that	that	SCONJ
ejpam-3889	206	11	s−1(m	s−1(m	PROPN
ejpam-3889	206	12	/	/	SYM
ejpam-3889	206	13	n	n	CCONJ
ejpam-3889	206	14	)	)	PUNCT
ejpam-3889	206	15	is	be	AUX
ejpam-3889	206	16	hopfian	hopfian	ADJ
ejpam-3889	206	17	.	.	PUNCT
ejpam-3889	207	1	let	let	VERB
ejpam-3889	207	2	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	207	3	)	)	PUNCT
ejpam-3889	207	4	:	:	PUNCT
ejpam-3889	208	1	s−1(m	s−1(m	PROPN
ejpam-3889	208	2	/	/	SYM
ejpam-3889	208	3	n	n	CCONJ
ejpam-3889	208	4	)	)	PUNCT
ejpam-3889	208	5	−→	−→	NOUN
ejpam-3889	208	6	s−1(m	s−1(m	PROPN
ejpam-3889	208	7	/	/	SYM
ejpam-3889	208	8	n	n	CCONJ
ejpam-3889	208	9	)	)	PUNCT
ejpam-3889	208	10	be	be	AUX
ejpam-3889	208	11	an	an	DET
ejpam-3889	208	12	epimorphism	epimorphism	NOUN
ejpam-3889	208	13	of	of	ADP
ejpam-3889	208	14	left	left	ADJ
ejpam-3889	208	15	s−1a	s−1a	NOUN
ejpam-3889	208	16	-	-	PUNCT
ejpam-3889	208	17	module	module	NOUN
ejpam-3889	208	18	,	,	PUNCT
ejpam-3889	208	19	as	as	SCONJ
ejpam-3889	208	20	m	m	PROPN
ejpam-3889	208	21	is	be	AUX
ejpam-3889	208	22	quasi	quasi	ADJ
ejpam-3889	208	23	-	-	ADJ
ejpam-3889	208	24	projective	projective	ADJ
ejpam-3889	208	25	,	,	PUNCT
ejpam-3889	208	26	then	then	ADV
ejpam-3889	208	27	s−1(m	s−1(m	PROPN
ejpam-3889	208	28	)	)	PUNCT
ejpam-3889	208	29	is	be	AUX
ejpam-3889	208	30	quasi	quasi	ADJ
ejpam-3889	208	31	-	-	ADJ
ejpam-3889	208	32	projective	projective	ADJ
ejpam-3889	208	33	.	.	PUNCT
ejpam-3889	209	1	consider	consider	VERB
ejpam-3889	209	2	s−1(π	s−1(π	PROPN
ejpam-3889	209	3	)	)	PUNCT
ejpam-3889	209	4	:	:	PUNCT
ejpam-3889	210	1	s−1(m	s−1(m	PROPN
ejpam-3889	210	2	)	)	PUNCT
ejpam-3889	210	3	−→	−→	NOUN
ejpam-3889	210	4	s−1(m	s−1(m	PROPN
ejpam-3889	210	5	/	/	SYM
ejpam-3889	210	6	n	n	CCONJ
ejpam-3889	210	7	)	)	PUNCT
ejpam-3889	210	8	,	,	PUNCT
ejpam-3889	210	9	then	then	ADV
ejpam-3889	210	10	there	there	PRON
ejpam-3889	210	11	exists	exist	VERB
ejpam-3889	210	12	s−1(f	s−1(f	PROPN
ejpam-3889	210	13	)	)	PUNCT
ejpam-3889	210	14	∈	∈	PROPN
ejpam-3889	210	15	end(s−1(m	end(s−1(m	PROPN
ejpam-3889	210	16	)	)	PUNCT
ejpam-3889	210	17	)	)	PUNCT
ejpam-3889	210	18	such	such	ADJ
ejpam-3889	210	19	that	that	DET
ejpam-3889	210	20	s−1(π	s−1(π	PROPN
ejpam-3889	210	21	◦	◦	NOUN
ejpam-3889	210	22	f	f	X
ejpam-3889	210	23	)	)	PUNCT
ejpam-3889	210	24	=	=	SYM
ejpam-3889	210	25	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	210	26	◦	◦	PROPN
ejpam-3889	210	27	π	π	PROPN
ejpam-3889	210	28	)	)	PUNCT
ejpam-3889	210	29	.	.	PUNCT
ejpam-3889	211	1	since	since	SCONJ
ejpam-3889	211	2	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	211	3	)	)	PUNCT
ejpam-3889	211	4	is	be	AUX
ejpam-3889	211	5	an	an	DET
ejpam-3889	211	6	epimorphism	epimorphism	NOUN
ejpam-3889	211	7	,	,	PUNCT
ejpam-3889	211	8	∀xs	∀xs	PROPN
ejpam-3889	211	9	∈	∈	NOUN
ejpam-3889	211	10	s	s	PART
ejpam-3889	211	11	−1(m	−1(m	NOUN
ejpam-3889	211	12	/	/	SYM
ejpam-3889	211	13	n),∃(yt	n),∃(yt	NOUN
ejpam-3889	211	14	)	)	PUNCT
ejpam-3889	211	15	∈	∈	PROPN
ejpam-3889	211	16	s	s	PART
ejpam-3889	211	17	−1(m	−1(m	NUM
ejpam-3889	211	18	/	/	SYM
ejpam-3889	211	19	n	n	CCONJ
ejpam-3889	211	20	)	)	PUNCT
ejpam-3889	211	21	such	such	ADJ
ejpam-3889	211	22	that	that	SCONJ
ejpam-3889	212	1	[	[	X
ejpam-3889	212	2	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	212	3	)	)	PUNCT
ejpam-3889	213	1	=	=	PUNCT
ejpam-3889	214	1	x	x	SYM
ejpam-3889	214	2	s	s	X
ejpam-3889	214	3	=	=	X
ejpam-3889	215	1	[	[	X
ejpam-3889	215	2	s−1(ϕ)](π(y)t	s−1(ϕ)](π(y)t	PROPN
ejpam-3889	215	3	)	)	PUNCT
ejpam-3889	216	1	[	[	X
ejpam-3889	216	2	s−1(π	s−1(π	NOUN
ejpam-3889	216	3	◦	◦	NOUN
ejpam-3889	216	4	f)](yt	f)](yt	PUNCT
ejpam-3889	216	5	)	)	PUNCT
ejpam-3889	217	1	=	=	PUNCT
ejpam-3889	218	1	[	[	X
ejpam-3889	218	2	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	218	3	◦	◦	NOUN
ejpam-3889	218	4	π)](yt	π)](yt	X
ejpam-3889	218	5	)	)	PUNCT
ejpam-3889	219	1	[	[	X
ejpam-3889	219	2	s−1(π)](f(x)s	s−1(π)](f(x)s	NOUN
ejpam-3889	219	3	)	)	PUNCT
ejpam-3889	220	1	=	=	PUNCT
ejpam-3889	221	1	[	[	X
ejpam-3889	221	2	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	221	3	)	)	PUNCT
ejpam-3889	222	1	=	=	VERB
ejpam-3889	222	2	⇒	⇒	X
ejpam-3889	222	3	[	[	X
ejpam-3889	222	4	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	222	5	)	)	PUNCT
ejpam-3889	222	6	=	=	PUNCT
ejpam-3889	223	1	[	[	X
ejpam-3889	223	2	s−1(f)](yt	s−1(f)](yt	X
ejpam-3889	223	3	)	)	PUNCT
ejpam-3889	223	4	=	=	PUNCT
ejpam-3889	224	1	x	x	SYM
ejpam-3889	224	2	s	s	NOUN
ejpam-3889	224	3	=	=	NOUN
ejpam-3889	224	4	⇒	⇒	NOUN
ejpam-3889	224	5	f(y	f(y	NOUN
ejpam-3889	224	6	)	)	PUNCT
ejpam-3889	224	7	t	t	NOUN
ejpam-3889	225	1	−	−	NOUN
ejpam-3889	225	2	x	x	SYM
ejpam-3889	225	3	s	s	X
ejpam-3889	225	4	=	=	SYM
ejpam-3889	225	5	0	0	PUNCT
ejpam-3889	225	6	=	=	NOUN
ejpam-3889	225	7	⇒	⇒	PROPN
ejpam-3889	225	8	f(y	f(y	NOUN
ejpam-3889	225	9	)	)	PUNCT
ejpam-3889	225	10	t	t	NOUN
ejpam-3889	226	1	−	−	NOUN
ejpam-3889	227	1	x	x	SYM
ejpam-3889	227	2	s	s	X
ejpam-3889	227	3	∈	∈	NOUN
ejpam-3889	227	4	s	s	PART
ejpam-3889	227	5	−1(n	−1(n	NOUN
ejpam-3889	227	6	)	)	PUNCT
ejpam-3889	227	7	,	,	PUNCT
ejpam-3889	227	8	then	then	ADV
ejpam-3889	227	9	s−1(m	s−1(m	PROPN
ejpam-3889	227	10	)	)	PUNCT
ejpam-3889	227	11	=	=	SYM
ejpam-3889	227	12	im(s−1(f	im(s−1(f	PROPN
ejpam-3889	227	13	)	)	PUNCT
ejpam-3889	227	14	)	)	PUNCT
ejpam-3889	228	1	+	+	CCONJ
ejpam-3889	228	2	s−1(n	s−1(n	NOUN
ejpam-3889	228	3	)	)	PUNCT
ejpam-3889	228	4	,	,	PUNCT
ejpam-3889	228	5	as	as	ADP
ejpam-3889	228	6	s−1(n	s−1(n	PROPN
ejpam-3889	228	7	)	)	PUNCT
ejpam-3889	228	8	is	be	AUX
ejpam-3889	228	9	superfluous	superfluous	ADJ
ejpam-3889	228	10	,	,	PUNCT
ejpam-3889	228	11	then	then	ADV
ejpam-3889	228	12	im(s−1(f	im(s−1(f	PROPN
ejpam-3889	228	13	)	)	PUNCT
ejpam-3889	228	14	)	)	PUNCT
ejpam-3889	229	1	=	=	SYM
ejpam-3889	229	2	s−1(m	s−1(m	PROPN
ejpam-3889	229	3	)	)	PUNCT
ejpam-3889	230	1	=	=	NOUN
ejpam-3889	230	2	⇒	⇒	NOUN
ejpam-3889	230	3	s−(f	s−(f	PROPN
ejpam-3889	230	4	)	)	PUNCT
ejpam-3889	230	5	is	be	AUX
ejpam-3889	230	6	an	an	DET
ejpam-3889	230	7	epimorphism	epimorphism	NOUN
ejpam-3889	230	8	.	.	PUNCT
ejpam-3889	231	1	so	so	ADV
ejpam-3889	231	2	,	,	PUNCT
ejpam-3889	231	3	s−1f	s−1f	NOUN
ejpam-3889	231	4	is	be	AUX
ejpam-3889	231	5	an	an	DET
ejpam-3889	231	6	automorphism	automorphism	NOUN
ejpam-3889	231	7	,	,	PUNCT
ejpam-3889	231	8	because	because	SCONJ
ejpam-3889	231	9	s−1(m	s−1(m	PROPN
ejpam-3889	231	10	)	)	PUNCT
ejpam-3889	231	11	is	be	AUX
ejpam-3889	231	12	hopfian	hopfian	ADJ
ejpam-3889	231	13	.	.	PUNCT
ejpam-3889	232	1	so	so	ADV
ejpam-3889	232	2	the	the	DET
ejpam-3889	232	3	restriction	restriction	NOUN
ejpam-3889	232	4	of	of	ADP
ejpam-3889	232	5	s−1(f	s−1(f	PROPN
ejpam-3889	232	6	)	)	PUNCT
ejpam-3889	232	7	over	over	ADP
ejpam-3889	232	8	s−1(n	s−1(n	PROPN
ejpam-3889	232	9	)	)	PUNCT
ejpam-3889	232	10	is	be	AUX
ejpam-3889	232	11	a	a	DET
ejpam-3889	232	12	automorphism	automorphism	NOUN
ejpam-3889	232	13	of	of	ADP
ejpam-3889	232	14	s−1(n	s−1(n	PROPN
ejpam-3889	232	15	)	)	PUNCT
ejpam-3889	232	16	.	.	PUNCT
ejpam-3889	233	1	if	if	SCONJ
ejpam-3889	233	2	[	[	X
ejpam-3889	233	3	s−1(ϕ)](xs	s−1(ϕ)](xs	X
ejpam-3889	233	4	)	)	PUNCT
ejpam-3889	233	5	=	=	PUNCT
ejpam-3889	234	1	[	[	X
ejpam-3889	234	2	s−1(f)](xs	s−1(f)](xs	PROPN
ejpam-3889	234	3	)	)	PUNCT
ejpam-3889	234	4	=	=	SYM
ejpam-3889	234	5	0	0	NUM
ejpam-3889	234	6	,	,	PUNCT
ejpam-3889	234	7	then	then	ADV
ejpam-3889	234	8	[	[	X
ejpam-3889	234	9	s−1(f)](xs	s−1(f)](xs	PROPN
ejpam-3889	234	10	)	)	PUNCT
ejpam-3889	234	11	∈	∈	PROPN
ejpam-3889	234	12	s−1(n	s−1(n	PROPN
ejpam-3889	234	13	)	)	PUNCT
ejpam-3889	234	14	,	,	PUNCT
ejpam-3889	234	15	or	or	CCONJ
ejpam-3889	234	16	s−1(n	s−1(n	NOUN
ejpam-3889	234	17	)	)	PUNCT
ejpam-3889	234	18	is	be	AUX
ejpam-3889	234	19	completely	completely	ADV
ejpam-3889	234	20	invariant	invariant	ADJ
ejpam-3889	234	21	,	,	PUNCT
ejpam-3889	234	22	then	then	ADV
ejpam-3889	234	23	x	x	X
ejpam-3889	234	24	s	s	PROPN
ejpam-3889	234	25	∈	∈	PROPN
ejpam-3889	234	26	s	s	PART
ejpam-3889	234	27	−1(n	−1(n	NOUN
ejpam-3889	234	28	)	)	PUNCT
ejpam-3889	234	29	,	,	PUNCT
ejpam-3889	234	30	so	so	ADV
ejpam-3889	234	31	x	x	X
ejpam-3889	235	1	s	s	AUX
ejpam-3889	235	2	=	=	SYM
ejpam-3889	235	3	0	0	PUNCT
ejpam-3889	235	4	=	=	NOUN
ejpam-3889	235	5	⇒	⇒	NOUN
ejpam-3889	235	6	ker(s−1(ϕ	ker(s−1(ϕ	PROPN
ejpam-3889	235	7	)	)	PUNCT
ejpam-3889	235	8	)	)	PUNCT
ejpam-3889	236	1	=	=	SYM
ejpam-3889	236	2	s−1(n	s−1(n	PROPN
ejpam-3889	236	3	)	)	PUNCT
ejpam-3889	236	4	=	=	SYM
ejpam-3889	236	5	0	0	PUNCT
ejpam-3889	237	1	=	=	NOUN
ejpam-3889	237	2	⇒	⇒	NOUN
ejpam-3889	237	3	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	237	4	)	)	PUNCT
ejpam-3889	237	5	is	be	AUX
ejpam-3889	237	6	a	a	DET
ejpam-3889	237	7	monomorphism	monomorphism	NOUN
ejpam-3889	237	8	=	=	PRON
ejpam-3889	237	9	⇒	⇒	X
ejpam-3889	237	10	ϕ	ϕ	NOUN
ejpam-3889	237	11	is	be	AUX
ejpam-3889	237	12	an	an	DET
ejpam-3889	237	13	automorphism	automorphism	NOUN
ejpam-3889	237	14	,	,	PUNCT
ejpam-3889	237	15	lastly	lastly	ADV
ejpam-3889	237	16	s−1(m	s−1(m	PROPN
ejpam-3889	237	17	/	/	SYM
ejpam-3889	237	18	n	n	CCONJ
ejpam-3889	237	19	)	)	PUNCT
ejpam-3889	237	20	is	be	AUX
ejpam-3889	237	21	hopfian	hopfian	ADJ
ejpam-3889	237	22	,	,	PUNCT
ejpam-3889	237	23	hence	hence	ADV
ejpam-3889	237	24	s−1(m	s−1(m	PROPN
ejpam-3889	237	25	/	/	SYM
ejpam-3889	237	26	n	n	CCONJ
ejpam-3889	237	27	)	)	PUNCT
ejpam-3889	237	28	is	be	AUX
ejpam-3889	237	29	a	a	DET
ejpam-3889	237	30	hopfian	hopfian	NOUN
ejpam-3889	237	31	left	leave	VERB
ejpam-3889	237	32	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	237	33	.	.	PUNCT
ejpam-3889	238	1	4	4	X
ejpam-3889	238	2	.	.	X
ejpam-3889	238	3	localization	localization	NOUN
ejpam-3889	238	4	of	of	ADP
ejpam-3889	238	5	hopfian	hopfian	ADJ
ejpam-3889	238	6	and	and	CCONJ
ejpam-3889	238	7	cohopfian	cohopfian	ADJ
ejpam-3889	238	8	objects	object	NOUN
ejpam-3889	238	9	in	in	ADP
ejpam-3889	238	10	the	the	DET
ejpam-3889	238	11	category	category	NOUN
ejpam-3889	238	12	of	of	ADP
ejpam-3889	238	13	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	238	14	)	)	PUNCT
ejpam-3889	238	15	definition	definition	NOUN
ejpam-3889	238	16	7	7	NUM
ejpam-3889	238	17	.	.	PUNCT
ejpam-3889	239	1	let	let	VERB
ejpam-3889	239	2	m	m	PRON
ejpam-3889	239	3	a	a	DET
ejpam-3889	239	4	graded	grade	VERB
ejpam-3889	239	5	left	leave	VERB
ejpam-3889	239	6	a	a	DET
ejpam-3889	239	7	-	-	PUNCT
ejpam-3889	239	8	module	module	NOUN
ejpam-3889	239	9	.	.	PUNCT
ejpam-3889	240	1	then	then	ADV
ejpam-3889	240	2	m	m	NOUN
ejpam-3889	240	3	is	be	AUX
ejpam-3889	240	4	said	say	VERB
ejpam-3889	240	5	hopfian	hopfian	ADJ
ejpam-3889	240	6	(	(	PUNCT
ejpam-3889	240	7	respectively	respectively	ADV
ejpam-3889	240	8	cohopfian	cohopfian	ADJ
ejpam-3889	240	9	)	)	PUNCT
ejpam-3889	240	10	,	,	PUNCT
ejpam-3889	240	11	if	if	SCONJ
ejpam-3889	240	12	any	any	DET
ejpam-3889	240	13	epimorphism	epimorphism	NOUN
ejpam-3889	240	14	(	(	PUNCT
ejpam-3889	240	15	respectively	respectively	ADV
ejpam-3889	240	16	monomorphism	monomorphism	NOUN
ejpam-3889	240	17	)	)	PUNCT
ejpam-3889	240	18	of	of	ADP
ejpam-3889	240	19	m	m	PROPN
ejpam-3889	240	20	is	be	AUX
ejpam-3889	240	21	an	an	DET
ejpam-3889	240	22	automorphism	automorphism	NOUN
ejpam-3889	240	23	of	of	ADP
ejpam-3889	240	24	m	m	PROPN
ejpam-3889	240	25	.	.	PUNCT
ejpam-3889	241	1	lemma	lemma	PROPN
ejpam-3889	241	2	1	1	X
ejpam-3889	241	3	.	.	PUNCT
ejpam-3889	241	4	let	let	VERB
ejpam-3889	241	5	a	a	DET
ejpam-3889	241	6	=	=	SYM
ejpam-3889	241	7	⊕	⊕	PROPN
ejpam-3889	241	8	n∈z	n∈z	VERB
ejpam-3889	241	9	an	an	DET
ejpam-3889	241	10	be	be	AUX
ejpam-3889	241	11	a	a	DET
ejpam-3889	241	12	graded	grade	VERB
ejpam-3889	241	13	ring	ring	NOUN
ejpam-3889	241	14	,	,	PUNCT
ejpam-3889	241	15	m	m	VERB
ejpam-3889	241	16	=	=	ADJ
ejpam-3889	241	17	⊕	⊕	PROPN
ejpam-3889	241	18	n∈z	n∈z	VERB
ejpam-3889	241	19	mn	mn	PROPN
ejpam-3889	241	20	a	a	DET
ejpam-3889	241	21	graded	grade	VERB
ejpam-3889	241	22	left	leave	VERB
ejpam-3889	241	23	a	a	DET
ejpam-3889	241	24	-	-	PUNCT
ejpam-3889	241	25	module	module	NOUN
ejpam-3889	241	26	,	,	PUNCT
ejpam-3889	241	27	then	then	ADV
ejpam-3889	241	28	m	m	NOUN
ejpam-3889	241	29	is	be	AUX
ejpam-3889	241	30	hopfian(respectively	hopfian(respectively	ADV
ejpam-3889	241	31	cohopfian	cohopfian	ADJ
ejpam-3889	241	32	)	)	PUNCT
ejpam-3889	241	33	if	if	SCONJ
ejpam-3889	241	34	,	,	PUNCT
ejpam-3889	241	35	and	and	CCONJ
ejpam-3889	241	36	only	only	ADV
ejpam-3889	241	37	if	if	SCONJ
ejpam-3889	241	38	,	,	PUNCT
ejpam-3889	241	39	mn	mn	PROPN
ejpam-3889	241	40	is	be	AUX
ejpam-3889	241	41	a	a	DET
ejpam-3889	241	42	hopfian(respectiveley	hopfian(respectiveley	PROPN
ejpam-3889	241	43	cohopfien	cohopfien	NOUN
ejpam-3889	241	44	)	)	PUNCT
ejpam-3889	241	45	groupe	groupe	NOUN
ejpam-3889	241	46	.	.	PUNCT
ejpam-3889	242	1	proof	proof	NOUN
ejpam-3889	242	2	.	.	PUNCT
ejpam-3889	243	1	suppose	suppose	VERB
ejpam-3889	243	2	that	that	SCONJ
ejpam-3889	243	3	m	m	PROPN
ejpam-3889	243	4	=	=	SYM
ejpam-3889	243	5	⊕	⊕	PROPN
ejpam-3889	243	6	n∈z	n∈z	ADJ
ejpam-3889	243	7	mn	mn	PROPN
ejpam-3889	243	8	is	be	AUX
ejpam-3889	243	9	hopfian	hopfian	ADJ
ejpam-3889	243	10	,	,	PUNCT
ejpam-3889	243	11	show	show	VERB
ejpam-3889	243	12	that	that	SCONJ
ejpam-3889	243	13	mn	mn	PROPN
ejpam-3889	243	14	is	be	AUX
ejpam-3889	243	15	a	a	DET
ejpam-3889	243	16	hopfian	hopfian	ADJ
ejpam-3889	243	17	group	group	NOUN
ejpam-3889	243	18	.	.	PUNCT
ejpam-3889	244	1	let	let	VERB
ejpam-3889	244	2	f	f	NOUN
ejpam-3889	244	3	:	:	PUNCT
ejpam-3889	244	4	m	m	VERB
ejpam-3889	244	5	=	=	SYM
ejpam-3889	244	6	⊕	⊕	PROPN
ejpam-3889	244	7	n∈z	n∈z	VERB
ejpam-3889	245	1	mn	mn	PROPN
ejpam-3889	245	2	−→	−→	NOUN
ejpam-3889	246	1	m	m	PROPN
ejpam-3889	246	2	=	=	PROPN
ejpam-3889	246	3	⊕	⊕	PROPN
ejpam-3889	246	4	n∈z	n∈z	ADJ
ejpam-3889	247	1	mn	mn	PROPN
ejpam-3889	247	2	be	be	AUX
ejpam-3889	247	3	a	a	DET
ejpam-3889	247	4	graded	grade	VERB
ejpam-3889	247	5	morphism	morphism	NOUN
ejpam-3889	247	6	,	,	PUNCT
ejpam-3889	247	7	so	so	SCONJ
ejpam-3889	247	8	we	we	PRON
ejpam-3889	247	9	have	have	VERB
ejpam-3889	247	10	the	the	DET
ejpam-3889	247	11	induce	induce	ADJ
ejpam-3889	247	12	morphism	morphism	NOUN
ejpam-3889	247	13	of	of	ADP
ejpam-3889	247	14	groups	group	NOUN
ejpam-3889	247	15	:	:	PUNCT
ejpam-3889	247	16	fn	fn	X
ejpam-3889	247	17	:	:	PUNCT
ejpam-3889	247	18	mn	mn	PROPN
ejpam-3889	247	19	−→	−→	PROPN
ejpam-3889	247	20	mn	mn	PROPN
ejpam-3889	247	21	,	,	PUNCT
ejpam-3889	247	22	for	for	ADP
ejpam-3889	247	23	all	all	DET
ejpam-3889	247	24	x	x	PROPN
ejpam-3889	247	25	∈	∈	PROPN
ejpam-3889	247	26	mn	mn	PROPN
ejpam-3889	247	27	,	,	PUNCT
ejpam-3889	247	28	we	we	PRON
ejpam-3889	247	29	have	have	VERB
ejpam-3889	247	30	fn(x	fn(x	NOUN
ejpam-3889	247	31	)	)	PUNCT
ejpam-3889	247	32	=	=	SYM
ejpam-3889	247	33	f(xn	f(xn	X
ejpam-3889	247	34	)	)	PUNCT
ejpam-3889	247	35	.	.	PUNCT
ejpam-3889	248	1	we	we	PRON
ejpam-3889	248	2	see	see	VERB
ejpam-3889	248	3	that	that	DET
ejpam-3889	248	4	fn	fn	NOUN
ejpam-3889	248	5	is	be	AUX
ejpam-3889	248	6	well	well	ADV
ejpam-3889	248	7	defined	define	VERB
ejpam-3889	248	8	because	because	SCONJ
ejpam-3889	248	9	f	f	PROPN
ejpam-3889	248	10	is	be	AUX
ejpam-3889	248	11	graded	grade	VERB
ejpam-3889	248	12	,	,	PUNCT
ejpam-3889	248	13	moreover	moreover	ADV
ejpam-3889	248	14	,	,	PUNCT
ejpam-3889	248	15	for	for	ADP
ejpam-3889	248	16	all	all	DET
ejpam-3889	248	17	xn	xn	PROPN
ejpam-3889	248	18	and	and	CCONJ
ejpam-3889	248	19	yn	yn	PROPN
ejpam-3889	248	20	∈	∈	PROPN
ejpam-3889	248	21	mn	mn	PROPN
ejpam-3889	248	22	,	,	PUNCT
ejpam-3889	248	23	we	we	PRON
ejpam-3889	248	24	have	have	VERB
ejpam-3889	248	25	fn(xn	fn(xn	NOUN
ejpam-3889	248	26	+	+	CCONJ
ejpam-3889	248	27	yn	yn	NOUN
ejpam-3889	248	28	)	)	PUNCT
ejpam-3889	248	29	=	=	SYM
ejpam-3889	248	30	f(xn	f(xn	PROPN
ejpam-3889	248	31	+	+	CCONJ
ejpam-3889	248	32	yn	yn	NOUN
ejpam-3889	248	33	)	)	PUNCT
ejpam-3889	248	34	=	=	PUNCT
ejpam-3889	249	1	f(xn	f(xn	X
ejpam-3889	249	2	)	)	PUNCT
ejpam-3889	249	3	+	+	PUNCT
ejpam-3889	250	1	f(yn	f(yn	X
ejpam-3889	250	2	)	)	PUNCT
ejpam-3889	250	3	=	=	SYM
ejpam-3889	250	4	fn(xn	fn(xn	PROPN
ejpam-3889	250	5	)	)	PUNCT
ejpam-3889	251	1	+	+	SYM
ejpam-3889	251	2	fn(yn	fn(yn	ADJ
ejpam-3889	251	3	)	)	PUNCT
ejpam-3889	252	1	=	=	VERB
ejpam-3889	252	2	⇒	⇒	NOUN
ejpam-3889	252	3	fn	fn	NOUN
ejpam-3889	252	4	is	be	AUX
ejpam-3889	252	5	a	a	DET
ejpam-3889	252	6	morphism	morphism	NOUN
ejpam-3889	252	7	of	of	ADP
ejpam-3889	252	8	s.	s.	PROPN
ejpam-3889	252	9	a.	a.	PROPN
ejpam-3889	252	10	balde	balde	PROPN
ejpam-3889	252	11	,	,	PUNCT
ejpam-3889	252	12	m.	m.	PROPN
ejpam-3889	252	13	b.	b.	PROPN
ejpam-3889	252	14	maaouia	maaouia	PROPN
ejpam-3889	252	15	,	,	PUNCT
ejpam-3889	252	16	a.	a.	NOUN
ejpam-3889	252	17	o.	o.	NOUN
ejpam-3889	252	18	chbih	chbih	PROPN
ejpam-3889	252	19	/	/	SYM
ejpam-3889	252	20	eur	eur	PROPN
ejpam-3889	252	21	.	.	PUNCT
ejpam-3889	253	1	j.	j.	PROPN
ejpam-3889	253	2	pure	pure	PROPN
ejpam-3889	253	3	appl	appl	PROPN
ejpam-3889	253	4	.	.	PROPN
ejpam-3889	253	5	math	math	PROPN
ejpam-3889	253	6	,	,	PUNCT
ejpam-3889	253	7	14	14	NUM
ejpam-3889	253	8	(	(	PUNCT
ejpam-3889	253	9	2	2	NUM
ejpam-3889	253	10	)	)	PUNCT
ejpam-3889	253	11	(	(	PUNCT
ejpam-3889	253	12	2021	2021	NUM
ejpam-3889	253	13	)	)	PUNCT
ejpam-3889	253	14	,	,	PUNCT
ejpam-3889	253	15	404	404	NUM
ejpam-3889	253	16	-	-	SYM
ejpam-3889	253	17	422	422	NUM
ejpam-3889	253	18	413	413	NUM
ejpam-3889	253	19	groups	group	NOUN
ejpam-3889	253	20	.	.	PUNCT
ejpam-3889	254	1	let	let	VERB
ejpam-3889	254	2	h	h	NOUN
ejpam-3889	254	3	:	:	PUNCT
ejpam-3889	254	4	mn	mn	PROPN
ejpam-3889	254	5	−→mn	−→mn	ADV
ejpam-3889	254	6	be	be	AUX
ejpam-3889	254	7	an	an	DET
ejpam-3889	254	8	epimorphism	epimorphism	NOUN
ejpam-3889	254	9	of	of	ADP
ejpam-3889	254	10	groups	group	NOUN
ejpam-3889	254	11	.	.	PUNCT
ejpam-3889	255	1	put	put	VERB
ejpam-3889	255	2	for	for	ADP
ejpam-3889	255	3	all	all	DET
ejpam-3889	255	4	xi	xi	PROPN
ejpam-3889	255	5	,	,	PUNCT
ejpam-3889	255	6	i	i	PROPN
ejpam-3889	255	7	6=	6=	PROPN
ejpam-3889	255	8	n	n	CCONJ
ejpam-3889	255	9	,	,	PUNCT
ejpam-3889	255	10	f(xi	f(xi	PROPN
ejpam-3889	255	11	)	)	PUNCT
ejpam-3889	255	12	=	=	SYM
ejpam-3889	255	13	xi	xi	PROPN
ejpam-3889	255	14	and	and	CCONJ
ejpam-3889	255	15	for	for	ADP
ejpam-3889	255	16	all	all	DET
ejpam-3889	255	17	xn	xn	PROPN
ejpam-3889	255	18	∈	∈	PROPN
ejpam-3889	255	19	mn	mn	PROPN
ejpam-3889	255	20	,	,	PUNCT
ejpam-3889	255	21	f(xn	f(xn	PROPN
ejpam-3889	255	22	)	)	PUNCT
ejpam-3889	255	23	=	=	SYM
ejpam-3889	255	24	h(xn	h(xn	NOUN
ejpam-3889	255	25	)	)	PUNCT
ejpam-3889	255	26	,	,	PUNCT
ejpam-3889	255	27	it	it	PRON
ejpam-3889	255	28	is	be	AUX
ejpam-3889	255	29	easy	easy	ADJ
ejpam-3889	255	30	to	to	PART
ejpam-3889	255	31	prove	prove	VERB
ejpam-3889	255	32	that	that	SCONJ
ejpam-3889	255	33	f	f	PROPN
ejpam-3889	255	34	is	be	AUX
ejpam-3889	255	35	a	a	DET
ejpam-3889	255	36	graded	grade	VERB
ejpam-3889	255	37	epimorphism	epimorphism	NOUN
ejpam-3889	255	38	of	of	ADP
ejpam-3889	255	39	graded	grade	VERB
ejpam-3889	255	40	left	leave	VERB
ejpam-3889	255	41	a	a	DET
ejpam-3889	255	42	-	-	PUNCT
ejpam-3889	255	43	modules	module	NOUN
ejpam-3889	255	44	.	.	PUNCT
ejpam-3889	256	1	since	since	SCONJ
ejpam-3889	256	2	f	f	PROPN
ejpam-3889	256	3	is	be	AUX
ejpam-3889	256	4	an	an	DET
ejpam-3889	256	5	automorphism	automorphism	NOUN
ejpam-3889	256	6	because	because	SCONJ
ejpam-3889	256	7	m	m	PROPN
ejpam-3889	256	8	hopfian	hopfian	NOUN
ejpam-3889	256	9	by	by	ADP
ejpam-3889	256	10	hypothesis	hypothesis	NOUN
ejpam-3889	256	11	,	,	PUNCT
ejpam-3889	256	12	so	so	ADV
ejpam-3889	256	13	h	h	NOUN
ejpam-3889	256	14	is	be	AUX
ejpam-3889	256	15	an	an	DET
ejpam-3889	256	16	automorphism	automorphism	NOUN
ejpam-3889	256	17	,	,	PUNCT
ejpam-3889	256	18	thus	thus	ADV
ejpam-3889	256	19	mn	mn	PROPN
ejpam-3889	256	20	is	be	AUX
ejpam-3889	256	21	hopfien	hopfien	NOUN
ejpam-3889	256	22	.	.	PUNCT
ejpam-3889	257	1	suppose	suppose	VERB
ejpam-3889	257	2	that	that	SCONJ
ejpam-3889	257	3	mn	mn	PROPN
ejpam-3889	257	4	is	be	AUX
ejpam-3889	257	5	hopfian	hopfian	ADJ
ejpam-3889	257	6	,	,	PUNCT
ejpam-3889	257	7	show	show	VERB
ejpam-3889	257	8	that	that	SCONJ
ejpam-3889	257	9	m	m	VERB
ejpam-3889	257	10	=	=	SYM
ejpam-3889	257	11	⊕	⊕	PROPN
ejpam-3889	257	12	n∈z	n∈z	ADJ
ejpam-3889	257	13	mn	mn	PROPN
ejpam-3889	257	14	is	be	AUX
ejpam-3889	257	15	hopfian	hopfian	ADJ
ejpam-3889	257	16	.	.	PUNCT
ejpam-3889	258	1	let	let	VERB
ejpam-3889	258	2	f	f	NOUN
ejpam-3889	258	3	:	:	PUNCT
ejpam-3889	258	4	m	m	VERB
ejpam-3889	258	5	=	=	SYM
ejpam-3889	258	6	⊕	⊕	PROPN
ejpam-3889	258	7	n∈z	n∈z	VERB
ejpam-3889	259	1	mn	mn	PROPN
ejpam-3889	260	1	−→m	−→m	PROPN
ejpam-3889	261	1	=	=	PUNCT
ejpam-3889	261	2	⊕	⊕	PROPN
ejpam-3889	261	3	n∈z	n∈z	NOUN
ejpam-3889	261	4	mn	mn	PROPN
ejpam-3889	261	5	be	be	AUX
ejpam-3889	261	6	an	an	DET
ejpam-3889	261	7	epimorphism	epimorphism	NOUN
ejpam-3889	261	8	of	of	ADP
ejpam-3889	261	9	left	leave	VERB
ejpam-3889	261	10	a	a	DET
ejpam-3889	261	11	-	-	PUNCT
ejpam-3889	261	12	module	module	NOUN
ejpam-3889	261	13	.	.	PUNCT
ejpam-3889	262	1	show	show	VERB
ejpam-3889	262	2	that	that	SCONJ
ejpam-3889	262	3	f	f	PROPN
ejpam-3889	262	4	is	be	AUX
ejpam-3889	262	5	an	an	DET
ejpam-3889	262	6	automorphism	automorphism	NOUN
ejpam-3889	262	7	.	.	PUNCT
ejpam-3889	263	1	prove	prove	VERB
ejpam-3889	263	2	that	that	PRON
ejpam-3889	263	3	fn	fn	INTJ
ejpam-3889	263	4	:	:	PUNCT
ejpam-3889	263	5	mn	mn	PROPN
ejpam-3889	263	6	−→mn	−→mn	PROPN
ejpam-3889	263	7	is	be	AUX
ejpam-3889	263	8	epimorphism	epimorphism	NOUN
ejpam-3889	263	9	of	of	ADP
ejpam-3889	263	10	groups	group	NOUN
ejpam-3889	263	11	for	for	ADP
ejpam-3889	263	12	all	all	DET
ejpam-3889	263	13	n	n	PRON
ejpam-3889	263	14	∈	∈	NOUN
ejpam-3889	263	15	z	z	AUX
ejpam-3889	263	16	let	let	VERB
ejpam-3889	263	17	yn	yn	PRON
ejpam-3889	263	18	∈mn	∈mn	PROPN
ejpam-3889	263	19	,	,	PUNCT
ejpam-3889	263	20	then	then	ADV
ejpam-3889	263	21	there	there	PRON
ejpam-3889	263	22	exists	exist	VERB
ejpam-3889	263	23	x	x	X
ejpam-3889	263	24	∈m	∈m	NOUN
ejpam-3889	263	25	=	=	SYM
ejpam-3889	263	26	⊕	⊕	PROPN
ejpam-3889	263	27	n∈z	n∈z	VERB
ejpam-3889	263	28	an	an	PROPN
ejpam-3889	263	29	.	.	PUNCT
ejpam-3889	263	30	suppose	suppose	VERB
ejpam-3889	263	31	that	that	SCONJ
ejpam-3889	263	32	x	x	NOUN
ejpam-3889	263	33	=	=	PUNCT
ejpam-3889	263	34	∑	∑	PUNCT
ejpam-3889	263	35	finie	finie	PROPN
ejpam-3889	263	36	xt	xt	PROPN
ejpam-3889	263	37	,	,	PUNCT
ejpam-3889	263	38	or	or	CCONJ
ejpam-3889	263	39	for	for	ADP
ejpam-3889	263	40	all	all	DET
ejpam-3889	263	41	t	t	PROPN
ejpam-3889	263	42	,	,	PUNCT
ejpam-3889	263	43	f(xt	f(xt	NOUN
ejpam-3889	263	44	)	)	PUNCT
ejpam-3889	263	45	∈mt	∈mt	VERB
ejpam-3889	263	46	if	if	SCONJ
ejpam-3889	263	47	t	t	PROPN
ejpam-3889	263	48	6=	6=	NUM
ejpam-3889	263	49	n	n	CCONJ
ejpam-3889	263	50	,	,	PUNCT
ejpam-3889	263	51	f(xt	f(xt	NOUN
ejpam-3889	263	52	)	)	PUNCT
ejpam-3889	264	1	=	=	SYM
ejpam-3889	264	2	0	0	NUM
ejpam-3889	264	3	,	,	PUNCT
ejpam-3889	264	4	since	since	SCONJ
ejpam-3889	264	5	f(x	f(x	PROPN
ejpam-3889	264	6	)	)	PUNCT
ejpam-3889	265	1	=	=	PUNCT
ejpam-3889	265	2	yn	yn	PROPN
ejpam-3889	265	3	∈	∈	PROPN
ejpam-3889	265	4	mn	mn	PROPN
ejpam-3889	265	5	,	,	PUNCT
ejpam-3889	265	6	so	so	SCONJ
ejpam-3889	265	7	for	for	ADP
ejpam-3889	265	8	all	all	DET
ejpam-3889	265	9	yn	yn	PROPN
ejpam-3889	265	10	∈	∈	PROPN
ejpam-3889	265	11	mn	mn	PROPN
ejpam-3889	265	12	,	,	PUNCT
ejpam-3889	265	13	there	there	PRON
ejpam-3889	265	14	exists	exist	VERB
ejpam-3889	265	15	xn	xn	PROPN
ejpam-3889	265	16	∈	∈	PROPN
ejpam-3889	265	17	mn	mn	PROPN
ejpam-3889	265	18	such	such	ADJ
ejpam-3889	266	1	that	that	SCONJ
ejpam-3889	266	2	f(xn	f(xn	PROPN
ejpam-3889	266	3	)	)	PUNCT
ejpam-3889	266	4	=	=	SYM
ejpam-3889	266	5	yn	yn	PROPN
ejpam-3889	266	6	,	,	PUNCT
ejpam-3889	266	7	or	or	CCONJ
ejpam-3889	266	8	f(xn	f(xn	PROPN
ejpam-3889	266	9	)	)	PUNCT
ejpam-3889	266	10	=	=	SYM
ejpam-3889	266	11	fn(xn	fn(xn	NOUN
ejpam-3889	266	12	)	)	PUNCT
ejpam-3889	266	13	=	=	SYM
ejpam-3889	266	14	⇒	⇒	NOUN
ejpam-3889	266	15	fn(xn	fn(xn	NOUN
ejpam-3889	266	16	)	)	PUNCT
ejpam-3889	266	17	=	=	SYM
ejpam-3889	266	18	yn	yn	PROPN
ejpam-3889	266	19	,	,	PUNCT
ejpam-3889	266	20	so	so	ADV
ejpam-3889	266	21	fn	fn	NOUN
ejpam-3889	266	22	is	be	AUX
ejpam-3889	266	23	an	an	DET
ejpam-3889	266	24	epimorphism	epimorphism	NOUN
ejpam-3889	266	25	,	,	PUNCT
ejpam-3889	266	26	as	as	SCONJ
ejpam-3889	266	27	mn	mn	PROPN
ejpam-3889	266	28	is	be	AUX
ejpam-3889	266	29	hopfian	hopfian	ADJ
ejpam-3889	266	30	,	,	PUNCT
ejpam-3889	266	31	so	so	ADV
ejpam-3889	266	32	fn	fn	NOUN
ejpam-3889	266	33	is	be	AUX
ejpam-3889	266	34	an	an	DET
ejpam-3889	266	35	automorphism	automorphism	NOUN
ejpam-3889	266	36	=	=	PRON
ejpam-3889	266	37	⇒	⇒	X
ejpam-3889	266	38	f	f	X
ejpam-3889	266	39	is	be	AUX
ejpam-3889	266	40	also	also	ADV
ejpam-3889	266	41	an	an	DET
ejpam-3889	266	42	automorphism	automorphism	NOUN
ejpam-3889	266	43	,	,	PUNCT
ejpam-3889	266	44	consequentely	consequentely	ADV
ejpam-3889	266	45	,	,	PUNCT
ejpam-3889	266	46	m	m	VERB
ejpam-3889	266	47	is	be	AUX
ejpam-3889	266	48	hopfian	hopfian	ADJ
ejpam-3889	266	49	.	.	PUNCT
ejpam-3889	267	1	for	for	ADP
ejpam-3889	267	2	the	the	DET
ejpam-3889	267	3	hopfian	hopfian	ADJ
ejpam-3889	267	4	case	case	NOUN
ejpam-3889	267	5	,	,	PUNCT
ejpam-3889	267	6	the	the	DET
ejpam-3889	267	7	proof	proof	NOUN
ejpam-3889	267	8	is	be	AUX
ejpam-3889	267	9	similary	similary	ADJ
ejpam-3889	267	10	.	.	PUNCT
ejpam-3889	268	1	theorem	theorem	NOUN
ejpam-3889	268	2	9	9	NUM
ejpam-3889	268	3	.	.	PUNCT
ejpam-3889	269	1	let	let	VERB
ejpam-3889	269	2	a	a	DET
ejpam-3889	269	3	=	=	SYM
ejpam-3889	269	4	⊕	⊕	PROPN
ejpam-3889	269	5	n∈z	n∈z	VERB
ejpam-3889	269	6	an	an	DET
ejpam-3889	269	7	be	be	AUX
ejpam-3889	269	8	a	a	DET
ejpam-3889	269	9	graded	grade	VERB
ejpam-3889	269	10	ring	ring	NOUN
ejpam-3889	269	11	,	,	PUNCT
ejpam-3889	269	12	s	s	VERB
ejpam-3889	269	13	a	a	DET
ejpam-3889	269	14	saturated	saturate	VERB
ejpam-3889	269	15	multiplicative	multiplicative	ADJ
ejpam-3889	269	16	part	part	NOUN
ejpam-3889	269	17	formed	form	VERB
ejpam-3889	269	18	by	by	ADP
ejpam-3889	269	19	the	the	DET
ejpam-3889	269	20	non	non	ADJ
ejpam-3889	269	21	-	-	ADJ
ejpam-3889	269	22	zero	zero	ADJ
ejpam-3889	269	23	homogeneous	homogeneous	ADJ
ejpam-3889	269	24	elements	element	NOUN
ejpam-3889	269	25	of	of	ADP
ejpam-3889	269	26	a	a	DET
ejpam-3889	269	27	verifying	verifying	NOUN
ejpam-3889	269	28	the	the	DET
ejpam-3889	269	29	left	left	ADJ
ejpam-3889	269	30	ore	ore	NOUN
ejpam-3889	269	31	conditions	condition	NOUN
ejpam-3889	269	32	and	and	CCONJ
ejpam-3889	269	33	m	m	NOUN
ejpam-3889	269	34	=	=	PROPN
ejpam-3889	269	35	⊕	⊕	PROPN
ejpam-3889	269	36	n∈z	n∈z	VERB
ejpam-3889	269	37	mn	mn	PROPN
ejpam-3889	269	38	a	a	DET
ejpam-3889	269	39	graded	grade	VERB
ejpam-3889	269	40	left	leave	VERB
ejpam-3889	269	41	a	a	DET
ejpam-3889	269	42	-	-	PUNCT
ejpam-3889	269	43	module	module	NOUN
ejpam-3889	269	44	,	,	PUNCT
ejpam-3889	269	45	then	then	ADV
ejpam-3889	269	46	s−1(m	s−1(m	PROPN
ejpam-3889	269	47	)	)	PUNCT
ejpam-3889	270	1	=	=	PROPN
ejpam-3889	270	2	⊕	⊕	PROPN
ejpam-3889	270	3	i∈z	i∈z	PROPN
ejpam-3889	270	4	(	(	PUNCT
ejpam-3889	270	5	s−1m)i	s−1m)i	NOUN
ejpam-3889	270	6	is	be	AUX
ejpam-3889	270	7	hopfian(respectively	hopfian(respectively	ADV
ejpam-3889	270	8	cohopfian	cohopfian	ADJ
ejpam-3889	270	9	)	)	PUNCT
ejpam-3889	270	10	if	if	SCONJ
ejpam-3889	270	11	,	,	PUNCT
ejpam-3889	270	12	and	and	CCONJ
ejpam-3889	270	13	only	only	ADV
ejpam-3889	270	14	,	,	PUNCT
ejpam-3889	270	15	if	if	SCONJ
ejpam-3889	270	16	(	(	PUNCT
ejpam-3889	270	17	s−1m)i	s−1m)i	NOUN
ejpam-3889	270	18	is	be	AUX
ejpam-3889	270	19	a	a	DET
ejpam-3889	270	20	hopfian(respectiveley	hopfian(respectiveley	PROPN
ejpam-3889	270	21	cohopfien	cohopfien	NOUN
ejpam-3889	270	22	)	)	PUNCT
ejpam-3889	270	23	group	group	NOUN
ejpam-3889	270	24	.	.	PUNCT
ejpam-3889	271	1	proof	proof	NOUN
ejpam-3889	271	2	.	.	PUNCT
ejpam-3889	272	1	it	it	PRON
ejpam-3889	272	2	suffices	suffice	VERB
ejpam-3889	272	3	to	to	PART
ejpam-3889	272	4	prove	prove	VERB
ejpam-3889	272	5	that	that	SCONJ
ejpam-3889	272	6	(	(	PUNCT
ejpam-3889	272	7	s−1f	s−1f	NOUN
ejpam-3889	272	8	)	)	PUNCT
ejpam-3889	272	9	:	:	PUNCT
ejpam-3889	272	10	s−1(m	s−1(m	PROPN
ejpam-3889	272	11	)	)	PUNCT
ejpam-3889	273	1	=	=	PROPN
ejpam-3889	273	2	⊕	⊕	PROPN
ejpam-3889	273	3	i∈z	i∈z	PROPN
ejpam-3889	273	4	(	(	PUNCT
ejpam-3889	273	5	s−1m)i	s−1m)i	VERB
ejpam-3889	273	6	−→	−→	NOUN
ejpam-3889	273	7	s−1(m	s−1(m	PROPN
ejpam-3889	273	8	)	)	PUNCT
ejpam-3889	273	9	=	=	PROPN
ejpam-3889	273	10	⊕	⊕	PROPN
ejpam-3889	273	11	i∈z	i∈z	PROPN
ejpam-3889	273	12	(	(	PUNCT
ejpam-3889	273	13	s−1m)i	s−1m)i	PROPN
ejpam-3889	273	14	is	be	AUX
ejpam-3889	273	15	graded	grade	VERB
ejpam-3889	273	16	.	.	PUNCT
ejpam-3889	274	1	since	since	SCONJ
ejpam-3889	274	2	(	(	PUNCT
ejpam-3889	274	3	s−1m)i	s−1m)i	NOUN
ejpam-3889	274	4	=	=	X
ejpam-3889	274	5	{	{	PUNCT
ejpam-3889	274	6	ms	ms	PROPN
ejpam-3889	274	7	∈	∈	PROPN
ejpam-3889	274	8	s	s	PART
ejpam-3889	274	9	−1m,∃p	−1m,∃p	NOUN
ejpam-3889	274	10	,	,	PUNCT
ejpam-3889	274	11	m	m	PROPN
ejpam-3889	274	12	∈	∈	PROPN
ejpam-3889	274	13	mp	mp	NOUN
ejpam-3889	274	14	and	and	CCONJ
ejpam-3889	274	15	deg(s	deg(	VERB
ejpam-3889	274	16	)	)	PUNCT
ejpam-3889	274	17	=	=	PUNCT
ejpam-3889	274	18	p−	p−	X
ejpam-3889	274	19	i	i	NOUN
ejpam-3889	274	20	}	}	PUNCT
ejpam-3889	274	21	,	,	PUNCT
ejpam-3889	274	22	we	we	PRON
ejpam-3889	274	23	have	have	VERB
ejpam-3889	274	24	[	[	X
ejpam-3889	274	25	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	274	26	)	)	PUNCT
ejpam-3889	274	27	=	=	SYM
ejpam-3889	274	28	f(m	f(m	PROPN
ejpam-3889	274	29	)	)	PUNCT
ejpam-3889	274	30	s	s	PART
ejpam-3889	274	31	or	or	CCONJ
ejpam-3889	274	32	f	f	PROPN
ejpam-3889	274	33	is	be	AUX
ejpam-3889	274	34	graded	grade	VERB
ejpam-3889	274	35	=	=	PRON
ejpam-3889	274	36	⇒	⇒	NOUN
ejpam-3889	274	37	f(m	f(m	PROPN
ejpam-3889	274	38	)	)	PUNCT
ejpam-3889	274	39	∈mp	∈mp	NOUN
ejpam-3889	274	40	=	=	NOUN
ejpam-3889	274	41	⇒	⇒	NOUN
ejpam-3889	274	42	f(m	f(m	PROPN
ejpam-3889	274	43	)	)	PUNCT
ejpam-3889	274	44	s	s	PART
ejpam-3889	274	45	∈	∈	PROPN
ejpam-3889	274	46	(	(	PUNCT
ejpam-3889	274	47	s−1m)i	s−1m)i	PROPN
ejpam-3889	274	48	,	,	PUNCT
ejpam-3889	274	49	so	so	ADV
ejpam-3889	274	50	s−1(f	s−1(f	PROPN
ejpam-3889	274	51	)	)	PUNCT
ejpam-3889	274	52	is	be	AUX
ejpam-3889	274	53	a	a	DET
ejpam-3889	274	54	graded	grade	VERB
ejpam-3889	274	55	morphism	morphism	NOUN
ejpam-3889	274	56	.	.	PUNCT
ejpam-3889	275	1	then	then	ADV
ejpam-3889	275	2	,	,	PUNCT
ejpam-3889	275	3	by	by	ADP
ejpam-3889	275	4	lemma(1	lemma(1	NOUN
ejpam-3889	275	5	)	)	PUNCT
ejpam-3889	275	6	,	,	PUNCT
ejpam-3889	275	7	we	we	PRON
ejpam-3889	275	8	obtain	obtain	VERB
ejpam-3889	275	9	the	the	DET
ejpam-3889	275	10	result	result	NOUN
ejpam-3889	275	11	.	.	PUNCT
ejpam-3889	276	1	theorem	theorem	ADJ
ejpam-3889	276	2	10	10	NUM
ejpam-3889	276	3	.	.	PUNCT
ejpam-3889	277	1	let	let	VERB
ejpam-3889	277	2	a	a	DET
ejpam-3889	277	3	=	=	SYM
ejpam-3889	277	4	⊕	⊕	PROPN
ejpam-3889	277	5	n∈z	n∈z	VERB
ejpam-3889	277	6	an	an	DET
ejpam-3889	277	7	be	be	AUX
ejpam-3889	277	8	a	a	DET
ejpam-3889	277	9	graded	grade	VERB
ejpam-3889	277	10	ring	ring	NOUN
ejpam-3889	277	11	,	,	PUNCT
ejpam-3889	277	12	s	s	VERB
ejpam-3889	277	13	a	a	DET
ejpam-3889	277	14	saturated	saturate	VERB
ejpam-3889	277	15	multiplicative	multiplicative	ADJ
ejpam-3889	277	16	part	part	NOUN
ejpam-3889	277	17	formed	form	VERB
ejpam-3889	277	18	by	by	ADP
ejpam-3889	277	19	the	the	DET
ejpam-3889	277	20	non	non	ADJ
ejpam-3889	277	21	-	-	ADJ
ejpam-3889	277	22	zero	zero	ADJ
ejpam-3889	277	23	homogeneous	homogeneous	ADJ
ejpam-3889	277	24	elements	element	NOUN
ejpam-3889	277	25	of	of	ADP
ejpam-3889	277	26	a	a	DET
ejpam-3889	277	27	verifying	verifying	NOUN
ejpam-3889	277	28	the	the	DET
ejpam-3889	277	29	left	left	ADJ
ejpam-3889	277	30	ore	ore	NOUN
ejpam-3889	277	31	conditions	condition	NOUN
ejpam-3889	277	32	,	,	PUNCT
ejpam-3889	277	33	m	m	VERB
ejpam-3889	277	34	=	=	PROPN
ejpam-3889	277	35	⊕	⊕	PROPN
ejpam-3889	277	36	n∈z	n∈z	VERB
ejpam-3889	277	37	mn	mn	PROPN
ejpam-3889	277	38	a	a	DET
ejpam-3889	277	39	graded	grade	VERB
ejpam-3889	277	40	left	leave	VERB
ejpam-3889	277	41	a	a	DET
ejpam-3889	277	42	-	-	PUNCT
ejpam-3889	277	43	module	module	NOUN
ejpam-3889	277	44	.	.	PUNCT
ejpam-3889	278	1	then	then	ADV
ejpam-3889	278	2	,	,	PUNCT
ejpam-3889	278	3	if	if	SCONJ
ejpam-3889	278	4	s−1(m	s−1(m	PROPN
ejpam-3889	278	5	)	)	PUNCT
ejpam-3889	278	6	is	be	AUX
ejpam-3889	278	7	a	a	DET
ejpam-3889	278	8	hopfian	hopfian	ADJ
ejpam-3889	278	9	left	leave	VERB
ejpam-3889	278	10	graded	grade	VERB
ejpam-3889	278	11	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	278	12	,	,	PUNCT
ejpam-3889	278	13	implies	imply	VERB
ejpam-3889	278	14	that	that	SCONJ
ejpam-3889	278	15	m	m	PROPN
ejpam-3889	278	16	is	be	AUX
ejpam-3889	278	17	a	a	DET
ejpam-3889	278	18	hopfian	hopfian	ADJ
ejpam-3889	278	19	left	leave	VERB
ejpam-3889	278	20	graded	grade	VERB
ejpam-3889	278	21	a	a	DET
ejpam-3889	278	22	-	-	PUNCT
ejpam-3889	278	23	module	module	NOUN
ejpam-3889	278	24	.	.	PUNCT
ejpam-3889	279	1	s.	s.	PROPN
ejpam-3889	279	2	a.	a.	PROPN
ejpam-3889	279	3	balde	balde	PROPN
ejpam-3889	279	4	,	,	PUNCT
ejpam-3889	279	5	m.	m.	PROPN
ejpam-3889	279	6	b.	b.	PROPN
ejpam-3889	279	7	maaouia	maaouia	PROPN
ejpam-3889	279	8	,	,	PUNCT
ejpam-3889	279	9	a.	a.	NOUN
ejpam-3889	279	10	o.	o.	NOUN
ejpam-3889	279	11	chbih	chbih	PROPN
ejpam-3889	279	12	/	/	SYM
ejpam-3889	279	13	eur	eur	PROPN
ejpam-3889	279	14	.	.	PUNCT
ejpam-3889	280	1	j.	j.	PROPN
ejpam-3889	280	2	pure	pure	PROPN
ejpam-3889	280	3	appl	appl	PROPN
ejpam-3889	280	4	.	.	PROPN
ejpam-3889	280	5	math	math	PROPN
ejpam-3889	280	6	,	,	PUNCT
ejpam-3889	280	7	14	14	NUM
ejpam-3889	280	8	(	(	PUNCT
ejpam-3889	280	9	2	2	NUM
ejpam-3889	280	10	)	)	PUNCT
ejpam-3889	280	11	(	(	PUNCT
ejpam-3889	280	12	2021	2021	NUM
ejpam-3889	280	13	)	)	PUNCT
ejpam-3889	280	14	,	,	PUNCT
ejpam-3889	280	15	404	404	NUM
ejpam-3889	280	16	-	-	SYM
ejpam-3889	280	17	422	422	NUM
ejpam-3889	280	18	414	414	NUM
ejpam-3889	280	19	proof	proof	NOUN
ejpam-3889	280	20	.	.	PUNCT
ejpam-3889	281	1	let	let	VERB
ejpam-3889	281	2	f	f	NOUN
ejpam-3889	281	3	:	:	PUNCT
ejpam-3889	281	4	m	m	VERB
ejpam-3889	281	5	−→m	−→m	VERB
ejpam-3889	281	6	be	be	AUX
ejpam-3889	281	7	a	a	DET
ejpam-3889	281	8	graded	grade	VERB
ejpam-3889	281	9	epimorphism	epimorphism	NOUN
ejpam-3889	281	10	ofm	ofm	PROPN
ejpam-3889	281	11	.	.	PUNCT
ejpam-3889	282	1	then	then	ADV
ejpam-3889	282	2	s−1f	s−1f	NOUN
ejpam-3889	282	3	:	:	PUNCT
ejpam-3889	282	4	s−1(m	s−1(m	PROPN
ejpam-3889	282	5	)	)	PUNCT
ejpam-3889	283	1	=	=	PROPN
ejpam-3889	283	2	⊕	⊕	PROPN
ejpam-3889	283	3	i∈z	i∈z	PROPN
ejpam-3889	283	4	(	(	PUNCT
ejpam-3889	283	5	s−1m)i	s−1m)i	VERB
ejpam-3889	283	6	−→	−→	NOUN
ejpam-3889	283	7	s−1(m	s−1(m	PROPN
ejpam-3889	283	8	)	)	PUNCT
ejpam-3889	283	9	=	=	PROPN
ejpam-3889	283	10	⊕	⊕	PROPN
ejpam-3889	283	11	i∈z	i∈z	PROPN
ejpam-3889	283	12	(	(	PUNCT
ejpam-3889	283	13	s−1m)i	s−1m)i	NOUN
ejpam-3889	283	14	defined	define	VERB
ejpam-3889	283	15	by	by	ADP
ejpam-3889	283	16	[	[	X
ejpam-3889	283	17	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	283	18	)	)	PUNCT
ejpam-3889	283	19	=	=	SYM
ejpam-3889	283	20	f(m	f(m	PROPN
ejpam-3889	283	21	)	)	PUNCT
ejpam-3889	283	22	s	s	VERB
ejpam-3889	283	23	is	be	AUX
ejpam-3889	283	24	a	a	DET
ejpam-3889	283	25	graded	grade	VERB
ejpam-3889	283	26	endomorphism	endomorphism	NOUN
ejpam-3889	283	27	of	of	ADP
ejpam-3889	283	28	graded	grade	VERB
ejpam-3889	283	29	left	leave	VERB
ejpam-3889	283	30	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	283	31	,	,	PUNCT
ejpam-3889	283	32	because	because	SCONJ
ejpam-3889	283	33	[	[	X
ejpam-3889	283	34	s−1(f)](mi	s−1(f)](mi	PROPN
ejpam-3889	283	35	s	s	PART
ejpam-3889	283	36	)	)	PUNCT
ejpam-3889	283	37	=	=	SYM
ejpam-3889	283	38	f(mi	f(mi	PROPN
ejpam-3889	283	39	)	)	PUNCT
ejpam-3889	283	40	s	s	PART
ejpam-3889	283	41	,	,	PUNCT
ejpam-3889	283	42	since	since	SCONJ
ejpam-3889	283	43	f	f	PROPN
ejpam-3889	283	44	is	be	AUX
ejpam-3889	283	45	graded	grade	VERB
ejpam-3889	283	46	,	,	PUNCT
ejpam-3889	283	47	then	then	ADV
ejpam-3889	283	48	f(mi	f(mi	NOUN
ejpam-3889	283	49	)	)	PUNCT
ejpam-3889	283	50	∈	∈	PROPN
ejpam-3889	283	51	mi	mi	PROPN
ejpam-3889	283	52	,	,	PUNCT
ejpam-3889	283	53	thus	thus	ADV
ejpam-3889	283	54	f(xi	f(xi	X
ejpam-3889	283	55	)	)	PUNCT
ejpam-3889	283	56	s	s	PART
ejpam-3889	283	57	∈	∈	PROPN
ejpam-3889	283	58	(	(	PUNCT
ejpam-3889	283	59	s−1m)i	s−1m)i	PROPN
ejpam-3889	283	60	.	.	PUNCT
ejpam-3889	284	1	let	let	VERB
ejpam-3889	284	2	m′	m′	NOUN
ejpam-3889	284	3	s	s	PART
ejpam-3889	284	4	∈	∈	NOUN
ejpam-3889	284	5	s	s	PART
ejpam-3889	284	6	−1(m	−1(m	ADJ
ejpam-3889	284	7	)	)	PUNCT
ejpam-3889	284	8	,	,	PUNCT
ejpam-3889	284	9	as	as	SCONJ
ejpam-3889	284	10	f	f	PROPN
ejpam-3889	284	11	is	be	AUX
ejpam-3889	284	12	a	a	DET
ejpam-3889	284	13	graded	grade	VERB
ejpam-3889	284	14	epimorphism	epimorphism	NOUN
ejpam-3889	284	15	,	,	PUNCT
ejpam-3889	284	16	then	then	ADV
ejpam-3889	284	17	there	there	PRON
ejpam-3889	284	18	exists	exist	VERB
ejpam-3889	284	19	m	m	VERB
ejpam-3889	284	20	∈m	∈m	NOUN
ejpam-3889	284	21	,	,	PUNCT
ejpam-3889	284	22	f(m	f(m	PROPN
ejpam-3889	284	23	)	)	PUNCT
ejpam-3889	284	24	=	=	SYM
ejpam-3889	284	25	m′.	m′.	PROPN
ejpam-3889	285	1	so	so	SCONJ
ejpam-3889	286	1	[	[	X
ejpam-3889	286	2	s−1(f)](ms	s−1(f)](ms	PROPN
ejpam-3889	286	3	)	)	PUNCT
ejpam-3889	286	4	=	=	SYM
ejpam-3889	286	5	f(m	f(m	PROPN
ejpam-3889	286	6	)	)	PUNCT
ejpam-3889	286	7	s	s	PART
ejpam-3889	286	8	=	=	PUNCT
ejpam-3889	286	9	m′	m′	X
ejpam-3889	286	10	s	s	NOUN
ejpam-3889	286	11	,	,	PUNCT
ejpam-3889	286	12	hence	hence	ADV
ejpam-3889	286	13	s−1(f	s−1(f	PROPN
ejpam-3889	286	14	)	)	PUNCT
ejpam-3889	286	15	is	be	AUX
ejpam-3889	286	16	a	a	DET
ejpam-3889	286	17	graded	grade	VERB
ejpam-3889	286	18	epimorphism	epimorphism	NOUN
ejpam-3889	286	19	,	,	PUNCT
ejpam-3889	286	20	since	since	SCONJ
ejpam-3889	286	21	s−1(m	s−1(m	PROPN
ejpam-3889	286	22	)	)	PUNCT
ejpam-3889	286	23	is	be	AUX
ejpam-3889	286	24	hopfian	hopfian	ADJ
ejpam-3889	286	25	,	,	PUNCT
ejpam-3889	286	26	then	then	ADV
ejpam-3889	286	27	s−1(f	s−1(f	PROPN
ejpam-3889	286	28	)	)	PUNCT
ejpam-3889	286	29	is	be	AUX
ejpam-3889	286	30	a	a	DET
ejpam-3889	286	31	graded	grade	VERB
ejpam-3889	286	32	automorphism	automorphism	NOUN
ejpam-3889	286	33	of	of	ADP
ejpam-3889	286	34	s−1(m	s−1(m	PROPN
ejpam-3889	286	35	)	)	PUNCT
ejpam-3889	286	36	.	.	PUNCT
ejpam-3889	287	1	let	let	VERB
ejpam-3889	287	2	m1	m1	PROPN
ejpam-3889	287	3	and	and	CCONJ
ejpam-3889	287	4	m2	m2	PROPN
ejpam-3889	287	5	∈m	∈m	NOUN
ejpam-3889	287	6	such	such	ADJ
ejpam-3889	287	7	that	that	DET
ejpam-3889	287	8	f(m1	f(m1	NOUN
ejpam-3889	287	9	)	)	PUNCT
ejpam-3889	287	10	=	=	SYM
ejpam-3889	287	11	f(m2	f(m2	NOUN
ejpam-3889	287	12	)	)	PUNCT
ejpam-3889	288	1	=	=	NOUN
ejpam-3889	288	2	⇒	⇒	NOUN
ejpam-3889	288	3	f(m1	f(m1	NOUN
ejpam-3889	288	4	)	)	PUNCT
ejpam-3889	288	5	1	1	NUM
ejpam-3889	288	6	=	=	SYM
ejpam-3889	288	7	f(m2	f(m2	NOUN
ejpam-3889	288	8	)	)	PUNCT
ejpam-3889	288	9	1	1	NUM
ejpam-3889	289	1	=	=	NOUN
ejpam-3889	289	2	⇒	⇒	NOUN
ejpam-3889	289	3	[	[	X
ejpam-3889	289	4	s−1(f)](m1	s−1(f)](m1	NOUN
ejpam-3889	289	5	)	)	PUNCT
ejpam-3889	289	6	=	=	PUNCT
ejpam-3889	290	1	[	[	X
ejpam-3889	290	2	s−1(f)](m2	s−1(f)](m2	NOUN
ejpam-3889	290	3	)	)	PUNCT
ejpam-3889	290	4	=	=	NOUN
ejpam-3889	290	5	⇒	⇒	NOUN
ejpam-3889	290	6	m1	m1	PROPN
ejpam-3889	290	7	=	=	SYM
ejpam-3889	290	8	m2	m2	PROPN
ejpam-3889	290	9	thus	thus	ADV
ejpam-3889	290	10	f	f	PROPN
ejpam-3889	290	11	is	be	AUX
ejpam-3889	290	12	a	a	DET
ejpam-3889	290	13	graded	grade	VERB
ejpam-3889	290	14	automorphism	automorphism	NOUN
ejpam-3889	290	15	of	of	ADP
ejpam-3889	290	16	m	m	PRON
ejpam-3889	290	17	,	,	PUNCT
ejpam-3889	290	18	so	so	ADV
ejpam-3889	290	19	m	m	VERB
ejpam-3889	290	20	is	be	AUX
ejpam-3889	290	21	a	a	DET
ejpam-3889	290	22	hopfian	hopfian	NOUN
ejpam-3889	290	23	graded	grade	VERB
ejpam-3889	290	24	left	leave	VERB
ejpam-3889	290	25	a	a	DET
ejpam-3889	290	26	-	-	PUNCT
ejpam-3889	290	27	module	module	NOUN
ejpam-3889	290	28	.	.	PUNCT
ejpam-3889	291	1	corollary	corollary	ADJ
ejpam-3889	291	2	1	1	NUM
ejpam-3889	291	3	.	.	PUNCT
ejpam-3889	292	1	under	under	ADP
ejpam-3889	292	2	the	the	DET
ejpam-3889	292	3	same	same	ADJ
ejpam-3889	292	4	conditions	condition	NOUN
ejpam-3889	292	5	of	of	ADP
ejpam-3889	292	6	the	the	DET
ejpam-3889	292	7	previous	previous	ADJ
ejpam-3889	292	8	theorem	theorem	NOUN
ejpam-3889	292	9	.	.	PUNCT
ejpam-3889	293	1	if	if	SCONJ
ejpam-3889	293	2	s−1(m	s−1(m	PROPN
ejpam-3889	293	3	)	)	PUNCT
ejpam-3889	293	4	is	be	AUX
ejpam-3889	293	5	a	a	DET
ejpam-3889	293	6	hopfian	hopfian	NOUN
ejpam-3889	293	7	graded	grade	VERB
ejpam-3889	293	8	left	leave	VERB
ejpam-3889	293	9	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	293	10	,	,	PUNCT
ejpam-3889	293	11	then	then	ADV
ejpam-3889	293	12	mn	mn	PROPN
ejpam-3889	293	13	for	for	ADP
ejpam-3889	293	14	all	all	DET
ejpam-3889	293	15	n	n	PRON
ejpam-3889	293	16	∈	∈	PROPN
ejpam-3889	293	17	z	z	NOUN
ejpam-3889	293	18	is	be	AUX
ejpam-3889	293	19	a	a	DET
ejpam-3889	293	20	hopfian	hopfian	ADJ
ejpam-3889	293	21	group	group	NOUN
ejpam-3889	293	22	.	.	PUNCT
ejpam-3889	294	1	proof	proof	NOUN
ejpam-3889	294	2	.	.	PUNCT
ejpam-3889	295	1	it	it	PRON
ejpam-3889	295	2	’s	’	VERB
ejpam-3889	295	3	obvious	obvious	ADJ
ejpam-3889	295	4	by	by	ADP
ejpam-3889	295	5	theorem(10	theorem(10	NOUN
ejpam-3889	295	6	)	)	PUNCT
ejpam-3889	295	7	theorem	theorem	VERB
ejpam-3889	295	8	11	11	NUM
ejpam-3889	295	9	.	.	PUNCT
ejpam-3889	296	1	let	let	VERB
ejpam-3889	296	2	a	a	DET
ejpam-3889	296	3	graded	grade	VERB
ejpam-3889	296	4	ring	ring	NOUN
ejpam-3889	296	5	,	,	PUNCT
ejpam-3889	296	6	s	s	VERB
ejpam-3889	296	7	a	a	DET
ejpam-3889	296	8	saturated	saturate	VERB
ejpam-3889	296	9	multiplicative	multiplicative	ADJ
ejpam-3889	296	10	part	part	NOUN
ejpam-3889	296	11	formed	form	VERB
ejpam-3889	296	12	by	by	ADP
ejpam-3889	296	13	the	the	DET
ejpam-3889	296	14	non	non	ADJ
ejpam-3889	296	15	-	-	ADJ
ejpam-3889	296	16	zero	zero	ADJ
ejpam-3889	296	17	homogeneous	homogeneous	ADJ
ejpam-3889	296	18	elements	element	NOUN
ejpam-3889	296	19	of	of	ADP
ejpam-3889	296	20	a	a	DET
ejpam-3889	296	21	verifying	verifying	NOUN
ejpam-3889	296	22	the	the	DET
ejpam-3889	296	23	left	left	ADJ
ejpam-3889	296	24	ore	ore	NOUN
ejpam-3889	296	25	conditions	condition	NOUN
ejpam-3889	296	26	,	,	PUNCT
ejpam-3889	296	27	m	m	AUX
ejpam-3889	296	28	left	leave	VERB
ejpam-3889	296	29	graded	grade	VERB
ejpam-3889	296	30	a	a	DET
ejpam-3889	296	31	-	-	PUNCT
ejpam-3889	296	32	module	module	NOUN
ejpam-3889	296	33	.	.	PUNCT
ejpam-3889	297	1	then	then	ADV
ejpam-3889	297	2	,	,	PUNCT
ejpam-3889	297	3	if	if	SCONJ
ejpam-3889	297	4	m	m	NOUN
ejpam-3889	297	5	is	be	AUX
ejpam-3889	297	6	a	a	DET
ejpam-3889	297	7	cohopfian	cohopfian	ADJ
ejpam-3889	297	8	and	and	CCONJ
ejpam-3889	297	9	completely	completely	ADV
ejpam-3889	297	10	invariant	invariant	ADJ
ejpam-3889	297	11	submodule	submodule	NOUN
ejpam-3889	297	12	of	of	ADP
ejpam-3889	297	13	the	the	DET
ejpam-3889	297	14	left	left	ADJ
ejpam-3889	297	15	a	a	DET
ejpam-3889	297	16	-	-	PUNCT
ejpam-3889	297	17	module	module	NOUN
ejpam-3889	297	18	s−1(m	s−1(m	PROPN
ejpam-3889	297	19	)	)	PUNCT
ejpam-3889	297	20	,	,	PUNCT
ejpam-3889	297	21	implies	imply	VERB
ejpam-3889	297	22	that	that	SCONJ
ejpam-3889	297	23	,	,	PUNCT
ejpam-3889	297	24	s−1(m	s−1(m	PROPN
ejpam-3889	297	25	)	)	PUNCT
ejpam-3889	297	26	is	be	AUX
ejpam-3889	297	27	a	a	DET
ejpam-3889	297	28	left	left	ADJ
ejpam-3889	297	29	cohopfian	cohopfian	ADJ
ejpam-3889	297	30	graded	grade	VERB
ejpam-3889	297	31	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	297	32	.	.	PUNCT
ejpam-3889	298	1	proof	proof	NOUN
ejpam-3889	298	2	.	.	PUNCT
ejpam-3889	299	1	let	let	VERB
ejpam-3889	299	2	g	g	NOUN
ejpam-3889	299	3	:	:	PUNCT
ejpam-3889	299	4	s−1(m	s−1(m	PROPN
ejpam-3889	299	5	)	)	PUNCT
ejpam-3889	299	6	−→	−→	NOUN
ejpam-3889	299	7	s−1(m	s−1(m	PROPN
ejpam-3889	299	8	)	)	PUNCT
ejpam-3889	299	9	a	a	DET
ejpam-3889	299	10	graded	grade	VERB
ejpam-3889	299	11	s−1(a)-morphisme	s−1(a)-morphisme	NOUN
ejpam-3889	299	12	,	,	PUNCT
ejpam-3889	299	13	we	we	PRON
ejpam-3889	299	14	remark	remark	VERB
ejpam-3889	299	15	also	also	ADV
ejpam-3889	299	16	that	that	SCONJ
ejpam-3889	299	17	g	g	PROPN
ejpam-3889	299	18	is	be	AUX
ejpam-3889	299	19	a	a	DET
ejpam-3889	299	20	graded	grade	VERB
ejpam-3889	299	21	a	a	DET
ejpam-3889	299	22	-	-	PUNCT
ejpam-3889	299	23	morphisme	morphisme	ADJ
ejpam-3889	299	24	.	.	PUNCT
ejpam-3889	300	1	as	as	SCONJ
ejpam-3889	300	2	m	m	NOUN
ejpam-3889	300	3	is	be	AUX
ejpam-3889	300	4	completely	completely	ADV
ejpam-3889	300	5	invariant	invariant	ADJ
ejpam-3889	300	6	as	as	SCONJ
ejpam-3889	300	7	graded	grade	VERB
ejpam-3889	300	8	submodule	submodule	NOUN
ejpam-3889	300	9	of	of	ADP
ejpam-3889	300	10	graded	grade	VERB
ejpam-3889	300	11	left	leave	VERB
ejpam-3889	300	12	a	a	DET
ejpam-3889	300	13	-	-	PUNCT
ejpam-3889	300	14	module	module	NOUN
ejpam-3889	300	15	s−1	s−1	PROPN
ejpam-3889	300	16	m	m	PRON
ejpam-3889	300	17	,	,	PUNCT
ejpam-3889	300	18	so	so	ADV
ejpam-3889	300	19	g(m	g(m	ADJ
ejpam-3889	300	20	)	)	PUNCT
ejpam-3889	300	21	⊂m	⊂m	PROPN
ejpam-3889	300	22	.	.	PUNCT
ejpam-3889	301	1	suppose	suppose	VERB
ejpam-3889	301	2	that	that	SCONJ
ejpam-3889	301	3	g	g	PROPN
ejpam-3889	301	4	is	be	AUX
ejpam-3889	301	5	a	a	DET
ejpam-3889	301	6	graded	grade	VERB
ejpam-3889	301	7	monomorphism	monomorphism	NOUN
ejpam-3889	301	8	.	.	PUNCT
ejpam-3889	302	1	so	so	ADV
ejpam-3889	302	2	the	the	DET
ejpam-3889	302	3	induce	induce	NOUN
ejpam-3889	302	4	graded	grade	VERB
ejpam-3889	302	5	morphism	morphism	NOUN
ejpam-3889	302	6	gind	gind	NOUN
ejpam-3889	302	7	:	:	PUNCT
ejpam-3889	302	8	m	m	VERB
ejpam-3889	302	9	−→m	−→m	NOUN
ejpam-3889	302	10	is	be	AUX
ejpam-3889	302	11	a	a	DET
ejpam-3889	302	12	graded	grade	VERB
ejpam-3889	302	13	monomorphism	monomorphism	NOUN
ejpam-3889	302	14	of	of	ADP
ejpam-3889	302	15	m	m	PROPN
ejpam-3889	302	16	and	and	CCONJ
ejpam-3889	302	17	,	,	PUNCT
ejpam-3889	302	18	as	as	SCONJ
ejpam-3889	302	19	m	m	PROPN
ejpam-3889	302	20	is	be	AUX
ejpam-3889	302	21	cohopfian	cohopfian	ADJ
ejpam-3889	302	22	,	,	PUNCT
ejpam-3889	302	23	then	then	ADV
ejpam-3889	302	24	gind	gind	NOUN
ejpam-3889	302	25	is	be	AUX
ejpam-3889	302	26	a	a	DET
ejpam-3889	302	27	graded	grade	VERB
ejpam-3889	302	28	automorphism	automorphism	NOUN
ejpam-3889	302	29	of	of	ADP
ejpam-3889	302	30	m	m	PROPN
ejpam-3889	302	31	.	.	PUNCT
ejpam-3889	303	1	consider	consider	VERB
ejpam-3889	303	2	s−1(gind	s−1(gind	NOUN
ejpam-3889	303	3	)	)	PUNCT
ejpam-3889	303	4	:	:	PUNCT
ejpam-3889	304	1	s−1(m	s−1(m	PROPN
ejpam-3889	304	2	)	)	PUNCT
ejpam-3889	304	3	−→	−→	NOUN
ejpam-3889	304	4	s−1(m	s−1(m	PROPN
ejpam-3889	304	5	)	)	PUNCT
ejpam-3889	304	6	m	m	PROPN
ejpam-3889	304	7	s	s	PROPN
ejpam-3889	304	8	7−→	7−→	NOUN
ejpam-3889	304	9	g(m	g(m	VERB
ejpam-3889	304	10	)	)	PUNCT
ejpam-3889	304	11	s	s	VERB
ejpam-3889	304	12	thus	thus	ADV
ejpam-3889	304	13	g(m	g(m	VERB
ejpam-3889	304	14	)	)	PUNCT
ejpam-3889	304	15	s	s	PART
ejpam-3889	304	16	=	=	SYM
ejpam-3889	304	17	1	1	NUM
ejpam-3889	304	18	s	s	NOUN
ejpam-3889	304	19	g(m	g(m	NOUN
ejpam-3889	304	20	)	)	PUNCT
ejpam-3889	304	21	1	1	NUM
ejpam-3889	304	22	.	.	PUNCT
ejpam-3889	304	23	or	or	CCONJ
ejpam-3889	304	24	g	g	PROPN
ejpam-3889	304	25	is	be	AUX
ejpam-3889	304	26	a	a	DET
ejpam-3889	304	27	graded	grade	VERB
ejpam-3889	304	28	s−1(a)-morphism	s−1(a)-morphism	PROPN
ejpam-3889	304	29	,	,	PUNCT
ejpam-3889	304	30	so	so	ADV
ejpam-3889	304	31	1	1	NUM
ejpam-3889	304	32	s	s	NOUN
ejpam-3889	304	33	=	=	NOUN
ejpam-3889	304	34	g(m	g(m	VERB
ejpam-3889	304	35	)	)	PUNCT
ejpam-3889	304	36	1	1	NUM
ejpam-3889	304	37	=	=	SYM
ejpam-3889	304	38	g(1s	g(1s	NUM
ejpam-3889	304	39	m	m	VERB
ejpam-3889	304	40	1	1	NUM
ejpam-3889	304	41	)	)	PUNCT
ejpam-3889	304	42	=	=	SYM
ejpam-3889	304	43	g(ms	g(ms	X
ejpam-3889	304	44	)	)	PUNCT
ejpam-3889	305	1	=	=	VERB
ejpam-3889	305	2	⇒	⇒	NOUN
ejpam-3889	305	3	s−1(gind	s−1(gind	NOUN
ejpam-3889	305	4	)	)	PUNCT
ejpam-3889	305	5	=	=	PUNCT
ejpam-3889	305	6	g	g	PROPN
ejpam-3889	305	7	g	g	PROPN
ejpam-3889	305	8	is	be	AUX
ejpam-3889	305	9	a	a	DET
ejpam-3889	305	10	graded	grade	VERB
ejpam-3889	305	11	monomorphism	monomorphism	NOUN
ejpam-3889	305	12	,	,	PUNCT
ejpam-3889	305	13	then	then	ADV
ejpam-3889	305	14	gind	gind	NOUN
ejpam-3889	305	15	is	be	AUX
ejpam-3889	305	16	a	a	DET
ejpam-3889	305	17	graded	grade	VERB
ejpam-3889	305	18	monomorphism	monomorphism	NOUN
ejpam-3889	305	19	.	.	PUNCT
ejpam-3889	306	1	since	since	SCONJ
ejpam-3889	306	2	m	m	PROPN
ejpam-3889	306	3	is	be	AUX
ejpam-3889	306	4	cohopfian	cohopfian	ADJ
ejpam-3889	306	5	,	,	PUNCT
ejpam-3889	306	6	then	then	ADV
ejpam-3889	306	7	gind	gind	NOUN
ejpam-3889	306	8	is	be	AUX
ejpam-3889	306	9	a	a	DET
ejpam-3889	306	10	graded	grade	VERB
ejpam-3889	306	11	automorphism	automorphism	NOUN
ejpam-3889	306	12	of	of	ADP
ejpam-3889	306	13	m	m	PROPN
ejpam-3889	306	14	.	.	PUNCT
ejpam-3889	307	1	s.	s.	PROPN
ejpam-3889	307	2	a.	a.	PROPN
ejpam-3889	307	3	balde	balde	PROPN
ejpam-3889	307	4	,	,	PUNCT
ejpam-3889	307	5	m.	m.	PROPN
ejpam-3889	307	6	b.	b.	PROPN
ejpam-3889	307	7	maaouia	maaouia	PROPN
ejpam-3889	307	8	,	,	PUNCT
ejpam-3889	307	9	a.	a.	NOUN
ejpam-3889	307	10	o.	o.	NOUN
ejpam-3889	307	11	chbih	chbih	PROPN
ejpam-3889	307	12	/	/	SYM
ejpam-3889	307	13	eur	eur	PROPN
ejpam-3889	307	14	.	.	PUNCT
ejpam-3889	308	1	j.	j.	PROPN
ejpam-3889	308	2	pure	pure	PROPN
ejpam-3889	308	3	appl	appl	PROPN
ejpam-3889	308	4	.	.	PROPN
ejpam-3889	308	5	math	math	PROPN
ejpam-3889	308	6	,	,	PUNCT
ejpam-3889	308	7	14	14	NUM
ejpam-3889	308	8	(	(	PUNCT
ejpam-3889	308	9	2	2	NUM
ejpam-3889	308	10	)	)	PUNCT
ejpam-3889	308	11	(	(	PUNCT
ejpam-3889	308	12	2021	2021	NUM
ejpam-3889	308	13	)	)	PUNCT
ejpam-3889	308	14	,	,	PUNCT
ejpam-3889	308	15	404	404	NUM
ejpam-3889	308	16	-	-	SYM
ejpam-3889	308	17	422	422	NUM
ejpam-3889	308	18	415	415	NUM
ejpam-3889	308	19	let	let	VERB
ejpam-3889	308	20	m′	m′	NOUN
ejpam-3889	308	21	s	s	PART
ejpam-3889	308	22	∈	∈	NOUN
ejpam-3889	308	23	s	s	PART
ejpam-3889	308	24	−1(m	−1(m	ADJ
ejpam-3889	308	25	)	)	PUNCT
ejpam-3889	308	26	=	=	VERB
ejpam-3889	308	27	⇒	⇒	VERB
ejpam-3889	308	28	m′	m′	NOUN
ejpam-3889	308	29	∈m	∈m	NOUN
ejpam-3889	308	30	,	,	PUNCT
ejpam-3889	308	31	so	so	CCONJ
ejpam-3889	308	32	there	there	PRON
ejpam-3889	308	33	exists	exist	VERB
ejpam-3889	308	34	m	m	VERB
ejpam-3889	308	35	∈m	∈m	NOUN
ejpam-3889	308	36	:	:	PUNCT
ejpam-3889	308	37	gind(m	gind(m	NOUN
ejpam-3889	308	38	)	)	PUNCT
ejpam-3889	308	39	=	=	PUNCT
ejpam-3889	308	40	m′	m′	NOUN
ejpam-3889	308	41	hence	hence	ADV
ejpam-3889	308	42	m	m	PROPN
ejpam-3889	308	43	s	s	NOUN
ejpam-3889	308	44	∈	∈	PROPN
ejpam-3889	308	45	s	s	PART
ejpam-3889	308	46	−1(a	−1(a	NOUN
ejpam-3889	308	47	)	)	PUNCT
ejpam-3889	308	48	,	,	PUNCT
ejpam-3889	309	1	[	[	X
ejpam-3889	309	2	s−1(gind	s−1(gind	NOUN
ejpam-3889	309	3	)	)	PUNCT
ejpam-3889	309	4	]	]	PUNCT
ejpam-3889	309	5	(	(	PUNCT
ejpam-3889	309	6	m	m	PROPN
ejpam-3889	309	7	s	s	PART
ejpam-3889	309	8	)	)	PUNCT
ejpam-3889	309	9	=	=	SYM
ejpam-3889	309	10	g(m	g(m	VERB
ejpam-3889	309	11	)	)	PUNCT
ejpam-3889	309	12	s	s	PART
ejpam-3889	309	13	=	=	PUNCT
ejpam-3889	309	14	m′	m′	NOUN
ejpam-3889	309	15	s	s	PART
ejpam-3889	309	16	so	so	ADV
ejpam-3889	309	17	(	(	PUNCT
ejpam-3889	309	18	g(ms	g(ms	PROPN
ejpam-3889	309	19	)	)	PUNCT
ejpam-3889	309	20	)	)	PUNCT
ejpam-3889	310	1	=	=	PUNCT
ejpam-3889	311	1	m′	m′	NOUN
ejpam-3889	311	2	s	s	PART
ejpam-3889	311	3	.	.	PUNCT
ejpam-3889	312	1	thus	thus	ADV
ejpam-3889	312	2	s−1(m	s−1(m	PROPN
ejpam-3889	312	3	)	)	PUNCT
ejpam-3889	312	4	is	be	AUX
ejpam-3889	312	5	a	a	DET
ejpam-3889	312	6	cohopfian	cohopfian	ADJ
ejpam-3889	312	7	left	leave	VERB
ejpam-3889	312	8	s−1a	s−1a	NOUN
ejpam-3889	312	9	-	-	PUNCT
ejpam-3889	312	10	module	module	NOUN
ejpam-3889	312	11	.	.	PUNCT
ejpam-3889	313	1	corollary	corollary	ADJ
ejpam-3889	313	2	2	2	NUM
ejpam-3889	313	3	.	.	PUNCT
ejpam-3889	314	1	under	under	ADP
ejpam-3889	314	2	the	the	DET
ejpam-3889	314	3	same	same	ADJ
ejpam-3889	314	4	conditions	condition	NOUN
ejpam-3889	314	5	of	of	ADP
ejpam-3889	314	6	the	the	DET
ejpam-3889	314	7	previous	previous	ADJ
ejpam-3889	314	8	theorem	theorem	NOUN
ejpam-3889	314	9	if	if	SCONJ
ejpam-3889	314	10	m	m	NOUN
ejpam-3889	314	11	is	be	AUX
ejpam-3889	314	12	a	a	DET
ejpam-3889	314	13	cohopfian	cohopfian	ADJ
ejpam-3889	314	14	and	and	CCONJ
ejpam-3889	314	15	completely	completely	ADV
ejpam-3889	314	16	invariant	invariant	ADJ
ejpam-3889	314	17	submodule	submodule	NOUN
ejpam-3889	314	18	of	of	ADP
ejpam-3889	314	19	the	the	DET
ejpam-3889	314	20	left	left	ADJ
ejpam-3889	314	21	a	a	DET
ejpam-3889	314	22	-	-	PUNCT
ejpam-3889	314	23	module	module	NOUN
ejpam-3889	314	24	s−1	s−1	PROPN
ejpam-3889	314	25	m	m	PROPN
ejpam-3889	314	26	,	,	PUNCT
ejpam-3889	314	27	then	then	ADV
ejpam-3889	314	28	(	(	PUNCT
ejpam-3889	314	29	s−1m)i	s−1m)i	NOUN
ejpam-3889	314	30	is	be	AUX
ejpam-3889	314	31	a	a	DET
ejpam-3889	314	32	cohopfian	cohopfian	ADJ
ejpam-3889	314	33	group	group	NOUN
ejpam-3889	314	34	.	.	PUNCT
ejpam-3889	315	1	proof	proof	NOUN
ejpam-3889	315	2	.	.	PUNCT
ejpam-3889	316	1	by	by	ADP
ejpam-3889	316	2	theorem(11	theorem(11	NOUN
ejpam-3889	316	3	)	)	PUNCT
ejpam-3889	316	4	theorem	theorem	NOUN
ejpam-3889	316	5	12	12	NUM
ejpam-3889	316	6	.	.	PUNCT
ejpam-3889	317	1	let	let	VERB
ejpam-3889	317	2	m	m	PRON
ejpam-3889	317	3	be	be	AUX
ejpam-3889	317	4	a	a	DET
ejpam-3889	317	5	noetherian	noetherian	ADJ
ejpam-3889	317	6	quasi	quasi	ADJ
ejpam-3889	317	7	-	-	ADJ
ejpam-3889	317	8	injective	injective	ADJ
ejpam-3889	317	9	graded	grade	VERB
ejpam-3889	317	10	left	leave	VERB
ejpam-3889	317	11	a	a	DET
ejpam-3889	317	12	-	-	PUNCT
ejpam-3889	317	13	module	module	NOUN
ejpam-3889	317	14	,	,	PUNCT
ejpam-3889	317	15	n	n	PRON
ejpam-3889	317	16	be	be	VERB
ejpam-3889	317	17	an	an	DET
ejpam-3889	317	18	essential	essential	ADJ
ejpam-3889	317	19	and	and	CCONJ
ejpam-3889	317	20	completely	completely	ADV
ejpam-3889	317	21	invariant	invariant	ADJ
ejpam-3889	317	22	graded	grade	VERB
ejpam-3889	317	23	submodule	submodule	NOUN
ejpam-3889	317	24	of	of	ADP
ejpam-3889	317	25	m	m	PROPN
ejpam-3889	317	26	and	and	CCONJ
ejpam-3889	317	27	s	s	VERB
ejpam-3889	317	28	a	a	DET
ejpam-3889	317	29	saturated	saturate	VERB
ejpam-3889	317	30	multiplicative	multiplicative	ADJ
ejpam-3889	317	31	part	part	NOUN
ejpam-3889	317	32	formed	form	VERB
ejpam-3889	317	33	by	by	ADP
ejpam-3889	317	34	the	the	DET
ejpam-3889	317	35	non	non	ADJ
ejpam-3889	317	36	-	-	ADJ
ejpam-3889	317	37	zero	zero	ADJ
ejpam-3889	317	38	homogeneous	homogeneous	ADJ
ejpam-3889	317	39	elements	element	NOUN
ejpam-3889	317	40	of	of	ADP
ejpam-3889	317	41	a	a	DET
ejpam-3889	317	42	verifying	verifying	NOUN
ejpam-3889	317	43	the	the	DET
ejpam-3889	317	44	left	left	ADJ
ejpam-3889	317	45	ore	ore	NOUN
ejpam-3889	317	46	conditions	condition	NOUN
ejpam-3889	317	47	.	.	PUNCT
ejpam-3889	318	1	then	then	ADV
ejpam-3889	318	2	,	,	PUNCT
ejpam-3889	318	3	the	the	DET
ejpam-3889	318	4	garded	garde	VERB
ejpam-3889	318	5	left	leave	VERB
ejpam-3889	318	6	s−1amodules	s−1amodule	NOUN
ejpam-3889	318	7	s−1(m	s−1(m	PROPN
ejpam-3889	318	8	)	)	PUNCT
ejpam-3889	318	9	is	be	AUX
ejpam-3889	318	10	cohopfian	cohopfian	ADJ
ejpam-3889	318	11	if	if	SCONJ
ejpam-3889	318	12	,	,	PUNCT
ejpam-3889	318	13	and	and	CCONJ
ejpam-3889	318	14	only	only	ADV
ejpam-3889	318	15	,	,	PUNCT
ejpam-3889	318	16	if	if	SCONJ
ejpam-3889	318	17	s−1(n	s−1(n	NOUN
ejpam-3889	318	18	)	)	PUNCT
ejpam-3889	318	19	is	be	AUX
ejpam-3889	318	20	cohopfian	cohopfian	ADJ
ejpam-3889	318	21	garded	garde	VERB
ejpam-3889	318	22	left	leave	VERB
ejpam-3889	318	23	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	318	24	.	.	PUNCT
ejpam-3889	319	1	proof	proof	NOUN
ejpam-3889	319	2	.	.	PUNCT
ejpam-3889	320	1	suppose	suppose	VERB
ejpam-3889	320	2	that	that	SCONJ
ejpam-3889	320	3	s−1(m	s−1(m	PROPN
ejpam-3889	320	4	)	)	PUNCT
ejpam-3889	320	5	is	be	AUX
ejpam-3889	320	6	cohopfian	cohopfian	ADJ
ejpam-3889	320	7	and	and	CCONJ
ejpam-3889	320	8	let	let	VERB
ejpam-3889	320	9	s−1(f	s−1(f	PROPN
ejpam-3889	320	10	)	)	PUNCT
ejpam-3889	320	11	:	:	PUNCT
ejpam-3889	320	12	s−1(n	s−1(n	PROPN
ejpam-3889	320	13	)	)	PUNCT
ejpam-3889	320	14	−→	−→	NOUN
ejpam-3889	320	15	s−1(n	s−1(n	PROPN
ejpam-3889	320	16	)	)	PUNCT
ejpam-3889	320	17	be	be	VERB
ejpam-3889	320	18	a	a	DET
ejpam-3889	320	19	graded	grade	VERB
ejpam-3889	320	20	monomorphism	monomorphism	NOUN
ejpam-3889	320	21	.	.	PUNCT
ejpam-3889	321	1	as	as	ADP
ejpam-3889	321	2	s−1(m	s−1(m	PROPN
ejpam-3889	321	3	)	)	PUNCT
ejpam-3889	321	4	is	be	AUX
ejpam-3889	321	5	quasi	quasi	ADJ
ejpam-3889	321	6	-	-	ADJ
ejpam-3889	321	7	injective	injective	ADJ
ejpam-3889	321	8	because	because	SCONJ
ejpam-3889	321	9	m	m	NOUN
ejpam-3889	321	10	is	be	AUX
ejpam-3889	321	11	noetherian	noetherian	ADJ
ejpam-3889	321	12	.	.	PUNCT
ejpam-3889	322	1	then	then	ADV
ejpam-3889	322	2	,	,	PUNCT
ejpam-3889	322	3	there	there	PRON
ejpam-3889	322	4	exists	exist	VERB
ejpam-3889	322	5	a	a	DET
ejpam-3889	322	6	graded	grade	VERB
ejpam-3889	322	7	morphism	morphism	NOUN
ejpam-3889	322	8	s−1(g	s−1(g	PROPN
ejpam-3889	322	9	)	)	PUNCT
ejpam-3889	322	10	∈	∈	PROPN
ejpam-3889	322	11	end(s−1(m	end(s−1(m	PROPN
ejpam-3889	322	12	)	)	PUNCT
ejpam-3889	322	13	)	)	PUNCT
ejpam-3889	322	14	such	such	ADJ
ejpam-3889	322	15	that	that	SCONJ
ejpam-3889	322	16	s−1(g|s−1(n	s−1(g|s−1(n	NOUN
ejpam-3889	322	17	)	)	PUNCT
ejpam-3889	322	18	)	)	PUNCT
ejpam-3889	323	1	=	=	SYM
ejpam-3889	323	2	s−1(f	s−1(f	PROPN
ejpam-3889	323	3	)	)	PUNCT
ejpam-3889	323	4	.	.	PUNCT
ejpam-3889	324	1	s−1(g	s−1(g	PROPN
ejpam-3889	324	2	)	)	PUNCT
ejpam-3889	324	3	is	be	AUX
ejpam-3889	324	4	injective	injective	ADJ
ejpam-3889	324	5	since	since	SCONJ
ejpam-3889	324	6	s−1(n	s−1(n	PROPN
ejpam-3889	324	7	)	)	PUNCT
ejpam-3889	324	8	is	be	AUX
ejpam-3889	324	9	essential	essential	ADJ
ejpam-3889	324	10	in	in	ADP
ejpam-3889	324	11	s−1(m	s−1(m	PROPN
ejpam-3889	324	12	)	)	PUNCT
ejpam-3889	324	13	,	,	PUNCT
ejpam-3889	324	14	and	and	CCONJ
ejpam-3889	324	15	as	as	ADP
ejpam-3889	324	16	s−1(m	s−1(m	PROPN
ejpam-3889	324	17	)	)	PUNCT
ejpam-3889	324	18	is	be	AUX
ejpam-3889	324	19	cohopfian	cohopfian	ADJ
ejpam-3889	324	20	,	,	PUNCT
ejpam-3889	324	21	s−1(g	s−1(g	PROPN
ejpam-3889	324	22	)	)	PUNCT
ejpam-3889	324	23	is	be	AUX
ejpam-3889	324	24	invertible	invertible	ADJ
ejpam-3889	324	25	.	.	PUNCT
ejpam-3889	325	1	let	let	VERB
ejpam-3889	325	2	x	x	SYM
ejpam-3889	325	3	s	s	PART
ejpam-3889	325	4	∈	∈	PROPN
ejpam-3889	325	5	s	s	PART
ejpam-3889	325	6	−1(n	−1(n	NOUN
ejpam-3889	325	7	)	)	PUNCT
ejpam-3889	325	8	,	,	PUNCT
ejpam-3889	325	9	there	there	PRON
ejpam-3889	325	10	exists	exist	VERB
ejpam-3889	325	11	y	y	PROPN
ejpam-3889	325	12	t	t	PROPN
ejpam-3889	325	13	∈	∈	PROPN
ejpam-3889	325	14	s	s	PART
ejpam-3889	325	15	−1(m	−1(m	NOUN
ejpam-3889	325	16	)	)	PUNCT
ejpam-3889	325	17	such	such	ADJ
ejpam-3889	325	18	that	that	SCONJ
ejpam-3889	325	19	x	x	PRON
ejpam-3889	325	20	s	s	NOUN
ejpam-3889	325	21	=	=	X
ejpam-3889	326	1	[	[	X
ejpam-3889	326	2	s−1(g)](yt	s−1(g)](yt	PROPN
ejpam-3889	326	3	)	)	PUNCT
ejpam-3889	326	4	.	.	PUNCT
ejpam-3889	327	1	or	or	CCONJ
ejpam-3889	327	2	s−1(g−1	s−1(g−1	X
ejpam-3889	327	3	)	)	PUNCT
ejpam-3889	327	4	∈	∈	PROPN
ejpam-3889	327	5	end(s−1(n	end(s−1(n	PROPN
ejpam-3889	327	6	)	)	PUNCT
ejpam-3889	327	7	)	)	PUNCT
ejpam-3889	327	8	and	and	CCONJ
ejpam-3889	327	9	s−1(n	s−1(n	NOUN
ejpam-3889	327	10	)	)	PUNCT
ejpam-3889	327	11	is	be	AUX
ejpam-3889	327	12	completely	completely	ADV
ejpam-3889	327	13	invariant	invariant	ADJ
ejpam-3889	327	14	,	,	PUNCT
ejpam-3889	327	15	so	so	ADV
ejpam-3889	327	16	y	y	PROPN
ejpam-3889	327	17	t	t	PROPN
ejpam-3889	328	1	=	=	PUNCT
ejpam-3889	329	1	[	[	X
ejpam-3889	329	2	s−1(g−1)](xs	s−1(g−1)](xs	ADP
ejpam-3889	329	3	)	)	PUNCT
ejpam-3889	329	4	∈	∈	PROPN
ejpam-3889	329	5	s−1(n	s−1(n	PROPN
ejpam-3889	329	6	)	)	PUNCT
ejpam-3889	329	7	,	,	PUNCT
ejpam-3889	329	8	thus	thus	ADV
ejpam-3889	329	9	s−1(f	s−1(f	PROPN
ejpam-3889	329	10	)	)	PUNCT
ejpam-3889	329	11	is	be	AUX
ejpam-3889	329	12	an	an	DET
ejpam-3889	329	13	automorphisme	automorphisme	NOUN
ejpam-3889	329	14	,	,	PUNCT
ejpam-3889	329	15	consequentely	consequentely	ADV
ejpam-3889	329	16	s−1(n	s−1(n	PROPN
ejpam-3889	329	17	)	)	PUNCT
ejpam-3889	329	18	is	be	AUX
ejpam-3889	329	19	cohopfian	cohopfian	ADJ
ejpam-3889	329	20	.	.	PUNCT
ejpam-3889	330	1	reciprocally	reciprocally	PROPN
ejpam-3889	330	2	,	,	PUNCT
ejpam-3889	330	3	suppose	suppose	VERB
ejpam-3889	330	4	that	that	SCONJ
ejpam-3889	330	5	s−1(n	s−1(n	PROPN
ejpam-3889	330	6	)	)	PUNCT
ejpam-3889	330	7	is	be	AUX
ejpam-3889	330	8	cohopfian	cohopfian	ADJ
ejpam-3889	330	9	and	and	CCONJ
ejpam-3889	330	10	let	let	VERB
ejpam-3889	330	11	s−1(f	s−1(f	PROPN
ejpam-3889	330	12	)	)	PUNCT
ejpam-3889	330	13	:	:	PUNCT
ejpam-3889	331	1	s−1(m	s−1(m	PROPN
ejpam-3889	331	2	)	)	PUNCT
ejpam-3889	331	3	−→	−→	NOUN
ejpam-3889	331	4	s−1(m	s−1(m	PROPN
ejpam-3889	331	5	)	)	PUNCT
ejpam-3889	331	6	a	a	DET
ejpam-3889	331	7	graded	grade	VERB
ejpam-3889	331	8	monomorphism	monomorphism	NOUN
ejpam-3889	331	9	.	.	PUNCT
ejpam-3889	332	1	then	then	ADV
ejpam-3889	332	2	s−1(f|s−1(n	s−1(f|s−1(n	PROPN
ejpam-3889	332	3	)	)	PUNCT
ejpam-3889	332	4	)	)	PUNCT
ejpam-3889	332	5	is	be	AUX
ejpam-3889	332	6	a	a	DET
ejpam-3889	332	7	graded	grade	VERB
ejpam-3889	332	8	monomorphism	monomorphism	NOUN
ejpam-3889	332	9	of	of	ADP
ejpam-3889	332	10	s−1(n	s−1(n	PROPN
ejpam-3889	332	11	)	)	PUNCT
ejpam-3889	332	12	.	.	PUNCT
ejpam-3889	333	1	thus	thus	ADV
ejpam-3889	333	2	,	,	PUNCT
ejpam-3889	333	3	s−1(f	s−1(f	PROPN
ejpam-3889	333	4	)	)	PUNCT
ejpam-3889	333	5	∈	∈	PROPN
ejpam-3889	333	6	aut(s−1(n	aut(s−1(n	PROPN
ejpam-3889	333	7	)	)	PUNCT
ejpam-3889	333	8	)	)	PUNCT
ejpam-3889	333	9	,	,	PUNCT
ejpam-3889	333	10	hence	hence	ADV
ejpam-3889	333	11	[	[	X
ejpam-3889	333	12	s−1(f)](s−1(n	s−1(f)](s−1(n	NOUN
ejpam-3889	333	13	)	)	PUNCT
ejpam-3889	333	14	)	)	PUNCT
ejpam-3889	334	1	=	=	SYM
ejpam-3889	334	2	s−1(n	s−1(n	PROPN
ejpam-3889	334	3	)	)	PUNCT
ejpam-3889	334	4	.	.	PUNCT
ejpam-3889	335	1	as	as	SCONJ
ejpam-3889	335	2	s−1(m	s−1(m	PROPN
ejpam-3889	335	3	)	)	PUNCT
ejpam-3889	335	4	is	be	AUX
ejpam-3889	335	5	quasiinjective	quasiinjective	ADJ
ejpam-3889	335	6	,	,	PUNCT
ejpam-3889	335	7	then	then	ADV
ejpam-3889	335	8	there	there	PRON
ejpam-3889	335	9	exists	exist	VERB
ejpam-3889	335	10	s−1(l	s−1(l	PROPN
ejpam-3889	335	11	)	)	PUNCT
ejpam-3889	335	12	a	a	DET
ejpam-3889	335	13	submodule	submodule	NOUN
ejpam-3889	335	14	of	of	ADP
ejpam-3889	335	15	s−1(m	s−1(m	PROPN
ejpam-3889	335	16	)	)	PUNCT
ejpam-3889	335	17	such	such	ADJ
ejpam-3889	335	18	that	that	SCONJ
ejpam-3889	335	19	s−1(m	s−1(m	PROPN
ejpam-3889	335	20	)	)	PUNCT
ejpam-3889	335	21	=	=	PUNCT
ejpam-3889	336	1	[	[	X
ejpam-3889	336	2	s−1(f)](s−1(m))⊕	s−1(f)](s−1(m))⊕	VERB
ejpam-3889	336	3	s−1(l	s−1(l	PROPN
ejpam-3889	336	4	)	)	PUNCT
ejpam-3889	336	5	.	.	PUNCT
ejpam-3889	337	1	thus	thus	ADV
ejpam-3889	337	2	,	,	PUNCT
ejpam-3889	337	3	we	we	PRON
ejpam-3889	337	4	have	have	VERB
ejpam-3889	337	5	0	0	NUM
ejpam-3889	337	6	=	=	SYM
ejpam-3889	338	1	[	[	X
ejpam-3889	338	2	s−1(f)](s−1(n))∩s−1(l	s−1(f)](s−1(n))∩s−1(l	NOUN
ejpam-3889	338	3	)	)	PUNCT
ejpam-3889	338	4	=	=	SYM
ejpam-3889	338	5	s−1(n)∩s−1(l	s−1(n)∩s−1(l	NOUN
ejpam-3889	338	6	)	)	PUNCT
ejpam-3889	338	7	,	,	PUNCT
ejpam-3889	338	8	since	since	SCONJ
ejpam-3889	338	9	s−1(n	s−1(n	PROPN
ejpam-3889	338	10	)	)	PUNCT
ejpam-3889	338	11	is	be	AUX
ejpam-3889	338	12	essential	essential	ADJ
ejpam-3889	338	13	,	,	PUNCT
ejpam-3889	338	14	then	then	ADV
ejpam-3889	338	15	s−1(l	s−1(l	PROPN
ejpam-3889	338	16	)	)	PUNCT
ejpam-3889	339	1	=	=	SYM
ejpam-3889	339	2	0	0	NUM
ejpam-3889	339	3	,	,	PUNCT
ejpam-3889	339	4	hence	hence	ADV
ejpam-3889	339	5	s−1(m	s−1(m	PROPN
ejpam-3889	339	6	)	)	PUNCT
ejpam-3889	339	7	=	=	PUNCT
ejpam-3889	339	8	s−1[(f)](s−1(m	s−1[(f)](s−1(m	NOUN
ejpam-3889	339	9	)	)	PUNCT
ejpam-3889	339	10	)	)	PUNCT
ejpam-3889	339	11	,	,	PUNCT
ejpam-3889	339	12	thus	thus	ADV
ejpam-3889	339	13	s−1(f	s−1(f	PROPN
ejpam-3889	339	14	)	)	PUNCT
ejpam-3889	339	15	is	be	AUX
ejpam-3889	339	16	an	an	DET
ejpam-3889	339	17	epimorphism	epimorphism	NOUN
ejpam-3889	339	18	,	,	PUNCT
ejpam-3889	339	19	so	so	ADV
ejpam-3889	339	20	s−1(m	s−1(m	PROPN
ejpam-3889	339	21	)	)	PUNCT
ejpam-3889	339	22	is	be	AUX
ejpam-3889	339	23	cohopfian	cohopfian	ADJ
ejpam-3889	339	24	left	leave	VERB
ejpam-3889	339	25	s−1a	s−1a	NOUN
ejpam-3889	339	26	-	-	PUNCT
ejpam-3889	339	27	module	module	NOUN
ejpam-3889	339	28	.	.	PUNCT
ejpam-3889	340	1	theorem	theorem	NOUN
ejpam-3889	340	2	13	13	NUM
ejpam-3889	340	3	.	.	PUNCT
ejpam-3889	341	1	let	let	VERB
ejpam-3889	341	2	m	m	PRON
ejpam-3889	341	3	be	be	AUX
ejpam-3889	341	4	a	a	DET
ejpam-3889	341	5	graded	grade	VERB
ejpam-3889	341	6	quasi	quasi	NOUN
ejpam-3889	341	7	-	-	NOUN
ejpam-3889	341	8	projective	projective	ADJ
ejpam-3889	341	9	left	leave	VERB
ejpam-3889	341	10	a	a	DET
ejpam-3889	341	11	-	-	PUNCT
ejpam-3889	341	12	module	module	NOUN
ejpam-3889	341	13	,	,	PUNCT
ejpam-3889	341	14	n	n	CCONJ
ejpam-3889	341	15	a	a	PRON
ejpam-3889	341	16	superfluous	superfluous	ADJ
ejpam-3889	341	17	and	and	CCONJ
ejpam-3889	341	18	completely	completely	ADV
ejpam-3889	341	19	invariant	invariant	ADJ
ejpam-3889	341	20	graded	grade	VERB
ejpam-3889	341	21	sub	sub	NOUN
ejpam-3889	341	22	-	-	NOUN
ejpam-3889	341	23	module	module	NOUN
ejpam-3889	341	24	of	of	ADP
ejpam-3889	341	25	m	m	PROPN
ejpam-3889	341	26	and	and	CCONJ
ejpam-3889	341	27	s	s	VERB
ejpam-3889	341	28	a	a	DET
ejpam-3889	341	29	saturated	saturate	VERB
ejpam-3889	341	30	multiplicative	multiplicative	ADJ
ejpam-3889	341	31	part	part	NOUN
ejpam-3889	341	32	formed	form	VERB
ejpam-3889	341	33	by	by	ADP
ejpam-3889	341	34	the	the	DET
ejpam-3889	341	35	non	non	ADJ
ejpam-3889	341	36	-	-	ADJ
ejpam-3889	341	37	zero	zero	ADJ
ejpam-3889	341	38	homogeneous	homogeneous	ADJ
ejpam-3889	341	39	elements	element	NOUN
ejpam-3889	341	40	of	of	ADP
ejpam-3889	341	41	a	a	DET
ejpam-3889	341	42	verifying	verifying	NOUN
ejpam-3889	341	43	the	the	DET
ejpam-3889	341	44	left	left	ADJ
ejpam-3889	341	45	ore	ore	NOUN
ejpam-3889	341	46	conditions	condition	NOUN
ejpam-3889	341	47	.	.	PUNCT
ejpam-3889	342	1	then	then	ADV
ejpam-3889	342	2	the	the	DET
ejpam-3889	342	3	left	left	ADJ
ejpam-3889	342	4	s−1(a)-modules	s−1(a)-modules	NUM
ejpam-3889	342	5	s−1(n	s−1(n	NOUN
ejpam-3889	342	6	)	)	PUNCT
ejpam-3889	342	7	is	be	AUX
ejpam-3889	342	8	hopfian	hopfian	ADJ
ejpam-3889	342	9	if	if	SCONJ
ejpam-3889	342	10	,	,	PUNCT
ejpam-3889	342	11	and	and	CCONJ
ejpam-3889	342	12	only	only	ADV
ejpam-3889	342	13	if	if	SCONJ
ejpam-3889	342	14	,	,	PUNCT
ejpam-3889	342	15	s−1(m	s−1(m	PROPN
ejpam-3889	342	16	/	/	SYM
ejpam-3889	342	17	n	n	CCONJ
ejpam-3889	342	18	)	)	PUNCT
ejpam-3889	342	19	is	be	AUX
ejpam-3889	342	20	hopfian	hopfian	ADJ
ejpam-3889	342	21	.	.	PUNCT
ejpam-3889	343	1	proof	proof	NOUN
ejpam-3889	343	2	.	.	PUNCT
ejpam-3889	344	1	suppose	suppose	VERB
ejpam-3889	344	2	that	that	SCONJ
ejpam-3889	344	3	s−1(m	s−1(m	PROPN
ejpam-3889	344	4	/	/	SYM
ejpam-3889	344	5	n	n	CCONJ
ejpam-3889	344	6	)	)	PUNCT
ejpam-3889	344	7	is	be	AUX
ejpam-3889	344	8	hopfian	hopfian	ADJ
ejpam-3889	344	9	and	and	CCONJ
ejpam-3889	344	10	let	let	VERB
ejpam-3889	344	11	s−1(f	s−1(f	PROPN
ejpam-3889	344	12	)	)	PUNCT
ejpam-3889	344	13	:	:	PUNCT
ejpam-3889	345	1	s−1(m	s−1(m	PROPN
ejpam-3889	345	2	)	)	PUNCT
ejpam-3889	345	3	−→	−→	NOUN
ejpam-3889	345	4	s−1(m	s−1(m	PROPN
ejpam-3889	345	5	)	)	PUNCT
ejpam-3889	345	6	a	a	DET
ejpam-3889	345	7	graded	grade	VERB
ejpam-3889	345	8	epimorphism	epimorphism	NOUN
ejpam-3889	345	9	.	.	PUNCT
ejpam-3889	346	1	as	as	SCONJ
ejpam-3889	346	2	s−1(n	s−1(n	PROPN
ejpam-3889	346	3	)	)	PUNCT
ejpam-3889	346	4	is	be	AUX
ejpam-3889	346	5	completely	completely	ADV
ejpam-3889	346	6	invariant	invariant	ADJ
ejpam-3889	346	7	,	,	PUNCT
ejpam-3889	346	8	then	then	ADV
ejpam-3889	346	9	[	[	X
ejpam-3889	346	10	s−1(f)](n	s−1(f)](n	NOUN
ejpam-3889	346	11	)	)	PUNCT
ejpam-3889	346	12	⊂	⊂	PROPN
ejpam-3889	346	13	s−1(s−1(n	s−1(s−1(n	NOUN
ejpam-3889	346	14	)	)	PUNCT
ejpam-3889	346	15	)	)	PUNCT
ejpam-3889	346	16	,	,	PUNCT
ejpam-3889	346	17	implies	imply	VERB
ejpam-3889	346	18	s−1(f	s−1(f	PROPN
ejpam-3889	346	19	)	)	PUNCT
ejpam-3889	346	20	s.	s.	PROPN
ejpam-3889	346	21	a.	a.	PROPN
ejpam-3889	346	22	balde	balde	PROPN
ejpam-3889	346	23	,	,	PUNCT
ejpam-3889	346	24	m.	m.	PROPN
ejpam-3889	346	25	b.	b.	PROPN
ejpam-3889	346	26	maaouia	maaouia	PROPN
ejpam-3889	346	27	,	,	PUNCT
ejpam-3889	346	28	a.	a.	NOUN
ejpam-3889	346	29	o.	o.	NOUN
ejpam-3889	346	30	chbih	chbih	PROPN
ejpam-3889	346	31	/	/	SYM
ejpam-3889	346	32	eur	eur	PROPN
ejpam-3889	346	33	.	.	PUNCT
ejpam-3889	347	1	j.	j.	PROPN
ejpam-3889	347	2	pure	pure	PROPN
ejpam-3889	347	3	appl	appl	PROPN
ejpam-3889	347	4	.	.	PROPN
ejpam-3889	347	5	math	math	PROPN
ejpam-3889	347	6	,	,	PUNCT
ejpam-3889	347	7	14	14	NUM
ejpam-3889	347	8	(	(	PUNCT
ejpam-3889	347	9	2	2	NUM
ejpam-3889	347	10	)	)	PUNCT
ejpam-3889	347	11	(	(	PUNCT
ejpam-3889	347	12	2021	2021	NUM
ejpam-3889	347	13	)	)	PUNCT
ejpam-3889	347	14	,	,	PUNCT
ejpam-3889	347	15	404	404	NUM
ejpam-3889	347	16	-	-	SYM
ejpam-3889	347	17	422	422	NUM
ejpam-3889	347	18	416	416	NUM
ejpam-3889	347	19	induces	induce	NOUN
ejpam-3889	347	20	a	a	DET
ejpam-3889	347	21	graded	grade	VERB
ejpam-3889	347	22	epimorphism	epimorphism	NOUN
ejpam-3889	347	23	s−1((f	s−1((f	NOUN
ejpam-3889	347	24	)	)	PUNCT
ejpam-3889	347	25	)	)	PUNCT
ejpam-3889	347	26	:	:	PUNCT
ejpam-3889	348	1	s−1(m	s−1(m	PROPN
ejpam-3889	348	2	/	/	SYM
ejpam-3889	348	3	n	n	CCONJ
ejpam-3889	348	4	)	)	PUNCT
ejpam-3889	348	5	−→	−→	NOUN
ejpam-3889	348	6	s−1(m	s−1(m	PROPN
ejpam-3889	348	7	/	/	SYM
ejpam-3889	348	8	n	n	CCONJ
ejpam-3889	348	9	)	)	PUNCT
ejpam-3889	348	10	,	,	PUNCT
ejpam-3889	348	11	since	since	SCONJ
ejpam-3889	348	12	s−1(m	s−1(m	PROPN
ejpam-3889	348	13	/	/	SYM
ejpam-3889	348	14	n	n	CCONJ
ejpam-3889	348	15	)	)	PUNCT
ejpam-3889	348	16	is	be	AUX
ejpam-3889	348	17	hopfian	hopfian	ADJ
ejpam-3889	348	18	,	,	PUNCT
ejpam-3889	348	19	then	then	ADV
ejpam-3889	348	20	s−1((f	s−1((f	PROPN
ejpam-3889	348	21	)	)	PUNCT
ejpam-3889	348	22	)	)	PUNCT
ejpam-3889	349	1	is	be	AUX
ejpam-3889	349	2	a	a	DET
ejpam-3889	349	3	graded	grade	VERB
ejpam-3889	349	4	automorphism	automorphism	NOUN
ejpam-3889	349	5	.	.	PUNCT
ejpam-3889	350	1	put	put	VERB
ejpam-3889	350	2	s−1(k	s−1(k	PROPN
ejpam-3889	350	3	)	)	PUNCT
ejpam-3889	350	4	=	=	SYM
ejpam-3889	350	5	ker(s−1(f	ker(s−1(f	PROPN
ejpam-3889	350	6	)	)	PUNCT
ejpam-3889	350	7	)	)	PUNCT
ejpam-3889	350	8	and	and	CCONJ
ejpam-3889	350	9	s−1(π	s−1(π	PROPN
ejpam-3889	350	10	)	)	PUNCT
ejpam-3889	350	11	:	:	PUNCT
ejpam-3889	351	1	s−1(m	s−1(m	PROPN
ejpam-3889	351	2	)	)	PUNCT
ejpam-3889	352	1	−→	−→	NOUN
ejpam-3889	352	2	s−1(m	s−1(m	PROPN
ejpam-3889	352	3	/	/	SYM
ejpam-3889	352	4	n	n	CCONJ
ejpam-3889	352	5	)	)	PUNCT
ejpam-3889	352	6	the	the	DET
ejpam-3889	352	7	canonical	canonical	ADJ
ejpam-3889	352	8	projection	projection	NOUN
ejpam-3889	352	9	,	,	PUNCT
ejpam-3889	352	10	we	we	PRON
ejpam-3889	352	11	have	have	AUX
ejpam-3889	352	12	:	:	PUNCT
ejpam-3889	352	13	s−1((f	s−1((f	ADJ
ejpam-3889	352	14	)	)	PUNCT
ejpam-3889	352	15	)	)	PUNCT
ejpam-3889	353	1	◦	◦	NOUN
ejpam-3889	354	1	[	[	X
ejpam-3889	354	2	s−1(π)](s−1(k	s−1(π)](s−1(k	NOUN
ejpam-3889	354	3	)	)	PUNCT
ejpam-3889	354	4	)	)	PUNCT
ejpam-3889	355	1	=	=	PUNCT
ejpam-3889	355	2	s−1(π	s−1(π	PROPN
ejpam-3889	355	3	◦	◦	NOUN
ejpam-3889	355	4	f(k	f(k	VERB
ejpam-3889	355	5	)	)	PUNCT
ejpam-3889	355	6	)	)	PUNCT
ejpam-3889	356	1	=	=	SYM
ejpam-3889	356	2	0	0	PUNCT
ejpam-3889	356	3	indeed	indeed	ADV
ejpam-3889	356	4	,	,	PUNCT
ejpam-3889	356	5	∀xs	∀xs	PROPN
ejpam-3889	356	6	∈	∈	PROPN
ejpam-3889	356	7	s	s	PART
ejpam-3889	356	8	−1(k	−1(k	NOUN
ejpam-3889	356	9	)	)	PUNCT
ejpam-3889	356	10	,	,	PUNCT
ejpam-3889	356	11	we	we	PRON
ejpam-3889	356	12	have	have	VERB
ejpam-3889	356	13	:	:	PUNCT
ejpam-3889	356	14	[	[	X
ejpam-3889	356	15	s−1(f	s−1(f	X
ejpam-3889	356	16	◦	◦	NOUN
ejpam-3889	356	17	π)](xs	π)](xs	X
ejpam-3889	356	18	)	)	PUNCT
ejpam-3889	356	19	=	=	PUNCT
ejpam-3889	357	1	[	[	X
ejpam-3889	357	2	s−1(π	s−1(π	NOUN
ejpam-3889	357	3	◦	◦	NOUN
ejpam-3889	357	4	f)](xs	f)](xs	ADP
ejpam-3889	357	5	)	)	PUNCT
ejpam-3889	357	6	,	,	PUNCT
ejpam-3889	357	7	so	so	CCONJ
ejpam-3889	357	8	[	[	X
ejpam-3889	357	9	s−1(π	s−1(π	NOUN
ejpam-3889	357	10	◦	◦	NOUN
ejpam-3889	357	11	f)](xs	f)](xs	ADP
ejpam-3889	357	12	)	)	PUNCT
ejpam-3889	357	13	=	=	PUNCT
ejpam-3889	358	1	[	[	X
ejpam-3889	358	2	s−1(π)](f(x)s	s−1(π)](f(x)s	NOUN
ejpam-3889	358	3	)	)	PUNCT
ejpam-3889	359	1	=	=	PUNCT
ejpam-3889	360	1	[	[	X
ejpam-3889	360	2	s−1(π)](0s	s−1(π)](0s	NOUN
ejpam-3889	360	3	)	)	PUNCT
ejpam-3889	360	4	=	=	SYM
ejpam-3889	360	5	0	0	NUM
ejpam-3889	360	6	,	,	PUNCT
ejpam-3889	360	7	thus	thus	ADV
ejpam-3889	360	8	[	[	X
ejpam-3889	360	9	s−1(f	s−1(f	PROPN
ejpam-3889	360	10	◦	◦	NOUN
ejpam-3889	360	11	π)](k	π)](k	PUNCT
ejpam-3889	360	12	)	)	PUNCT
ejpam-3889	360	13	=	=	SYM
ejpam-3889	361	1	0	0	X
ejpam-3889	361	2	.	.	PUNCT
ejpam-3889	362	1	we	we	PRON
ejpam-3889	362	2	have	have	VERB
ejpam-3889	362	3	:	:	PUNCT
ejpam-3889	363	1	[	[	X
ejpam-3889	363	2	s−1(f	s−1(f	PROPN
ejpam-3889	363	3	◦	◦	NOUN
ejpam-3889	363	4	π)](k	π)](k	PUNCT
ejpam-3889	363	5	)	)	PUNCT
ejpam-3889	364	1	=	=	PUNCT
ejpam-3889	365	1	s−1[π	s−1[π	VERB
ejpam-3889	365	2	◦	◦	NOUN
ejpam-3889	365	3	f	f	X
ejpam-3889	366	1	]	]	X
ejpam-3889	366	2	(	(	PUNCT
ejpam-3889	366	3	k	k	NOUN
ejpam-3889	366	4	)	)	PUNCT
ejpam-3889	366	5	=	=	SYM
ejpam-3889	366	6	0	0	PUNCT
ejpam-3889	367	1	=	=	NOUN
ejpam-3889	367	2	⇒	⇒	X
ejpam-3889	367	3	[	[	X
ejpam-3889	367	4	s−1(f)](π(k	s−1(f)](π(k	NUM
ejpam-3889	367	5	)	)	PUNCT
ejpam-3889	367	6	)	)	PUNCT
ejpam-3889	368	1	=	=	SYM
ejpam-3889	368	2	0	0	PUNCT
ejpam-3889	369	1	=	=	NOUN
ejpam-3889	369	2	⇒	⇒	NOUN
ejpam-3889	369	3	[	[	X
ejpam-3889	369	4	s−1(π)](k	s−1(π)](k	X
ejpam-3889	369	5	)	)	PUNCT
ejpam-3889	369	6	⊂	⊂	PROPN
ejpam-3889	369	7	s−1(n	s−1(n	PROPN
ejpam-3889	369	8	)	)	PUNCT
ejpam-3889	369	9	=	=	NOUN
ejpam-3889	369	10	⇒	⇒	NOUN
ejpam-3889	369	11	s−1(k	s−1(k	PROPN
ejpam-3889	369	12	)	)	PUNCT
ejpam-3889	369	13	⊂	⊂	PROPN
ejpam-3889	369	14	s−1(n	s−1(n	PROPN
ejpam-3889	369	15	)	)	PUNCT
ejpam-3889	369	16	since	since	SCONJ
ejpam-3889	369	17	m	m	PROPN
ejpam-3889	369	18	is	be	AUX
ejpam-3889	369	19	graded	grade	VERB
ejpam-3889	369	20	quasiprojective	quasiprojective	ADJ
ejpam-3889	369	21	=	=	NOUN
ejpam-3889	369	22	⇒	⇒	NOUN
ejpam-3889	369	23	s−1(m	s−1(m	PROPN
ejpam-3889	369	24	)	)	PUNCT
ejpam-3889	369	25	is	be	AUX
ejpam-3889	369	26	graded	grade	VERB
ejpam-3889	369	27	quasi	quasi	ADJ
ejpam-3889	369	28	-	-	NOUN
ejpam-3889	369	29	projective	projective	ADJ
ejpam-3889	369	30	,	,	PUNCT
ejpam-3889	369	31	there	there	PRON
ejpam-3889	369	32	exists	exist	VERB
ejpam-3889	369	33	a	a	DET
ejpam-3889	369	34	graded	grade	VERB
ejpam-3889	369	35	endomorphism	endomorphism	PROPN
ejpam-3889	369	36	s−1(s	s−1(s	PROPN
ejpam-3889	369	37	)	)	PUNCT
ejpam-3889	369	38	:	:	PUNCT
ejpam-3889	370	1	s−1(m	s−1(m	PROPN
ejpam-3889	370	2	)	)	PUNCT
ejpam-3889	370	3	−→	−→	NOUN
ejpam-3889	370	4	s−1(m	s−1(m	PROPN
ejpam-3889	370	5	)	)	PUNCT
ejpam-3889	370	6	such	such	ADJ
ejpam-3889	370	7	that	that	DET
ejpam-3889	370	8	s−1(f	s−1(f	PROPN
ejpam-3889	370	9	◦	◦	PROPN
ejpam-3889	370	10	s	s	PART
ejpam-3889	370	11	)	)	PUNCT
ejpam-3889	370	12	=	=	SYM
ejpam-3889	370	13	s−1(ids−1(m	s−1(ids−1(m	PROPN
ejpam-3889	370	14	)	)	PUNCT
ejpam-3889	370	15	)	)	PUNCT
ejpam-3889	370	16	,	,	PUNCT
ejpam-3889	370	17	this	this	PRON
ejpam-3889	370	18	implies	imply	VERB
ejpam-3889	370	19	s−1(m	s−1(m	PROPN
ejpam-3889	370	20	)	)	PUNCT
ejpam-3889	370	21	=	=	SYM
ejpam-3889	370	22	s−1(k	s−1(k	PROPN
ejpam-3889	370	23	⊕	⊕	PROPN
ejpam-3889	370	24	im(s	im(s	PROPN
ejpam-3889	370	25	)	)	PUNCT
ejpam-3889	370	26	)	)	PUNCT
ejpam-3889	370	27	,	,	PUNCT
ejpam-3889	370	28	or	or	CCONJ
ejpam-3889	370	29	k	k	X
ejpam-3889	370	30	=	=	PUNCT
ejpam-3889	370	31	n	n	PROPN
ejpam-3889	370	32	and	and	CCONJ
ejpam-3889	370	33	s−1(n	s−1(n	NOUN
ejpam-3889	370	34	)	)	PUNCT
ejpam-3889	370	35	is	be	AUX
ejpam-3889	370	36	superfluous	superfluous	ADJ
ejpam-3889	370	37	in	in	ADP
ejpam-3889	370	38	s−1(m	s−1(m	PROPN
ejpam-3889	370	39	)	)	PUNCT
ejpam-3889	370	40	,	,	PUNCT
ejpam-3889	370	41	then	then	ADV
ejpam-3889	370	42	s−1(m	s−1(m	PROPN
ejpam-3889	370	43	)	)	PUNCT
ejpam-3889	370	44	=	=	SYM
ejpam-3889	370	45	s−1(im(s	s−1(im(s	NOUN
ejpam-3889	370	46	)	)	PUNCT
ejpam-3889	370	47	)	)	PUNCT
ejpam-3889	370	48	,	,	PUNCT
ejpam-3889	370	49	so	so	ADV
ejpam-3889	370	50	s−1(k	s−1(k	PROPN
ejpam-3889	370	51	)	)	PUNCT
ejpam-3889	370	52	=	=	SYM
ejpam-3889	370	53	ker(s−1(f	ker(s−1(f	PROPN
ejpam-3889	370	54	)	)	PUNCT
ejpam-3889	370	55	)	)	PUNCT
ejpam-3889	371	1	=	=	SYM
ejpam-3889	371	2	0	0	NUM
ejpam-3889	371	3	,	,	PUNCT
ejpam-3889	371	4	thus	thus	ADV
ejpam-3889	371	5	s−1(f	s−1(f	PROPN
ejpam-3889	371	6	)	)	PUNCT
ejpam-3889	371	7	is	be	AUX
ejpam-3889	371	8	a	a	DET
ejpam-3889	371	9	monomorphism	monomorphism	NOUN
ejpam-3889	371	10	,	,	PUNCT
ejpam-3889	371	11	finally	finally	ADV
ejpam-3889	371	12	,	,	PUNCT
ejpam-3889	371	13	s−1(f	s−1(f	PROPN
ejpam-3889	371	14	)	)	PUNCT
ejpam-3889	371	15	is	be	AUX
ejpam-3889	371	16	a	a	DET
ejpam-3889	371	17	monomorphism	monomorphism	NOUN
ejpam-3889	371	18	,	,	PUNCT
ejpam-3889	371	19	so	so	ADV
ejpam-3889	371	20	s−1(m	s−1(m	PROPN
ejpam-3889	371	21	)	)	PUNCT
ejpam-3889	371	22	is	be	AUX
ejpam-3889	371	23	a	a	DET
ejpam-3889	371	24	graded	grade	VERB
ejpam-3889	371	25	hopfian	hopfian	NOUN
ejpam-3889	371	26	left	leave	VERB
ejpam-3889	371	27	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	371	28	.	.	PUNCT
ejpam-3889	372	1	reciprocally	reciprocally	PROPN
ejpam-3889	372	2	,	,	PUNCT
ejpam-3889	372	3	suppose	suppose	VERB
ejpam-3889	372	4	that	that	SCONJ
ejpam-3889	372	5	s−1(m	s−1(m	PROPN
ejpam-3889	372	6	)	)	PUNCT
ejpam-3889	372	7	is	be	AUX
ejpam-3889	372	8	hopfian	hopfian	ADJ
ejpam-3889	372	9	,	,	PUNCT
ejpam-3889	372	10	show	show	VERB
ejpam-3889	372	11	that	that	SCONJ
ejpam-3889	372	12	s−1(m	s−1(m	PROPN
ejpam-3889	372	13	/	/	SYM
ejpam-3889	372	14	n	n	CCONJ
ejpam-3889	372	15	)	)	PUNCT
ejpam-3889	372	16	is	be	AUX
ejpam-3889	372	17	hopfian	hopfian	ADJ
ejpam-3889	372	18	.	.	PUNCT
ejpam-3889	373	1	let	let	VERB
ejpam-3889	373	2	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	373	3	)	)	PUNCT
ejpam-3889	373	4	:	:	PUNCT
ejpam-3889	374	1	s−1(m	s−1(m	PROPN
ejpam-3889	374	2	/	/	SYM
ejpam-3889	374	3	n	n	CCONJ
ejpam-3889	374	4	)	)	PUNCT
ejpam-3889	374	5	−→	−→	NOUN
ejpam-3889	374	6	s−1(m	s−1(m	PROPN
ejpam-3889	374	7	/	/	SYM
ejpam-3889	374	8	n	n	CCONJ
ejpam-3889	374	9	)	)	PUNCT
ejpam-3889	374	10	a	a	DET
ejpam-3889	374	11	graded	grade	VERB
ejpam-3889	374	12	epimorphism	epimorphism	NOUN
ejpam-3889	374	13	of	of	ADP
ejpam-3889	374	14	left	left	ADJ
ejpam-3889	374	15	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	374	16	,	,	PUNCT
ejpam-3889	374	17	as	as	SCONJ
ejpam-3889	374	18	m	m	PROPN
ejpam-3889	374	19	is	be	AUX
ejpam-3889	374	20	quasi	quasi	ADJ
ejpam-3889	374	21	-	-	NOUN
ejpam-3889	374	22	projective	projective	ADJ
ejpam-3889	374	23	,	,	PUNCT
ejpam-3889	374	24	then	then	ADV
ejpam-3889	374	25	,	,	PUNCT
ejpam-3889	374	26	s−1(m	s−1(m	PROPN
ejpam-3889	374	27	)	)	PUNCT
ejpam-3889	374	28	is	be	AUX
ejpam-3889	374	29	quasi	quasi	ADJ
ejpam-3889	374	30	-	-	ADJ
ejpam-3889	374	31	projective	projective	ADJ
ejpam-3889	374	32	.	.	PUNCT
ejpam-3889	375	1	consider	consider	VERB
ejpam-3889	375	2	s−1(π	s−1(π	PROPN
ejpam-3889	375	3	)	)	PUNCT
ejpam-3889	375	4	:	:	PUNCT
ejpam-3889	376	1	s−1(m	s−1(m	PROPN
ejpam-3889	376	2	)	)	PUNCT
ejpam-3889	376	3	−→	−→	NOUN
ejpam-3889	376	4	s−1(m	s−1(m	PROPN
ejpam-3889	376	5	/	/	SYM
ejpam-3889	376	6	n	n	CCONJ
ejpam-3889	376	7	)	)	PUNCT
ejpam-3889	376	8	,	,	PUNCT
ejpam-3889	376	9	then	then	ADV
ejpam-3889	376	10	there	there	PRON
ejpam-3889	376	11	exists	exist	VERB
ejpam-3889	376	12	s−1(f	s−1(f	PROPN
ejpam-3889	376	13	)	)	PUNCT
ejpam-3889	376	14	∈	∈	PROPN
ejpam-3889	376	15	end(s−1(m	end(s−1(m	PROPN
ejpam-3889	376	16	)	)	PUNCT
ejpam-3889	376	17	)	)	PUNCT
ejpam-3889	376	18	such	such	ADJ
ejpam-3889	376	19	that	that	DET
ejpam-3889	376	20	s−1(π	s−1(π	PROPN
ejpam-3889	376	21	◦	◦	NOUN
ejpam-3889	376	22	f	f	X
ejpam-3889	376	23	)	)	PUNCT
ejpam-3889	376	24	=	=	SYM
ejpam-3889	376	25	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	376	26	◦	◦	PROPN
ejpam-3889	376	27	π	π	PROPN
ejpam-3889	376	28	)	)	PUNCT
ejpam-3889	376	29	.	.	PUNCT
ejpam-3889	377	1	since	since	SCONJ
ejpam-3889	377	2	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	377	3	)	)	PUNCT
ejpam-3889	377	4	is	be	AUX
ejpam-3889	377	5	an	an	DET
ejpam-3889	377	6	epimorphism	epimorphism	NOUN
ejpam-3889	377	7	,	,	PUNCT
ejpam-3889	377	8	∀	∀	X
ejpam-3889	377	9	(	(	PUNCT
ejpam-3889	378	1	xs	xs	NOUN
ejpam-3889	378	2	)	)	PUNCT
ejpam-3889	378	3	∈	∈	PROPN
ejpam-3889	378	4	s−1(m	s−1(m	PROPN
ejpam-3889	378	5	/	/	SYM
ejpam-3889	378	6	n),∃	n),∃	PROPN
ejpam-3889	378	7	(	(	PUNCT
ejpam-3889	378	8	yt	yt	PROPN
ejpam-3889	378	9	)	)	PUNCT
ejpam-3889	378	10	∈	∈	PROPN
ejpam-3889	378	11	s	s	PART
ejpam-3889	378	12	−1(m	−1(m	NUM
ejpam-3889	378	13	/	/	SYM
ejpam-3889	378	14	n	n	CCONJ
ejpam-3889	378	15	)	)	PUNCT
ejpam-3889	378	16	such	such	ADJ
ejpam-3889	378	17	that	that	SCONJ
ejpam-3889	378	18	[	[	X
ejpam-3889	378	19	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	378	20	)	)	PUNCT
ejpam-3889	379	1	=	=	PUNCT
ejpam-3889	380	1	x	x	SYM
ejpam-3889	380	2	s	s	X
ejpam-3889	380	3	=	=	X
ejpam-3889	381	1	[	[	X
ejpam-3889	381	2	s−1(ϕ)](π(y)t	s−1(ϕ)](π(y)t	PROPN
ejpam-3889	381	3	)	)	PUNCT
ejpam-3889	382	1	[	[	X
ejpam-3889	382	2	s−1(π	s−1(π	NOUN
ejpam-3889	382	3	◦	◦	NOUN
ejpam-3889	382	4	f)](yt	f)](yt	PUNCT
ejpam-3889	382	5	)	)	PUNCT
ejpam-3889	383	1	=	=	PUNCT
ejpam-3889	384	1	[	[	X
ejpam-3889	384	2	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	384	3	◦	◦	NOUN
ejpam-3889	384	4	π)](yt	π)](yt	X
ejpam-3889	384	5	)	)	PUNCT
ejpam-3889	385	1	[	[	X
ejpam-3889	385	2	s−1(π	s−1(π	NOUN
ejpam-3889	385	3	)	)	PUNCT
ejpam-3889	385	4	]	]	PUNCT
ejpam-3889	385	5	(	(	PUNCT
ejpam-3889	385	6	(	(	PUNCT
ejpam-3889	385	7	f(x))s	f(x))s	X
ejpam-3889	385	8	)	)	PUNCT
ejpam-3889	385	9	=	=	PUNCT
ejpam-3889	386	1	[	[	X
ejpam-3889	386	2	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	386	3	)	)	PUNCT
ejpam-3889	387	1	=	=	VERB
ejpam-3889	387	2	⇒	⇒	X
ejpam-3889	387	3	[	[	X
ejpam-3889	387	4	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	387	5	)	)	PUNCT
ejpam-3889	387	6	=	=	PUNCT
ejpam-3889	388	1	[	[	X
ejpam-3889	388	2	s−1(f)](yt	s−1(f)](yt	X
ejpam-3889	388	3	)	)	PUNCT
ejpam-3889	388	4	=	=	PUNCT
ejpam-3889	389	1	x	x	SYM
ejpam-3889	389	2	s	s	NOUN
ejpam-3889	389	3	=	=	NOUN
ejpam-3889	389	4	⇒	⇒	NOUN
ejpam-3889	389	5	(	(	PUNCT
ejpam-3889	389	6	f(y)t	f(y)t	PROPN
ejpam-3889	389	7	−	−	PUNCT
ejpam-3889	389	8	x	x	SYM
ejpam-3889	389	9	s	s	NOUN
ejpam-3889	389	10	=	=	SYM
ejpam-3889	389	11	0	0	PUNCT
ejpam-3889	389	12	=	=	NOUN
ejpam-3889	389	13	⇒	⇒	PROPN
ejpam-3889	389	14	f(y	f(y	NOUN
ejpam-3889	389	15	)	)	PUNCT
ejpam-3889	389	16	t	t	NOUN
ejpam-3889	389	17	−	−	NOUN
ejpam-3889	390	1	x	x	SYM
ejpam-3889	390	2	s	s	X
ejpam-3889	390	3	∈	∈	NOUN
ejpam-3889	390	4	s	s	PART
ejpam-3889	390	5	−1(n	−1(n	NOUN
ejpam-3889	390	6	)	)	PUNCT
ejpam-3889	390	7	,	,	PUNCT
ejpam-3889	390	8	then	then	ADV
ejpam-3889	390	9	s−1(m	s−1(m	PROPN
ejpam-3889	390	10	)	)	PUNCT
ejpam-3889	390	11	=	=	SYM
ejpam-3889	390	12	im(s−1(f	im(s−1(f	PROPN
ejpam-3889	390	13	)	)	PUNCT
ejpam-3889	390	14	)	)	PUNCT
ejpam-3889	391	1	+	+	CCONJ
ejpam-3889	391	2	s−1(n	s−1(n	NOUN
ejpam-3889	391	3	)	)	PUNCT
ejpam-3889	391	4	,	,	PUNCT
ejpam-3889	391	5	as	as	ADP
ejpam-3889	391	6	s−1(n	s−1(n	PROPN
ejpam-3889	391	7	)	)	PUNCT
ejpam-3889	391	8	is	be	AUX
ejpam-3889	391	9	superfluous	superfluous	ADJ
ejpam-3889	391	10	,	,	PUNCT
ejpam-3889	391	11	then	then	ADV
ejpam-3889	391	12	im(s−1(f	im(s−1(f	PROPN
ejpam-3889	391	13	)	)	PUNCT
ejpam-3889	391	14	)	)	PUNCT
ejpam-3889	392	1	=	=	SYM
ejpam-3889	392	2	s−1(m	s−1(m	PROPN
ejpam-3889	392	3	)	)	PUNCT
ejpam-3889	393	1	=	=	NOUN
ejpam-3889	393	2	⇒	⇒	PROPN
ejpam-3889	393	3	s−1(f	s−1(f	PROPN
ejpam-3889	393	4	)	)	PUNCT
ejpam-3889	393	5	is	be	AUX
ejpam-3889	393	6	a	a	DET
ejpam-3889	393	7	graded	grade	VERB
ejpam-3889	393	8	epimorphism	epimorphism	NOUN
ejpam-3889	393	9	.	.	PUNCT
ejpam-3889	394	1	so	so	ADV
ejpam-3889	394	2	,	,	PUNCT
ejpam-3889	394	3	s−1(f	s−1(f	PROPN
ejpam-3889	394	4	)	)	PUNCT
ejpam-3889	394	5	is	be	AUX
ejpam-3889	394	6	a	a	DET
ejpam-3889	394	7	graded	grade	VERB
ejpam-3889	394	8	automorphism	automorphism	NOUN
ejpam-3889	394	9	,	,	PUNCT
ejpam-3889	394	10	because	because	SCONJ
ejpam-3889	394	11	s−1(m	s−1(m	PROPN
ejpam-3889	394	12	)	)	PUNCT
ejpam-3889	394	13	is	be	AUX
ejpam-3889	394	14	hopfian	hopfian	ADJ
ejpam-3889	394	15	.	.	PUNCT
ejpam-3889	395	1	so	so	ADV
ejpam-3889	395	2	the	the	DET
ejpam-3889	395	3	restriction	restriction	NOUN
ejpam-3889	395	4	of	of	ADP
ejpam-3889	395	5	s−1(f	s−1(f	PROPN
ejpam-3889	395	6	)	)	PUNCT
ejpam-3889	395	7	over	over	ADP
ejpam-3889	395	8	s−1(n	s−1(n	PROPN
ejpam-3889	395	9	)	)	PUNCT
ejpam-3889	395	10	is	be	AUX
ejpam-3889	395	11	a	a	DET
ejpam-3889	395	12	graded	grade	VERB
ejpam-3889	395	13	automorphism	automorphism	NOUN
ejpam-3889	395	14	of	of	ADP
ejpam-3889	395	15	s−1(n	s−1(n	PROPN
ejpam-3889	395	16	)	)	PUNCT
ejpam-3889	395	17	.	.	PUNCT
ejpam-3889	396	1	if	if	SCONJ
ejpam-3889	396	2	[	[	X
ejpam-3889	396	3	s−1(ϕ)](xs	s−1(ϕ)](xs	X
ejpam-3889	396	4	)	)	PUNCT
ejpam-3889	396	5	=	=	PUNCT
ejpam-3889	397	1	[	[	X
ejpam-3889	397	2	s−1(f)](xs	s−1(f)](xs	PROPN
ejpam-3889	397	3	)	)	PUNCT
ejpam-3889	397	4	=	=	SYM
ejpam-3889	397	5	0	0	NUM
ejpam-3889	397	6	,	,	PUNCT
ejpam-3889	397	7	then	then	ADV
ejpam-3889	397	8	[	[	X
ejpam-3889	397	9	s−1(f)](xs	s−1(f)](xs	PROPN
ejpam-3889	397	10	)	)	PUNCT
ejpam-3889	397	11	∈	∈	PROPN
ejpam-3889	397	12	s−1(n	s−1(n	PROPN
ejpam-3889	397	13	)	)	PUNCT
ejpam-3889	397	14	,	,	PUNCT
ejpam-3889	397	15	or	or	CCONJ
ejpam-3889	397	16	s−1(n	s−1(n	NOUN
ejpam-3889	397	17	)	)	PUNCT
ejpam-3889	397	18	is	be	AUX
ejpam-3889	397	19	completely	completely	ADV
ejpam-3889	397	20	invariant	invariant	ADJ
ejpam-3889	397	21	,	,	PUNCT
ejpam-3889	397	22	then	then	ADV
ejpam-3889	397	23	x	x	X
ejpam-3889	397	24	s	s	PROPN
ejpam-3889	397	25	∈	∈	PROPN
ejpam-3889	397	26	s−1(n	s−1(n	PROPN
ejpam-3889	397	27	)	)	PUNCT
ejpam-3889	397	28	,	,	PUNCT
ejpam-3889	397	29	so	so	CCONJ
ejpam-3889	397	30	x	x	X
ejpam-3889	398	1	s	s	AUX
ejpam-3889	398	2	=	=	SYM
ejpam-3889	398	3	0	0	PUNCT
ejpam-3889	398	4	=	=	NOUN
ejpam-3889	398	5	⇒	⇒	NOUN
ejpam-3889	398	6	ker(s−1(ϕ	ker(s−1(ϕ	PROPN
ejpam-3889	398	7	)	)	PUNCT
ejpam-3889	398	8	)	)	PUNCT
ejpam-3889	399	1	=	=	SYM
ejpam-3889	399	2	s−1(n	s−1(n	PROPN
ejpam-3889	399	3	)	)	PUNCT
ejpam-3889	399	4	=	=	SYM
ejpam-3889	399	5	0	0	PUNCT
ejpam-3889	400	1	=	=	NOUN
ejpam-3889	400	2	⇒	⇒	NOUN
ejpam-3889	400	3	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	400	4	)	)	PUNCT
ejpam-3889	400	5	is	be	AUX
ejpam-3889	400	6	a	a	DET
ejpam-3889	400	7	monomorphism	monomorphism	NOUN
ejpam-3889	400	8	=	=	NOUN
ejpam-3889	400	9	⇒	⇒	NOUN
ejpam-3889	400	10	s−1(ϕ	s−1(ϕ	PROPN
ejpam-3889	400	11	)	)	PUNCT
ejpam-3889	400	12	is	be	AUX
ejpam-3889	400	13	an	an	DET
ejpam-3889	400	14	automorphism	automorphism	NOUN
ejpam-3889	400	15	,	,	PUNCT
ejpam-3889	400	16	lastly	lastly	ADV
ejpam-3889	400	17	s−1(m	s−1(m	PROPN
ejpam-3889	400	18	/	/	SYM
ejpam-3889	400	19	n	n	CCONJ
ejpam-3889	400	20	)	)	PUNCT
ejpam-3889	400	21	is	be	AUX
ejpam-3889	400	22	hopfian	hopfian	ADJ
ejpam-3889	400	23	,	,	PUNCT
ejpam-3889	400	24	hence	hence	ADV
ejpam-3889	400	25	s−1(m	s−1(m	PROPN
ejpam-3889	400	26	/	/	SYM
ejpam-3889	400	27	n	n	CCONJ
ejpam-3889	400	28	)	)	PUNCT
ejpam-3889	400	29	is	be	AUX
ejpam-3889	400	30	a	a	DET
ejpam-3889	400	31	hopfian	hopfian	PROPN
ejpam-3889	400	32	left	leave	VERB
ejpam-3889	400	33	s−1a	s−1a	NOUN
ejpam-3889	400	34	-	-	PUNCT
ejpam-3889	400	35	module	module	NOUN
ejpam-3889	400	36	.	.	PUNCT
ejpam-3889	401	1	5	5	X
ejpam-3889	401	2	.	.	X
ejpam-3889	401	3	localization	localization	NOUN
ejpam-3889	401	4	of	of	ADP
ejpam-3889	401	5	hopfian	hopfian	ADJ
ejpam-3889	401	6	and	and	CCONJ
ejpam-3889	401	7	cohopfian	cohopfian	ADJ
ejpam-3889	401	8	objects	object	NOUN
ejpam-3889	401	9	in	in	ADP
ejpam-3889	401	10	the	the	DET
ejpam-3889	401	11	comp	comp	NOUN
ejpam-3889	401	12	(	(	PUNCT
ejpam-3889	401	13	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	401	14	)	)	PUNCT
ejpam-3889	401	15	)	)	PUNCT
ejpam-3889	401	16	category	category	NOUN
ejpam-3889	401	17	definition	definition	NOUN
ejpam-3889	401	18	8	8	NUM
ejpam-3889	401	19	.	.	PUNCT
ejpam-3889	402	1	let	let	VERB
ejpam-3889	402	2	m∗	m∗	VERB
ejpam-3889	402	3	an	an	DET
ejpam-3889	402	4	object	object	NOUN
ejpam-3889	402	5	of	of	ADP
ejpam-3889	402	6	comp	comp	NOUN
ejpam-3889	402	7	(	(	PUNCT
ejpam-3889	402	8	agr(a−mod	agr(a−mod	PROPN
ejpam-3889	402	9	)	)	PUNCT
ejpam-3889	402	10	)	)	PUNCT
ejpam-3889	402	11	.	.	PUNCT
ejpam-3889	403	1	then	then	ADV
ejpam-3889	403	2	m∗	m∗	PROPN
ejpam-3889	403	3	is	be	AUX
ejpam-3889	403	4	said	say	VERB
ejpam-3889	403	5	to	to	PART
ejpam-3889	403	6	be	be	AUX
ejpam-3889	403	7	hopfian	hopfian	ADJ
ejpam-3889	403	8	(	(	PUNCT
ejpam-3889	403	9	resp	resp	NOUN
ejpam-3889	403	10	.	.	PUNCT
ejpam-3889	404	1	cohopfian	cohopfian	PROPN
ejpam-3889	404	2	)	)	PUNCT
ejpam-3889	405	1	s.	s.	PROPN
ejpam-3889	405	2	a.	a.	PROPN
ejpam-3889	405	3	balde	balde	PROPN
ejpam-3889	405	4	,	,	PUNCT
ejpam-3889	405	5	m.	m.	PROPN
ejpam-3889	405	6	b.	b.	PROPN
ejpam-3889	405	7	maaouia	maaouia	PROPN
ejpam-3889	405	8	,	,	PUNCT
ejpam-3889	405	9	a.	a.	NOUN
ejpam-3889	405	10	o.	o.	NOUN
ejpam-3889	405	11	chbih	chbih	PROPN
ejpam-3889	405	12	/	/	SYM
ejpam-3889	405	13	eur	eur	PROPN
ejpam-3889	405	14	.	.	PUNCT
ejpam-3889	406	1	j.	j.	PROPN
ejpam-3889	406	2	pure	pure	PROPN
ejpam-3889	406	3	appl	appl	PROPN
ejpam-3889	406	4	.	.	PROPN
ejpam-3889	406	5	math	math	PROPN
ejpam-3889	406	6	,	,	PUNCT
ejpam-3889	406	7	14	14	NUM
ejpam-3889	406	8	(	(	PUNCT
ejpam-3889	406	9	2	2	NUM
ejpam-3889	406	10	)	)	PUNCT
ejpam-3889	406	11	(	(	PUNCT
ejpam-3889	406	12	2021	2021	NUM
ejpam-3889	406	13	)	)	PUNCT
ejpam-3889	406	14	,	,	PUNCT
ejpam-3889	406	15	404	404	NUM
ejpam-3889	406	16	-	-	SYM
ejpam-3889	406	17	422	422	NUM
ejpam-3889	406	18	417	417	NUM
ejpam-3889	406	19	if	if	SCONJ
ejpam-3889	406	20	any	any	DET
ejpam-3889	406	21	epimorphism	epimorphism	NOUN
ejpam-3889	406	22	(	(	PUNCT
ejpam-3889	406	23	resp	resp	NOUN
ejpam-3889	406	24	.	.	PUNCT
ejpam-3889	407	1	monomorphism	monomorphism	NOUN
ejpam-3889	407	2	)	)	PUNCT
ejpam-3889	407	3	f∗	f∗	NOUN
ejpam-3889	407	4	of	of	ADP
ejpam-3889	407	5	m∗	m∗	NOUN
ejpam-3889	407	6	is	be	AUX
ejpam-3889	407	7	an	an	DET
ejpam-3889	407	8	automorphism	automorphism	NOUN
ejpam-3889	407	9	.	.	PUNCT
ejpam-3889	408	1	lemma	lemma	PROPN
ejpam-3889	408	2	2	2	X
ejpam-3889	408	3	.	.	PUNCT
ejpam-3889	409	1	let	let	VERB
ejpam-3889	409	2	m	m	PRON
ejpam-3889	409	3	be	be	AUX
ejpam-3889	409	4	a	a	DET
ejpam-3889	409	5	graded	grade	VERB
ejpam-3889	409	6	left	leave	VERB
ejpam-3889	409	7	a	a	DET
ejpam-3889	409	8	-	-	PUNCT
ejpam-3889	409	9	module	module	NOUN
ejpam-3889	409	10	,	,	PUNCT
ejpam-3889	409	11	f	f	X
ejpam-3889	409	12	:	:	PUNCT
ejpam-3889	409	13	m	m	VERB
ejpam-3889	409	14	−→	−→	ADJ
ejpam-3889	409	15	m	m	VERB
ejpam-3889	409	16	a	a	DET
ejpam-3889	409	17	graded	grade	VERB
ejpam-3889	409	18	morphism	morphism	NOUN
ejpam-3889	409	19	and	and	CCONJ
ejpam-3889	409	20	s	s	VERB
ejpam-3889	409	21	a	a	DET
ejpam-3889	409	22	saturated	saturate	VERB
ejpam-3889	409	23	multiplicative	multiplicative	ADJ
ejpam-3889	409	24	part	part	NOUN
ejpam-3889	409	25	formed	form	VERB
ejpam-3889	409	26	by	by	ADP
ejpam-3889	409	27	the	the	DET
ejpam-3889	409	28	non	non	ADJ
ejpam-3889	409	29	-	-	ADJ
ejpam-3889	409	30	zero	zero	ADJ
ejpam-3889	409	31	homogeneous	homogeneous	ADJ
ejpam-3889	409	32	elements	element	NOUN
ejpam-3889	409	33	of	of	ADP
ejpam-3889	409	34	a	a	DET
ejpam-3889	409	35	verifying	verifying	NOUN
ejpam-3889	409	36	the	the	DET
ejpam-3889	409	37	left	left	ADJ
ejpam-3889	409	38	ore	ore	NOUN
ejpam-3889	409	39	conditions	condition	NOUN
ejpam-3889	409	40	.	.	PUNCT
ejpam-3889	410	1	if	if	SCONJ
ejpam-3889	410	2	f	f	PROPN
ejpam-3889	410	3	is	be	AUX
ejpam-3889	410	4	an	an	DET
ejpam-3889	410	5	epimorphism	epimorphism	NOUN
ejpam-3889	410	6	(	(	PUNCT
ejpam-3889	410	7	respectively	respectively	ADV
ejpam-3889	410	8	a	a	DET
ejpam-3889	410	9	monomorphism	monomorphism	NOUN
ejpam-3889	410	10	)	)	PUNCT
ejpam-3889	410	11	,	,	PUNCT
ejpam-3889	410	12	then	then	ADV
ejpam-3889	410	13	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	410	14	)	)	PUNCT
ejpam-3889	410	15	)	)	PUNCT
ejpam-3889	410	16	is	be	AUX
ejpam-3889	410	17	an	an	DET
ejpam-3889	410	18	epimorphism	epimorphism	NOUN
ejpam-3889	410	19	(	(	PUNCT
ejpam-3889	410	20	respectively	respectively	ADV
ejpam-3889	410	21	monomorphism	monomorphism	NOUN
ejpam-3889	410	22	)	)	PUNCT
ejpam-3889	410	23	.	.	PUNCT
ejpam-3889	411	1	proof	proof	NOUN
ejpam-3889	411	2	.	.	PUNCT
ejpam-3889	412	1	since	since	SCONJ
ejpam-3889	412	2	f(n	f(n	PROPN
ejpam-3889	412	3	)	)	PUNCT
ejpam-3889	412	4	is	be	AUX
ejpam-3889	412	5	the	the	DET
ejpam-3889	412	6	induce	induce	NOUN
ejpam-3889	412	7	of	of	ADP
ejpam-3889	412	8	f	f	PROPN
ejpam-3889	412	9	,	,	PUNCT
ejpam-3889	412	10	then	then	ADV
ejpam-3889	412	11	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	412	12	)	)	PUNCT
ejpam-3889	412	13	)	)	PUNCT
ejpam-3889	413	1	is	be	AUX
ejpam-3889	413	2	an	an	DET
ejpam-3889	413	3	epimorphism	epimorphism	NOUN
ejpam-3889	413	4	(	(	PUNCT
ejpam-3889	413	5	respectively	respectively	ADV
ejpam-3889	413	6	a	a	DET
ejpam-3889	413	7	monomorphisme	monomorphisme	NOUN
ejpam-3889	413	8	)	)	PUNCT
ejpam-3889	413	9	.	.	PUNCT
ejpam-3889	414	1	lemma	lemma	PROPN
ejpam-3889	414	2	3	3	X
ejpam-3889	414	3	.	.	PUNCT
ejpam-3889	415	1	let	let	VERB
ejpam-3889	415	2	m	m	PRON
ejpam-3889	415	3	be	be	AUX
ejpam-3889	415	4	a	a	DET
ejpam-3889	415	5	graded	grade	VERB
ejpam-3889	415	6	left	leave	VERB
ejpam-3889	415	7	a	a	DET
ejpam-3889	415	8	-	-	PUNCT
ejpam-3889	415	9	module	module	NOUN
ejpam-3889	415	10	and	and	CCONJ
ejpam-3889	415	11	s	s	VERB
ejpam-3889	415	12	a	a	DET
ejpam-3889	415	13	saturated	saturate	VERB
ejpam-3889	415	14	multiplicative	multiplicative	ADJ
ejpam-3889	415	15	part	part	NOUN
ejpam-3889	415	16	formed	form	VERB
ejpam-3889	415	17	by	by	ADP
ejpam-3889	415	18	the	the	DET
ejpam-3889	415	19	non	non	ADJ
ejpam-3889	415	20	-	-	ADJ
ejpam-3889	415	21	zero	zero	ADJ
ejpam-3889	415	22	homogeneous	homogeneous	ADJ
ejpam-3889	415	23	elements	element	NOUN
ejpam-3889	415	24	of	of	ADP
ejpam-3889	415	25	a	a	DET
ejpam-3889	415	26	verifying	verifying	NOUN
ejpam-3889	415	27	the	the	DET
ejpam-3889	415	28	left	left	ADJ
ejpam-3889	415	29	ore	ore	NOUN
ejpam-3889	415	30	conditions	condition	NOUN
ejpam-3889	415	31	.	.	PUNCT
ejpam-3889	416	1	then	then	ADV
ejpam-3889	416	2	s−1(m	s−1(m	PROPN
ejpam-3889	416	3	)	)	PUNCT
ejpam-3889	416	4	is	be	AUX
ejpam-3889	416	5	a	a	DET
ejpam-3889	416	6	hopfian	hopfian	ADJ
ejpam-3889	416	7	(	(	PUNCT
ejpam-3889	416	8	respectively	respectively	ADV
ejpam-3889	416	9	cohopfian	cohopfian	ADJ
ejpam-3889	416	10	)	)	PUNCT
ejpam-3889	416	11	garded	garde	VERB
ejpam-3889	416	12	left	leave	VERB
ejpam-3889	416	13	a	a	DET
ejpam-3889	416	14	-	-	PUNCT
ejpam-3889	416	15	module	module	NOUN
ejpam-3889	416	16	if	if	SCONJ
ejpam-3889	416	17	any	any	DET
ejpam-3889	416	18	n	n	PRON
ejpam-3889	416	19	∈	∈	PROPN
ejpam-3889	416	20	z	z	PROPN
ejpam-3889	416	21	,	,	PUNCT
ejpam-3889	416	22	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	416	23	)	)	PUNCT
ejpam-3889	416	24	)	)	PUNCT
ejpam-3889	417	1	is	be	AUX
ejpam-3889	417	2	a	a	DET
ejpam-3889	417	3	hopfian	hopfian	ADJ
ejpam-3889	417	4	(	(	PUNCT
ejpam-3889	417	5	respectively	respectively	ADV
ejpam-3889	417	6	cohopfian	cohopfian	ADJ
ejpam-3889	417	7	)	)	PUNCT
ejpam-3889	417	8	graded	grade	VERB
ejpam-3889	417	9	left	leave	VERB
ejpam-3889	417	10	a	a	DET
ejpam-3889	417	11	-	-	PUNCT
ejpam-3889	417	12	module	module	NOUN
ejpam-3889	417	13	.	.	PUNCT
ejpam-3889	418	1	proof	proof	NOUN
ejpam-3889	418	2	.	.	PUNCT
ejpam-3889	419	1	let	let	VERB
ejpam-3889	419	2	s−1(g	s−1(g	PROPN
ejpam-3889	419	3	)	)	PUNCT
ejpam-3889	419	4	:	:	PUNCT
ejpam-3889	420	1	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	420	2	)	)	PUNCT
ejpam-3889	420	3	)	)	PUNCT
ejpam-3889	420	4	−→	−→	NOUN
ejpam-3889	420	5	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	420	6	)	)	PUNCT
ejpam-3889	420	7	)	)	PUNCT
ejpam-3889	420	8	be	be	AUX
ejpam-3889	420	9	a	a	DET
ejpam-3889	420	10	graded	grade	VERB
ejpam-3889	420	11	epimorphism	epimorphism	NOUN
ejpam-3889	420	12	(	(	PUNCT
ejpam-3889	420	13	respectively	respectively	ADV
ejpam-3889	420	14	a	a	DET
ejpam-3889	420	15	monomorphism	monomorphism	NOUN
ejpam-3889	420	16	)	)	PUNCT
ejpam-3889	420	17	,	,	PUNCT
ejpam-3889	420	18	since	since	SCONJ
ejpam-3889	420	19	s−1(m	s−1(m	PROPN
ejpam-3889	420	20	)	)	PUNCT
ejpam-3889	420	21	=	=	SYM
ejpam-3889	420	22	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	420	23	)	)	PUNCT
ejpam-3889	420	24	⊕	⊕	PROPN
ejpam-3889	421	1	k	k	PROPN
ejpam-3889	421	2	>	>	PROPN
ejpam-3889	421	3	n	n	PROPN
ejpam-3889	421	4	mn+k	mn+k	NOUN
ejpam-3889	421	5	)	)	PUNCT
ejpam-3889	421	6	∼=	∼=	PROPN
ejpam-3889	421	7	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	421	8	)	)	PUNCT
ejpam-3889	421	9	⊕	⊕	PROPN
ejpam-3889	422	1	k	k	PROPN
ejpam-3889	422	2	>	>	PROPN
ejpam-3889	422	3	n	n	PROPN
ejpam-3889	422	4	mn+k	mn+k	PROPN
ejpam-3889	422	5	)	)	PUNCT
ejpam-3889	422	6	.	.	PUNCT
ejpam-3889	423	1	put	put	VERB
ejpam-3889	423	2	s−1(f	s−1(f	PROPN
ejpam-3889	423	3	)	)	PUNCT
ejpam-3889	423	4	=	=	SYM
ejpam-3889	423	5	s−1(g	s−1(g	PROPN
ejpam-3889	423	6	)	)	PUNCT
ejpam-3889	424	1	+	+	CCONJ
ejpam-3889	424	2	s−1(idmn+k	s−1(idmn+k	PROPN
ejpam-3889	424	3	)	)	PUNCT
ejpam-3889	424	4	,	,	PUNCT
ejpam-3889	424	5	where	where	SCONJ
ejpam-3889	424	6	s−1(f	s−1(f	PROPN
ejpam-3889	424	7	)	)	PUNCT
ejpam-3889	424	8	is	be	AUX
ejpam-3889	424	9	an	an	DET
ejpam-3889	424	10	epimorphism	epimorphism	NOUN
ejpam-3889	424	11	(	(	PUNCT
ejpam-3889	424	12	respectively	respectively	ADV
ejpam-3889	424	13	a	a	DET
ejpam-3889	424	14	monomorphism	monomorphism	NOUN
ejpam-3889	424	15	)	)	PUNCT
ejpam-3889	424	16	of	of	ADP
ejpam-3889	424	17	s−1(m	s−1(m	PROPN
ejpam-3889	424	18	)	)	PUNCT
ejpam-3889	424	19	.	.	PUNCT
ejpam-3889	425	1	as	as	SCONJ
ejpam-3889	425	2	m	m	PROPN
ejpam-3889	425	3	is	be	AUX
ejpam-3889	425	4	hopfian	hopfian	ADJ
ejpam-3889	425	5	(	(	PUNCT
ejpam-3889	425	6	respectively	respectively	ADV
ejpam-3889	425	7	cohopfian	cohopfian	ADJ
ejpam-3889	425	8	)	)	PUNCT
ejpam-3889	425	9	this	this	PRON
ejpam-3889	425	10	implies	imply	VERB
ejpam-3889	425	11	that	that	SCONJ
ejpam-3889	425	12	f	f	PROPN
ejpam-3889	425	13	is	be	AUX
ejpam-3889	425	14	an	an	DET
ejpam-3889	425	15	isomorphism	isomorphism	NOUN
ejpam-3889	425	16	,	,	PUNCT
ejpam-3889	425	17	i.e	i.e	PROPN
ejpam-3889	425	18	s−1(f	s−1(f	PROPN
ejpam-3889	425	19	)	)	PUNCT
ejpam-3889	425	20	is	be	AUX
ejpam-3889	425	21	an	an	DET
ejpam-3889	425	22	isomorphism	isomorphism	NOUN
ejpam-3889	425	23	of	of	ADP
ejpam-3889	425	24	s−1(m	s−1(m	PROPN
ejpam-3889	425	25	)	)	PUNCT
ejpam-3889	425	26	,	,	PUNCT
ejpam-3889	425	27	thus	thus	ADV
ejpam-3889	425	28	s−1(g	s−1(g	PROPN
ejpam-3889	425	29	)	)	PUNCT
ejpam-3889	425	30	is	be	AUX
ejpam-3889	425	31	an	an	DET
ejpam-3889	425	32	isomorphism	isomorphism	NOUN
ejpam-3889	425	33	of	of	ADP
ejpam-3889	425	34	s−1m(n	s−1m(n	NOUN
ejpam-3889	425	35	)	)	PUNCT
ejpam-3889	425	36	)	)	PUNCT
ejpam-3889	425	37	,	,	PUNCT
ejpam-3889	425	38	so	so	ADV
ejpam-3889	425	39	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	425	40	)	)	PUNCT
ejpam-3889	425	41	)	)	PUNCT
ejpam-3889	425	42	is	be	AUX
ejpam-3889	425	43	hopfian	hopfian	NOUN
ejpam-3889	425	44	(	(	PUNCT
ejpam-3889	425	45	respectivement	respectivement	PROPN
ejpam-3889	425	46	cohopfian	cohopfian	NOUN
ejpam-3889	425	47	)	)	PUNCT
ejpam-3889	425	48	.	.	PUNCT
ejpam-3889	426	1	theorem	theorem	VERB
ejpam-3889	426	2	14	14	NUM
ejpam-3889	426	3	.	.	PUNCT
ejpam-3889	427	1	let	let	VERB
ejpam-3889	427	2	a	a	DET
ejpam-3889	427	3	be	be	AUX
ejpam-3889	427	4	a	a	DET
ejpam-3889	427	5	graded	grade	VERB
ejpam-3889	427	6	ring	ring	NOUN
ejpam-3889	427	7	,	,	PUNCT
ejpam-3889	427	8	s	s	VERB
ejpam-3889	427	9	a	a	DET
ejpam-3889	427	10	saturated	saturate	VERB
ejpam-3889	427	11	multiplicative	multiplicative	ADJ
ejpam-3889	427	12	part	part	NOUN
ejpam-3889	427	13	formed	form	VERB
ejpam-3889	427	14	by	by	ADP
ejpam-3889	427	15	the	the	DET
ejpam-3889	427	16	non	non	ADJ
ejpam-3889	427	17	-	-	ADJ
ejpam-3889	427	18	zero	zero	ADJ
ejpam-3889	427	19	homogeneous	homogeneous	ADJ
ejpam-3889	427	20	elements	element	NOUN
ejpam-3889	427	21	of	of	ADP
ejpam-3889	427	22	a	a	DET
ejpam-3889	427	23	verifying	verifying	NOUN
ejpam-3889	427	24	the	the	DET
ejpam-3889	427	25	left	left	ADJ
ejpam-3889	427	26	ore	ore	NOUN
ejpam-3889	427	27	conditions	condition	NOUN
ejpam-3889	427	28	,	,	PUNCT
ejpam-3889	427	29	m	m	VERB
ejpam-3889	427	30	a	a	DET
ejpam-3889	427	31	graded	grade	VERB
ejpam-3889	427	32	left	leave	VERB
ejpam-3889	427	33	a	a	DET
ejpam-3889	427	34	-	-	PUNCT
ejpam-3889	427	35	module	module	NOUN
ejpam-3889	427	36	and	and	CCONJ
ejpam-3889	427	37	m∗	m∗	VERB
ejpam-3889	427	38	the	the	DET
ejpam-3889	427	39	complex	complex	ADJ
ejpam-3889	427	40	sequence	sequence	NOUN
ejpam-3889	427	41	of	of	ADP
ejpam-3889	427	42	morphisms	morphism	NOUN
ejpam-3889	427	43	of	of	ADP
ejpam-3889	427	44	graded	grade	VERB
ejpam-3889	427	45	left	leave	VERB
ejpam-3889	427	46	a	a	DET
ejpam-3889	427	47	-	-	PUNCT
ejpam-3889	427	48	modules	module	NOUN
ejpam-3889	427	49	associated	associate	VERB
ejpam-3889	427	50	with	with	ADP
ejpam-3889	427	51	m	m	PROPN
ejpam-3889	427	52	.	.	PUNCT
ejpam-3889	428	1	then	then	ADV
ejpam-3889	428	2	,	,	PUNCT
ejpam-3889	428	3	if	if	SCONJ
ejpam-3889	428	4	the	the	DET
ejpam-3889	428	5	complexe	complexe	PROPN
ejpam-3889	428	6	complex	complex	PROPN
ejpam-3889	428	7	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	428	8	)	)	PUNCT
ejpam-3889	428	9	of	of	ADP
ejpam-3889	428	10	morphisms	morphism	NOUN
ejpam-3889	428	11	of	of	ADP
ejpam-3889	428	12	graded	grade	VERB
ejpam-3889	428	13	left	leave	VERB
ejpam-3889	428	14	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	428	15	is	be	AUX
ejpam-3889	428	16	hopfian	hopfian	ADJ
ejpam-3889	428	17	,	,	PUNCT
ejpam-3889	428	18	implies	imply	VERB
ejpam-3889	428	19	that	that	SCONJ
ejpam-3889	428	20	m∗	m∗	PROPN
ejpam-3889	428	21	is	be	AUX
ejpam-3889	428	22	hopfian	hopfian	ADJ
ejpam-3889	428	23	.	.	PUNCT
ejpam-3889	429	1	proof	proof	NOUN
ejpam-3889	429	2	.	.	PUNCT
ejpam-3889	430	1	let	let	VERB
ejpam-3889	430	2	f∗	f∗	NOUN
ejpam-3889	430	3	:	:	PUNCT
ejpam-3889	430	4	m∗	m∗	VERB
ejpam-3889	430	5	−→	−→	ADJ
ejpam-3889	430	6	m∗	m∗	NOUN
ejpam-3889	430	7	be	be	AUX
ejpam-3889	430	8	an	an	DET
ejpam-3889	430	9	epimorphism	epimorphism	NOUN
ejpam-3889	430	10	of	of	ADP
ejpam-3889	430	11	m∗.	m∗.	PROPN
ejpam-3889	430	12	then	then	ADV
ejpam-3889	430	13	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	430	14	)	)	PUNCT
ejpam-3889	430	15	)	)	PUNCT
ejpam-3889	430	16	:	:	PUNCT
ejpam-3889	431	1	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	431	2	)	)	PUNCT
ejpam-3889	431	3	)	)	PUNCT
ejpam-3889	431	4	−→	−→	NOUN
ejpam-3889	431	5	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	431	6	)	)	PUNCT
ejpam-3889	431	7	)	)	PUNCT
ejpam-3889	431	8	is	be	AUX
ejpam-3889	431	9	a	a	DET
ejpam-3889	431	10	graded	grade	VERB
ejpam-3889	431	11	epimorphism	epimorphism	NOUN
ejpam-3889	431	12	of	of	ADP
ejpam-3889	431	13	graded	grade	VERB
ejpam-3889	431	14	left	leave	VERB
ejpam-3889	431	15	s−1(a)-module	s−1(a)-module	PROPN
ejpam-3889	431	16	for	for	ADP
ejpam-3889	431	17	all	all	DET
ejpam-3889	431	18	n	n	PRON
ejpam-3889	431	19	∈	∈	PROPN
ejpam-3889	431	20	z.	z.	X
ejpam-3889	431	21	as	as	ADP
ejpam-3889	431	22	s−1m(n	s−1m(n	NOUN
ejpam-3889	431	23	)	)	PUNCT
ejpam-3889	431	24	is	be	AUX
ejpam-3889	431	25	hopfian	hopfian	ADJ
ejpam-3889	431	26	for	for	ADP
ejpam-3889	431	27	all	all	DET
ejpam-3889	431	28	n	n	PRON
ejpam-3889	431	29	∈	∈	PROPN
ejpam-3889	431	30	z	z	NOUN
ejpam-3889	431	31	,	,	PUNCT
ejpam-3889	431	32	so	so	ADV
ejpam-3889	431	33	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	431	34	)	)	PUNCT
ejpam-3889	431	35	)	)	PUNCT
ejpam-3889	431	36	is	be	AUX
ejpam-3889	431	37	a	a	DET
ejpam-3889	431	38	graded	grade	VERB
ejpam-3889	431	39	automorphism	automorphism	NOUN
ejpam-3889	431	40	of	of	ADP
ejpam-3889	431	41	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	431	42	)	)	PUNCT
ejpam-3889	431	43	)	)	PUNCT
ejpam-3889	431	44	.	.	PUNCT
ejpam-3889	432	1	let	let	VERB
ejpam-3889	432	2	m1	m1	PROPN
ejpam-3889	432	3	and	and	CCONJ
ejpam-3889	432	4	m2	m2	PROPN
ejpam-3889	432	5	∈m(n	∈m(n	PROPN
ejpam-3889	432	6	)	)	PUNCT
ejpam-3889	432	7	such	such	ADJ
ejpam-3889	432	8	that	that	SCONJ
ejpam-3889	432	9	f(n)(m1	f(n)(m1	NOUN
ejpam-3889	432	10	)	)	PUNCT
ejpam-3889	432	11	=	=	SYM
ejpam-3889	432	12	f(n)(m2	f(n)(m2	X
ejpam-3889	432	13	)	)	PUNCT
ejpam-3889	433	1	=	=	NOUN
ejpam-3889	433	2	⇒	⇒	NOUN
ejpam-3889	433	3	f(n)(m1	f(n)(m1	NOUN
ejpam-3889	433	4	)	)	PUNCT
ejpam-3889	433	5	1	1	NUM
ejpam-3889	433	6	=	=	SYM
ejpam-3889	433	7	f(n)(m2	f(n)(m2	ADJ
ejpam-3889	433	8	)	)	PUNCT
ejpam-3889	433	9	1	1	NUM
ejpam-3889	434	1	=	=	NOUN
ejpam-3889	434	2	⇒	⇒	NOUN
ejpam-3889	434	3	[	[	X
ejpam-3889	434	4	s−1(f(n))](m1	s−1(f(n))](m1	NOUN
ejpam-3889	434	5	)	)	PUNCT
ejpam-3889	434	6	=	=	SYM
ejpam-3889	435	1	[	[	X
ejpam-3889	435	2	s−1(f(n))](m2	s−1(f(n))](m2	NOUN
ejpam-3889	435	3	)	)	PUNCT
ejpam-3889	435	4	=	=	NOUN
ejpam-3889	435	5	⇒	⇒	NOUN
ejpam-3889	435	6	m1	m1	PROPN
ejpam-3889	435	7	=	=	SYM
ejpam-3889	435	8	m2	m2	PROPN
ejpam-3889	435	9	thus	thus	ADV
ejpam-3889	435	10	f(n	f(n	PROPN
ejpam-3889	435	11	)	)	PUNCT
ejpam-3889	435	12	is	be	AUX
ejpam-3889	435	13	a	a	DET
ejpam-3889	435	14	graded	grade	VERB
ejpam-3889	435	15	automorphism	automorphism	NOUN
ejpam-3889	435	16	of	of	ADP
ejpam-3889	435	17	m(n	m(n	PROPN
ejpam-3889	435	18	)	)	PUNCT
ejpam-3889	435	19	for	for	ADP
ejpam-3889	435	20	any	any	DET
ejpam-3889	435	21	n	n	NOUN
ejpam-3889	435	22	∈	∈	PROPN
ejpam-3889	435	23	z	z	NOUN
ejpam-3889	435	24	,	,	PUNCT
ejpam-3889	435	25	so	so	ADV
ejpam-3889	435	26	m	m	NOUN
ejpam-3889	435	27	is	be	AUX
ejpam-3889	435	28	hopfian	hopfian	ADJ
ejpam-3889	435	29	.	.	PUNCT
ejpam-3889	436	1	s.	s.	PROPN
ejpam-3889	436	2	a.	a.	PROPN
ejpam-3889	436	3	balde	balde	PROPN
ejpam-3889	436	4	,	,	PUNCT
ejpam-3889	436	5	m.	m.	PROPN
ejpam-3889	436	6	b.	b.	PROPN
ejpam-3889	436	7	maaouia	maaouia	PROPN
ejpam-3889	436	8	,	,	PUNCT
ejpam-3889	436	9	a.	a.	NOUN
ejpam-3889	436	10	o.	o.	NOUN
ejpam-3889	436	11	chbih	chbih	PROPN
ejpam-3889	436	12	/	/	SYM
ejpam-3889	436	13	eur	eur	PROPN
ejpam-3889	436	14	.	.	PUNCT
ejpam-3889	437	1	j.	j.	PROPN
ejpam-3889	437	2	pure	pure	PROPN
ejpam-3889	437	3	appl	appl	PROPN
ejpam-3889	437	4	.	.	PROPN
ejpam-3889	437	5	math	math	PROPN
ejpam-3889	437	6	,	,	PUNCT
ejpam-3889	437	7	14	14	NUM
ejpam-3889	437	8	(	(	PUNCT
ejpam-3889	437	9	2	2	NUM
ejpam-3889	437	10	)	)	PUNCT
ejpam-3889	437	11	(	(	PUNCT
ejpam-3889	437	12	2021	2021	NUM
ejpam-3889	437	13	)	)	PUNCT
ejpam-3889	437	14	,	,	PUNCT
ejpam-3889	437	15	404	404	NUM
ejpam-3889	437	16	-	-	SYM
ejpam-3889	437	17	422	422	NUM
ejpam-3889	437	18	418	418	NUM
ejpam-3889	437	19	theorem	theorem	NOUN
ejpam-3889	437	20	15	15	NUM
ejpam-3889	437	21	.	.	PUNCT
ejpam-3889	438	1	let	let	VERB
ejpam-3889	438	2	a	a	DET
ejpam-3889	438	3	be	be	AUX
ejpam-3889	438	4	a	a	DET
ejpam-3889	438	5	graded	grade	VERB
ejpam-3889	438	6	ring	ring	NOUN
ejpam-3889	438	7	,	,	PUNCT
ejpam-3889	438	8	s	s	VERB
ejpam-3889	438	9	a	a	DET
ejpam-3889	438	10	saturated	saturate	VERB
ejpam-3889	438	11	multiplicative	multiplicative	ADJ
ejpam-3889	438	12	part	part	NOUN
ejpam-3889	438	13	formed	form	VERB
ejpam-3889	438	14	by	by	ADP
ejpam-3889	438	15	the	the	DET
ejpam-3889	438	16	non	non	ADJ
ejpam-3889	438	17	-	-	ADJ
ejpam-3889	438	18	zero	zero	ADJ
ejpam-3889	438	19	homogeneous	homogeneous	ADJ
ejpam-3889	438	20	elements	element	NOUN
ejpam-3889	438	21	of	of	ADP
ejpam-3889	438	22	a	a	DET
ejpam-3889	438	23	verifying	verifying	NOUN
ejpam-3889	438	24	the	the	DET
ejpam-3889	438	25	left	left	ADJ
ejpam-3889	438	26	ore	ore	NOUN
ejpam-3889	438	27	conditions	condition	NOUN
ejpam-3889	438	28	,	,	PUNCT
ejpam-3889	438	29	m	m	VERB
ejpam-3889	438	30	a	a	DET
ejpam-3889	438	31	graded	grade	VERB
ejpam-3889	438	32	left	leave	VERB
ejpam-3889	438	33	a	a	DET
ejpam-3889	438	34	-	-	PUNCT
ejpam-3889	438	35	module	module	NOUN
ejpam-3889	438	36	.	.	PUNCT
ejpam-3889	439	1	if	if	SCONJ
ejpam-3889	439	2	m∗	m∗	VERB
ejpam-3889	439	3	the	the	DET
ejpam-3889	439	4	complex	complex	ADJ
ejpam-3889	439	5	sequence	sequence	NOUN
ejpam-3889	439	6	associated	associate	VERB
ejpam-3889	439	7	with	with	ADP
ejpam-3889	439	8	m	m	PRON
ejpam-3889	439	9	,	,	PUNCT
ejpam-3889	439	10	is	be	AUX
ejpam-3889	439	11	cohopfian	cohopfian	ADJ
ejpam-3889	439	12	and	and	CCONJ
ejpam-3889	439	13	completely	completely	ADV
ejpam-3889	439	14	invariant	invariant	ADJ
ejpam-3889	439	15	,	,	PUNCT
ejpam-3889	439	16	then	then	ADV
ejpam-3889	439	17	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	439	18	)	)	PUNCT
ejpam-3889	439	19	,	,	PUNCT
ejpam-3889	439	20	the	the	DET
ejpam-3889	439	21	complex	complex	ADJ
ejpam-3889	439	22	sequence	sequence	NOUN
ejpam-3889	439	23	associated	associate	VERB
ejpam-3889	439	24	with	with	ADP
ejpam-3889	439	25	s−1(m	s−1(m	PROPN
ejpam-3889	439	26	)	)	PUNCT
ejpam-3889	439	27	is	be	AUX
ejpam-3889	439	28	cohopfian	cohopfian	ADJ
ejpam-3889	439	29	.	.	PUNCT
ejpam-3889	440	1	proof	proof	NOUN
ejpam-3889	440	2	.	.	PUNCT
ejpam-3889	441	1	let	let	VERB
ejpam-3889	441	2	g	g	NOUN
ejpam-3889	441	3	:	:	PUNCT
ejpam-3889	441	4	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	441	5	)	)	PUNCT
ejpam-3889	441	6	−→	−→	NOUN
ejpam-3889	441	7	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	441	8	)	)	PUNCT
ejpam-3889	441	9	be	be	VERB
ejpam-3889	441	10	a	a	DET
ejpam-3889	441	11	graded	grade	VERB
ejpam-3889	441	12	s−1(a)-morphism	s−1(a)-morphism	PROPN
ejpam-3889	441	13	,	,	PUNCT
ejpam-3889	441	14	we	we	PRON
ejpam-3889	441	15	remark	remark	VERB
ejpam-3889	441	16	also	also	ADV
ejpam-3889	441	17	that	that	SCONJ
ejpam-3889	441	18	g	g	PROPN
ejpam-3889	441	19	is	be	AUX
ejpam-3889	441	20	a	a	DET
ejpam-3889	441	21	graded	grade	VERB
ejpam-3889	441	22	a	a	DET
ejpam-3889	441	23	-	-	PUNCT
ejpam-3889	441	24	morphism	morphism	NOUN
ejpam-3889	441	25	.	.	PUNCT
ejpam-3889	442	1	since	since	SCONJ
ejpam-3889	442	2	m∗	m∗	NOUN
ejpam-3889	442	3	is	be	AUX
ejpam-3889	442	4	completely	completely	ADV
ejpam-3889	442	5	invariant	invariant	ADJ
ejpam-3889	442	6	complex	complex	ADJ
ejpam-3889	442	7	sequence	sequence	NOUN
ejpam-3889	442	8	of	of	ADP
ejpam-3889	442	9	left	left	ADJ
ejpam-3889	442	10	s−1amodules	s−1amodule	NOUN
ejpam-3889	442	11	,	,	PUNCT
ejpam-3889	442	12	so	so	ADV
ejpam-3889	442	13	g(m∗	g(m∗	ADV
ejpam-3889	442	14	)	)	PUNCT
ejpam-3889	442	15	⊂m∗.	⊂m∗.	NOUN
ejpam-3889	442	16	suppose	suppose	VERB
ejpam-3889	442	17	that	that	SCONJ
ejpam-3889	442	18	g	g	PROPN
ejpam-3889	442	19	is	be	AUX
ejpam-3889	442	20	an	an	DET
ejpam-3889	442	21	monomorphism	monomorphism	NOUN
ejpam-3889	442	22	.	.	PUNCT
ejpam-3889	443	1	thus	thus	ADV
ejpam-3889	443	2	the	the	DET
ejpam-3889	443	3	induce	induce	ADJ
ejpam-3889	443	4	morphism	morphism	NOUN
ejpam-3889	443	5	gind	gind	NOUN
ejpam-3889	443	6	:	:	PUNCT
ejpam-3889	443	7	m∗	m∗	VERB
ejpam-3889	443	8	−→	−→	ADJ
ejpam-3889	443	9	m∗	m∗	NOUN
ejpam-3889	443	10	is	be	AUX
ejpam-3889	443	11	a	a	DET
ejpam-3889	443	12	monomorphism	monomorphism	NOUN
ejpam-3889	443	13	of	of	ADP
ejpam-3889	443	14	m∗	m∗	NOUN
ejpam-3889	443	15	and	and	CCONJ
ejpam-3889	443	16	since	since	SCONJ
ejpam-3889	443	17	m∗	m∗	NOUN
ejpam-3889	443	18	is	be	AUX
ejpam-3889	443	19	cohopfian	cohopfian	ADJ
ejpam-3889	443	20	,	,	PUNCT
ejpam-3889	443	21	then	then	ADV
ejpam-3889	443	22	gind	gind	NOUN
ejpam-3889	443	23	is	be	AUX
ejpam-3889	443	24	an	an	DET
ejpam-3889	443	25	automorphism	automorphism	NOUN
ejpam-3889	443	26	graded	grade	VERB
ejpam-3889	443	27	of	of	ADP
ejpam-3889	443	28	m∗.	m∗.	NOUN
ejpam-3889	443	29	consider	consider	VERB
ejpam-3889	443	30	s−1(gind(n	s−1(gind(n	NOUN
ejpam-3889	443	31	)	)	PUNCT
ejpam-3889	443	32	)	)	PUNCT
ejpam-3889	443	33	:	:	PUNCT
ejpam-3889	443	34	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	443	35	)	)	PUNCT
ejpam-3889	443	36	)	)	PUNCT
ejpam-3889	443	37	−→	−→	NOUN
ejpam-3889	443	38	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	443	39	)	)	PUNCT
ejpam-3889	443	40	)	)	PUNCT
ejpam-3889	444	1	m	m	PROPN
ejpam-3889	444	2	s	s	VERB
ejpam-3889	444	3	7−→	7−→	NOUN
ejpam-3889	444	4	g(n)(m	g(n)(m	NOUN
ejpam-3889	444	5	)	)	PUNCT
ejpam-3889	444	6	s	s	PART
ejpam-3889	445	1	so	so	ADV
ejpam-3889	445	2	g(n)(m	g(n)(m	NOUN
ejpam-3889	445	3	)	)	PUNCT
ejpam-3889	445	4	s	s	PART
ejpam-3889	445	5	=	=	SYM
ejpam-3889	445	6	1	1	NUM
ejpam-3889	445	7	s	s	NOUN
ejpam-3889	445	8	g(n)(m	g(n)(m	NOUN
ejpam-3889	445	9	)	)	PUNCT
ejpam-3889	445	10	1	1	NUM
ejpam-3889	445	11	.	.	PUNCT
ejpam-3889	445	12	or	or	CCONJ
ejpam-3889	445	13	g(n	g(n	PROPN
ejpam-3889	445	14	)	)	PUNCT
ejpam-3889	445	15	is	be	AUX
ejpam-3889	445	16	a	a	DET
ejpam-3889	445	17	graded	grade	VERB
ejpam-3889	445	18	s−1(a)-morphism	s−1(a)-morphism	PROPN
ejpam-3889	445	19	,	,	PUNCT
ejpam-3889	445	20	thus	thus	ADV
ejpam-3889	445	21	1	1	NUM
ejpam-3889	445	22	s	s	NOUN
ejpam-3889	445	23	=	=	PUNCT
ejpam-3889	445	24	g(n)(m	g(n)(m	PROPN
ejpam-3889	445	25	)	)	PUNCT
ejpam-3889	445	26	1	1	NUM
ejpam-3889	445	27	=	=	SYM
ejpam-3889	445	28	g(n)(1s	g(n)(1s	PROPN
ejpam-3889	445	29	m	m	VERB
ejpam-3889	445	30	1	1	NUM
ejpam-3889	445	31	)	)	PUNCT
ejpam-3889	445	32	=	=	SYM
ejpam-3889	445	33	g(n)(ms	g(n)(ms	ADJ
ejpam-3889	445	34	)	)	PUNCT
ejpam-3889	446	1	=	=	NOUN
ejpam-3889	446	2	⇒	⇒	NOUN
ejpam-3889	446	3	s−1(gind(n	s−1(gind(n	NOUN
ejpam-3889	446	4	)	)	PUNCT
ejpam-3889	446	5	)	)	PUNCT
ejpam-3889	447	1	=	=	SYM
ejpam-3889	447	2	g(n	g(n	PROPN
ejpam-3889	447	3	)	)	PUNCT
ejpam-3889	447	4	g(n	g(n	PROPN
ejpam-3889	447	5	)	)	PUNCT
ejpam-3889	447	6	is	be	AUX
ejpam-3889	447	7	a	a	DET
ejpam-3889	447	8	graded	grade	VERB
ejpam-3889	447	9	monomorphism	monomorphism	NOUN
ejpam-3889	447	10	,	,	PUNCT
ejpam-3889	447	11	then	then	ADV
ejpam-3889	447	12	gind(n	gind(n	NOUN
ejpam-3889	447	13	)	)	PUNCT
ejpam-3889	447	14	is	be	AUX
ejpam-3889	447	15	a	a	DET
ejpam-3889	447	16	graded	grade	VERB
ejpam-3889	447	17	monomorphism	monomorphism	NOUN
ejpam-3889	447	18	.	.	PUNCT
ejpam-3889	448	1	as	as	ADP
ejpam-3889	448	2	m(n	m(n	NOUN
ejpam-3889	448	3	)	)	PUNCT
ejpam-3889	448	4	is	be	AUX
ejpam-3889	448	5	cohopfian	cohopfian	ADJ
ejpam-3889	448	6	for	for	ADP
ejpam-3889	448	7	all	all	DET
ejpam-3889	448	8	n	n	PRON
ejpam-3889	448	9	∈	∈	PROPN
ejpam-3889	448	10	z	z	NOUN
ejpam-3889	448	11	,	,	PUNCT
ejpam-3889	448	12	then	then	ADV
ejpam-3889	448	13	gind(n	gind(n	NOUN
ejpam-3889	448	14	)	)	PUNCT
ejpam-3889	448	15	is	be	AUX
ejpam-3889	448	16	a	a	DET
ejpam-3889	448	17	graded	grade	VERB
ejpam-3889	448	18	automorphism	automorphism	NOUN
ejpam-3889	448	19	of	of	ADP
ejpam-3889	448	20	m(n	m(n	PROPN
ejpam-3889	448	21	)	)	PUNCT
ejpam-3889	448	22	.	.	PUNCT
ejpam-3889	449	1	let	let	VERB
ejpam-3889	449	2	m′	m′	NOUN
ejpam-3889	449	3	s	s	PART
ejpam-3889	449	4	∈	∈	PROPN
ejpam-3889	449	5	s	s	PART
ejpam-3889	449	6	−1(m(n	−1(m(n	NOUN
ejpam-3889	449	7	)	)	PUNCT
ejpam-3889	449	8	)	)	PUNCT
ejpam-3889	450	1	=	=	VERB
ejpam-3889	450	2	⇒	⇒	VERB
ejpam-3889	450	3	m′	m′	PROPN
ejpam-3889	450	4	∈m(n	∈m(n	PROPN
ejpam-3889	450	5	)	)	PUNCT
ejpam-3889	450	6	,	,	PUNCT
ejpam-3889	450	7	so	so	CCONJ
ejpam-3889	450	8	there	there	PRON
ejpam-3889	450	9	exists	exist	VERB
ejpam-3889	450	10	m	m	PROPN
ejpam-3889	450	11	∈m(n	∈m(n	PROPN
ejpam-3889	450	12	)	)	PUNCT
ejpam-3889	450	13	:	:	PUNCT
ejpam-3889	450	14	gind(n)(m	gind(n)(m	NUM
ejpam-3889	450	15	)	)	PUNCT
ejpam-3889	450	16	=	=	PUNCT
ejpam-3889	450	17	m′	m′	NOUN
ejpam-3889	450	18	hence	hence	ADV
ejpam-3889	450	19	m	m	NOUN
ejpam-3889	450	20	s	s	NOUN
ejpam-3889	450	21	∈	∈	PROPN
ejpam-3889	450	22	s	s	PART
ejpam-3889	450	23	−1(m(n	−1(m(n	NOUN
ejpam-3889	450	24	)	)	PUNCT
ejpam-3889	450	25	)	)	PUNCT
ejpam-3889	450	26	,	,	PUNCT
ejpam-3889	451	1	[	[	X
ejpam-3889	451	2	s−1(gind(n))](ms	s−1(gind(n))](ms	NUM
ejpam-3889	451	3	)	)	PUNCT
ejpam-3889	451	4	=	=	SYM
ejpam-3889	451	5	g(n)(m	g(n)(m	PROPN
ejpam-3889	451	6	)	)	PUNCT
ejpam-3889	451	7	s	s	PART
ejpam-3889	451	8	=	=	PUNCT
ejpam-3889	451	9	m′	m′	X
ejpam-3889	451	10	s	s	PART
ejpam-3889	451	11	thus	thus	ADV
ejpam-3889	451	12	(	(	PUNCT
ejpam-3889	451	13	g(n)(ms	g(n)(ms	INTJ
ejpam-3889	451	14	)	)	PUNCT
ejpam-3889	451	15	)	)	PUNCT
ejpam-3889	452	1	=	=	PUNCT
ejpam-3889	452	2	m′	m′	NOUN
ejpam-3889	452	3	s	s	PART
ejpam-3889	452	4	.	.	PUNCT
ejpam-3889	453	1	hence	hence	ADV
ejpam-3889	453	2	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	453	3	)	)	PUNCT
ejpam-3889	453	4	)	)	PUNCT
ejpam-3889	454	1	for	for	ADP
ejpam-3889	454	2	all	all	DET
ejpam-3889	454	3	n	n	PRON
ejpam-3889	454	4	∈	∈	PROPN
ejpam-3889	454	5	z	z	NOUN
ejpam-3889	454	6	is	be	AUX
ejpam-3889	454	7	cohopfian	cohopfian	ADJ
ejpam-3889	454	8	,	,	PUNCT
ejpam-3889	454	9	consequentely	consequentely	ADJ
ejpam-3889	454	10	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	454	11	)	)	PUNCT
ejpam-3889	454	12	is	be	AUX
ejpam-3889	454	13	cohopfian	cohopfian	ADJ
ejpam-3889	454	14	.	.	PUNCT
ejpam-3889	455	1	theorem	theorem	PROPN
ejpam-3889	455	2	16	16	NUM
ejpam-3889	455	3	.	.	PUNCT
ejpam-3889	456	1	let	let	VERB
ejpam-3889	456	2	m	m	PRON
ejpam-3889	456	3	be	be	AUX
ejpam-3889	456	4	a	a	DET
ejpam-3889	456	5	noetherian	noetherian	ADJ
ejpam-3889	456	6	graded	grade	VERB
ejpam-3889	456	7	left	leave	VERB
ejpam-3889	456	8	a	a	DET
ejpam-3889	456	9	-	-	PUNCT
ejpam-3889	456	10	module	module	NOUN
ejpam-3889	456	11	,	,	PUNCT
ejpam-3889	456	12	s	s	VERB
ejpam-3889	456	13	a	a	DET
ejpam-3889	456	14	saturated	saturate	VERB
ejpam-3889	456	15	multiplicative	multiplicative	ADJ
ejpam-3889	456	16	part	part	NOUN
ejpam-3889	456	17	formed	form	VERB
ejpam-3889	456	18	by	by	ADP
ejpam-3889	456	19	the	the	DET
ejpam-3889	456	20	non	non	ADJ
ejpam-3889	456	21	-	-	ADJ
ejpam-3889	456	22	zero	zero	ADJ
ejpam-3889	456	23	homogeneous	homogeneous	ADJ
ejpam-3889	456	24	elements	element	NOUN
ejpam-3889	456	25	of	of	ADP
ejpam-3889	456	26	a	a	DET
ejpam-3889	456	27	verifying	verifying	NOUN
ejpam-3889	456	28	the	the	DET
ejpam-3889	456	29	left	left	ADJ
ejpam-3889	456	30	ore	ore	NOUN
ejpam-3889	456	31	conditions	condition	NOUN
ejpam-3889	456	32	,	,	PUNCT
ejpam-3889	456	33	n	n	CCONJ
ejpam-3889	456	34	a	a	DET
ejpam-3889	456	35	submodule	submodule	NOUN
ejpam-3889	456	36	of	of	ADP
ejpam-3889	456	37	m	m	PROPN
ejpam-3889	456	38	,	,	PUNCT
ejpam-3889	456	39	m∗	m∗	PROPN
ejpam-3889	456	40	is	be	AUX
ejpam-3889	456	41	a	a	DET
ejpam-3889	456	42	noetherian	noetherian	ADJ
ejpam-3889	456	43	quasi	quasi	ADJ
ejpam-3889	456	44	-	-	ADJ
ejpam-3889	456	45	injective	injective	ADJ
ejpam-3889	456	46	complex	complex	ADJ
ejpam-3889	456	47	sequence	sequence	NOUN
ejpam-3889	456	48	associated	associate	VERB
ejpam-3889	456	49	with	with	ADP
ejpam-3889	456	50	m	m	NOUN
ejpam-3889	456	51	and	and	CCONJ
ejpam-3889	456	52	n∗	n∗	PROPN
ejpam-3889	456	53	is	be	AUX
ejpam-3889	456	54	an	an	DET
ejpam-3889	456	55	essential	essential	ADJ
ejpam-3889	456	56	and	and	CCONJ
ejpam-3889	456	57	completely	completely	ADV
ejpam-3889	456	58	invariant	invariant	ADJ
ejpam-3889	456	59	complex	complex	ADJ
ejpam-3889	456	60	sub	sub	NOUN
ejpam-3889	456	61	-	-	NOUN
ejpam-3889	456	62	sequence	sequence	NOUN
ejpam-3889	456	63	of	of	ADP
ejpam-3889	456	64	m∗.	m∗.	PROPN
ejpam-3889	456	65	then	then	ADV
ejpam-3889	456	66	,	,	PUNCT
ejpam-3889	456	67	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	456	68	)	)	PUNCT
ejpam-3889	456	69	the	the	DET
ejpam-3889	456	70	complex	complex	ADJ
ejpam-3889	456	71	sequence	sequence	NOUN
ejpam-3889	456	72	of	of	ADP
ejpam-3889	456	73	morphisms	morphism	NOUN
ejpam-3889	456	74	of	of	ADP
ejpam-3889	456	75	left	left	ADJ
ejpam-3889	456	76	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	456	77	is	be	AUX
ejpam-3889	456	78	cohopfian	cohopfian	ADJ
ejpam-3889	456	79	if	if	SCONJ
ejpam-3889	456	80	,	,	PUNCT
ejpam-3889	456	81	and	and	CCONJ
ejpam-3889	456	82	only	only	ADV
ejpam-3889	456	83	,	,	PUNCT
ejpam-3889	456	84	if	if	SCONJ
ejpam-3889	456	85	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	456	86	)	)	PUNCT
ejpam-3889	456	87	is	be	AUX
ejpam-3889	456	88	cohopfian	cohopfian	ADJ
ejpam-3889	456	89	.	.	PUNCT
ejpam-3889	457	1	proof	proof	NOUN
ejpam-3889	457	2	.	.	PUNCT
ejpam-3889	458	1	suppose	suppose	VERB
ejpam-3889	458	2	that	that	SCONJ
ejpam-3889	458	3	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	458	4	)	)	PUNCT
ejpam-3889	458	5	is	be	AUX
ejpam-3889	458	6	cohopfian	cohopfian	ADJ
ejpam-3889	458	7	and	and	CCONJ
ejpam-3889	458	8	let	let	VERB
ejpam-3889	458	9	s−1(f∗	s−1(f∗	NOUN
ejpam-3889	458	10	)	)	PUNCT
ejpam-3889	458	11	:	:	PUNCT
ejpam-3889	459	1	s−1(n∗	s−1(n∗	NUM
ejpam-3889	459	2	)	)	PUNCT
ejpam-3889	459	3	−→	−→	NOUN
ejpam-3889	459	4	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	459	5	)	)	PUNCT
ejpam-3889	459	6	be	be	AUX
ejpam-3889	459	7	a	a	DET
ejpam-3889	459	8	monomorphism	monomorphism	NOUN
ejpam-3889	459	9	.	.	PUNCT
ejpam-3889	460	1	as	as	SCONJ
ejpam-3889	460	2	m∗	m∗	NOUN
ejpam-3889	460	3	is	be	AUX
ejpam-3889	460	4	noetherian	noetherian	ADJ
ejpam-3889	460	5	,	,	PUNCT
ejpam-3889	460	6	then	then	ADV
ejpam-3889	460	7	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	460	8	)	)	PUNCT
ejpam-3889	460	9	is	be	AUX
ejpam-3889	460	10	noetherian	noetherian	ADJ
ejpam-3889	460	11	quasi	quasi	NOUN
ejpam-3889	460	12	-	-	ADJ
ejpam-3889	460	13	injective	injective	ADJ
ejpam-3889	460	14	,	,	PUNCT
ejpam-3889	460	15	so	so	SCONJ
ejpam-3889	460	16	for	for	ADP
ejpam-3889	460	17	all	all	DET
ejpam-3889	460	18	n	n	PRON
ejpam-3889	460	19	∈	∈	PROPN
ejpam-3889	460	20	z	z	PROPN
ejpam-3889	460	21	,	,	PUNCT
ejpam-3889	460	22	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	460	23	)	)	PUNCT
ejpam-3889	460	24	)	)	PUNCT
ejpam-3889	460	25	is	be	AUX
ejpam-3889	460	26	noetherian	noetherian	ADJ
ejpam-3889	460	27	and	and	CCONJ
ejpam-3889	460	28	quasi	quasi	ADJ
ejpam-3889	460	29	-	-	ADJ
ejpam-3889	460	30	injective	injective	ADJ
ejpam-3889	460	31	.	.	PUNCT
ejpam-3889	461	1	then	then	ADV
ejpam-3889	461	2	,	,	PUNCT
ejpam-3889	461	3	there	there	PRON
ejpam-3889	461	4	exists	exist	VERB
ejpam-3889	461	5	s−1(g(n	s−1(g(n	ADV
ejpam-3889	461	6	)	)	PUNCT
ejpam-3889	461	7	)	)	PUNCT
ejpam-3889	462	1	∈	∈	PROPN
ejpam-3889	462	2	end(s−1(m(n	end(s−1(m(n	NOUN
ejpam-3889	462	3	)	)	PUNCT
ejpam-3889	462	4	)	)	PUNCT
ejpam-3889	462	5	)	)	PUNCT
ejpam-3889	463	1	∀n	∀n	NUM
ejpam-3889	464	1	∈	∈	PROPN
ejpam-3889	464	2	z	z	NOUN
ejpam-3889	464	3	such	such	ADJ
ejpam-3889	464	4	that	that	DET
ejpam-3889	464	5	s−1(g|s−1(n(n	s−1(g|s−1(n(n	NOUN
ejpam-3889	464	6	)	)	PUNCT
ejpam-3889	464	7	)	)	PUNCT
ejpam-3889	464	8	)	)	PUNCT
ejpam-3889	465	1	=	=	SYM
ejpam-3889	465	2	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	465	3	)	)	PUNCT
ejpam-3889	465	4	)	)	PUNCT
ejpam-3889	465	5	.	.	PUNCT
ejpam-3889	466	1	s−1(g(n	s−1(g(n	ADV
ejpam-3889	466	2	)	)	PUNCT
ejpam-3889	466	3	)	)	PUNCT
ejpam-3889	466	4	is	be	AUX
ejpam-3889	466	5	injective	injective	ADJ
ejpam-3889	466	6	since	since	SCONJ
ejpam-3889	466	7	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	466	8	)	)	PUNCT
ejpam-3889	466	9	is	be	AUX
ejpam-3889	466	10	essential	essential	ADJ
ejpam-3889	466	11	in	in	ADP
ejpam-3889	466	12	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	466	13	)	)	PUNCT
ejpam-3889	466	14	,	,	PUNCT
ejpam-3889	466	15	hence	hence	ADV
ejpam-3889	466	16	∀n	∀n	X
ejpam-3889	466	17	∈	∈	PROPN
ejpam-3889	466	18	z	z	PROPN
ejpam-3889	466	19	,	,	PUNCT
ejpam-3889	466	20	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	466	21	)	)	PUNCT
ejpam-3889	466	22	)	)	PUNCT
ejpam-3889	466	23	is	be	AUX
ejpam-3889	466	24	essential	essential	ADJ
ejpam-3889	466	25	,	,	PUNCT
ejpam-3889	466	26	and	and	CCONJ
ejpam-3889	466	27	as	as	SCONJ
ejpam-3889	466	28	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	466	29	)	)	PUNCT
ejpam-3889	466	30	is	be	AUX
ejpam-3889	466	31	cohopfian	cohopfian	ADJ
ejpam-3889	466	32	,	,	PUNCT
ejpam-3889	466	33	s−1(g	s−1(g	PROPN
ejpam-3889	466	34	)	)	PUNCT
ejpam-3889	466	35	is	be	AUX
ejpam-3889	466	36	invertible	invertible	ADJ
ejpam-3889	466	37	.	.	PUNCT
ejpam-3889	467	1	let	let	VERB
ejpam-3889	467	2	x	x	SYM
ejpam-3889	467	3	s	s	PART
ejpam-3889	467	4	∈	∈	PROPN
ejpam-3889	467	5	s	s	PART
ejpam-3889	467	6	−1(n(n	−1(n(n	NOUN
ejpam-3889	467	7	)	)	PUNCT
ejpam-3889	467	8	)	)	PUNCT
ejpam-3889	467	9	,	,	PUNCT
ejpam-3889	467	10	there	there	PRON
ejpam-3889	467	11	exists	exist	VERB
ejpam-3889	467	12	y	y	PROPN
ejpam-3889	467	13	t	t	PROPN
ejpam-3889	467	14	∈	∈	PROPN
ejpam-3889	467	15	s	s	PART
ejpam-3889	467	16	−1(m(n	−1(m(n	NOUN
ejpam-3889	467	17	)	)	PUNCT
ejpam-3889	467	18	)	)	PUNCT
ejpam-3889	468	1	such	such	ADJ
ejpam-3889	468	2	that	that	SCONJ
ejpam-3889	468	3	x	x	PRON
ejpam-3889	468	4	s	s	X
ejpam-3889	468	5	=	=	X
ejpam-3889	468	6	[	[	X
ejpam-3889	468	7	s−1(g(n))](yt	s−1(g(n))](yt	NUM
ejpam-3889	468	8	)	)	PUNCT
ejpam-3889	468	9	.	.	PUNCT
ejpam-3889	469	1	or	or	CCONJ
ejpam-3889	469	2	s−1(g−1(n	s−1(g−1(n	NOUN
ejpam-3889	469	3	)	)	PUNCT
ejpam-3889	469	4	)	)	PUNCT
ejpam-3889	470	1	∈	∈	PROPN
ejpam-3889	470	2	end(s−1(n(n	end(s−1(n(n	PROPN
ejpam-3889	470	3	)	)	PUNCT
ejpam-3889	470	4	)	)	PUNCT
ejpam-3889	470	5	)	)	PUNCT
ejpam-3889	470	6	and	and	CCONJ
ejpam-3889	470	7	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	470	8	)	)	PUNCT
ejpam-3889	470	9	)	)	PUNCT
ejpam-3889	470	10	is	be	AUX
ejpam-3889	470	11	completely	completely	ADV
ejpam-3889	470	12	invariant	invariant	ADJ
ejpam-3889	470	13	,	,	PUNCT
ejpam-3889	470	14	so	so	ADV
ejpam-3889	470	15	y	y	PROPN
ejpam-3889	470	16	t	t	PROPN
ejpam-3889	470	17	=	=	PUNCT
ejpam-3889	470	18	s−1(g−1)(xs	s−1(g−1)(xs	X
ejpam-3889	470	19	)	)	PUNCT
ejpam-3889	470	20	∈	∈	PROPN
ejpam-3889	470	21	s−1(n	s−1(n	PROPN
ejpam-3889	470	22	)	)	PUNCT
ejpam-3889	470	23	,	,	PUNCT
ejpam-3889	470	24	thus	thus	ADV
ejpam-3889	470	25	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	470	26	)	)	PUNCT
ejpam-3889	470	27	)	)	PUNCT
ejpam-3889	471	1	is	be	AUX
ejpam-3889	471	2	an	an	DET
ejpam-3889	471	3	epimorphism	epimorphism	NOUN
ejpam-3889	471	4	for	for	ADP
ejpam-3889	471	5	all	all	DET
ejpam-3889	471	6	n	n	PRON
ejpam-3889	471	7	∈	∈	NOUN
ejpam-3889	471	8	z.	z.	X
ejpam-3889	471	9	thus	thus	ADV
ejpam-3889	471	10	s−1(f∗	s−1(f∗	X
ejpam-3889	471	11	)	)	PUNCT
ejpam-3889	471	12	is	be	AUX
ejpam-3889	471	13	an	an	DET
ejpam-3889	471	14	automorphisme	automorphisme	NOUN
ejpam-3889	471	15	,	,	PUNCT
ejpam-3889	471	16	consequentely	consequentely	ADV
ejpam-3889	471	17	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	471	18	)	)	PUNCT
ejpam-3889	471	19	s.	s.	PROPN
ejpam-3889	471	20	a.	a.	PROPN
ejpam-3889	471	21	balde	balde	PROPN
ejpam-3889	471	22	,	,	PUNCT
ejpam-3889	471	23	m.	m.	PROPN
ejpam-3889	471	24	b.	b.	PROPN
ejpam-3889	471	25	maaouia	maaouia	PROPN
ejpam-3889	471	26	,	,	PUNCT
ejpam-3889	471	27	a.	a.	NOUN
ejpam-3889	471	28	o.	o.	NOUN
ejpam-3889	471	29	chbih	chbih	PROPN
ejpam-3889	471	30	/	/	SYM
ejpam-3889	471	31	eur	eur	PROPN
ejpam-3889	471	32	.	.	PUNCT
ejpam-3889	472	1	j.	j.	PROPN
ejpam-3889	472	2	pure	pure	PROPN
ejpam-3889	472	3	appl	appl	PROPN
ejpam-3889	472	4	.	.	PROPN
ejpam-3889	472	5	math	math	PROPN
ejpam-3889	472	6	,	,	PUNCT
ejpam-3889	472	7	14	14	NUM
ejpam-3889	472	8	(	(	PUNCT
ejpam-3889	472	9	2	2	NUM
ejpam-3889	472	10	)	)	PUNCT
ejpam-3889	472	11	(	(	PUNCT
ejpam-3889	472	12	2021	2021	NUM
ejpam-3889	472	13	)	)	PUNCT
ejpam-3889	472	14	,	,	PUNCT
ejpam-3889	472	15	404	404	NUM
ejpam-3889	472	16	-	-	SYM
ejpam-3889	472	17	422	422	NUM
ejpam-3889	472	18	419	419	NUM
ejpam-3889	472	19	is	be	AUX
ejpam-3889	472	20	cohopfian	cohopfian	ADJ
ejpam-3889	472	21	.	.	PUNCT
ejpam-3889	473	1	reciprocally	reciprocally	PROPN
ejpam-3889	473	2	,	,	PUNCT
ejpam-3889	473	3	suppose	suppose	VERB
ejpam-3889	473	4	that	that	SCONJ
ejpam-3889	473	5	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	473	6	)	)	PUNCT
ejpam-3889	473	7	is	be	AUX
ejpam-3889	473	8	cohopfian	cohopfian	ADJ
ejpam-3889	473	9	and	and	CCONJ
ejpam-3889	473	10	let	let	VERB
ejpam-3889	473	11	s−1(f∗	s−1(f∗	NOUN
ejpam-3889	473	12	)	)	PUNCT
ejpam-3889	473	13	:	:	PUNCT
ejpam-3889	473	14	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	473	15	)	)	PUNCT
ejpam-3889	473	16	−→	−→	NOUN
ejpam-3889	473	17	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	473	18	)	)	PUNCT
ejpam-3889	473	19	a	a	DET
ejpam-3889	473	20	monomorphism	monomorphism	NOUN
ejpam-3889	473	21	.	.	PUNCT
ejpam-3889	474	1	then	then	ADV
ejpam-3889	474	2	s−1(f∗|s−1n∗	s−1(f∗|s−1n∗	ADV
ejpam-3889	474	3	)	)	PUNCT
ejpam-3889	474	4	is	be	AUX
ejpam-3889	474	5	a	a	DET
ejpam-3889	474	6	monomorphism	monomorphism	NOUN
ejpam-3889	474	7	of	of	ADP
ejpam-3889	474	8	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	474	9	)	)	PUNCT
ejpam-3889	474	10	,	,	PUNCT
ejpam-3889	474	11	so	so	SCONJ
ejpam-3889	474	12	for	for	ADP
ejpam-3889	474	13	all	all	DET
ejpam-3889	474	14	n	n	PRON
ejpam-3889	474	15	∈	∈	PROPN
ejpam-3889	474	16	z	z	PROPN
ejpam-3889	474	17	,	,	PUNCT
ejpam-3889	474	18	s−1(f(n)|s−1n(n	s−1(f(n)|s−1n(n	NOUN
ejpam-3889	474	19	)	)	PUNCT
ejpam-3889	474	20	)	)	PUNCT
ejpam-3889	474	21	is	be	AUX
ejpam-3889	474	22	a	a	DET
ejpam-3889	474	23	monomorphism	monomorphism	NOUN
ejpam-3889	474	24	of	of	ADP
ejpam-3889	474	25	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	474	26	)	)	PUNCT
ejpam-3889	474	27	)	)	PUNCT
ejpam-3889	474	28	.	.	PUNCT
ejpam-3889	475	1	thus	thus	ADV
ejpam-3889	475	2	,	,	PUNCT
ejpam-3889	475	3	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	475	4	)	)	PUNCT
ejpam-3889	475	5	)	)	PUNCT
ejpam-3889	475	6	∈	∈	PROPN
ejpam-3889	475	7	aut(s−1(n(n	aut(s−1(n(n	PROPN
ejpam-3889	475	8	)	)	PUNCT
ejpam-3889	475	9	)	)	PUNCT
ejpam-3889	475	10	)	)	PUNCT
ejpam-3889	475	11	,	,	PUNCT
ejpam-3889	475	12	hence	hence	ADV
ejpam-3889	475	13	[	[	X
ejpam-3889	475	14	s−1(f(n))](s−1(n(n	s−1(f(n))](s−1(n(n	ADJ
ejpam-3889	475	15	)	)	PUNCT
ejpam-3889	475	16	)	)	PUNCT
ejpam-3889	475	17	)	)	PUNCT
ejpam-3889	476	1	=	=	PUNCT
ejpam-3889	476	2	s−1(n(n	s−1(n(n	NOUN
ejpam-3889	476	3	)	)	PUNCT
ejpam-3889	476	4	)	)	PUNCT
ejpam-3889	476	5	.	.	PUNCT
ejpam-3889	477	1	as	as	ADP
ejpam-3889	477	2	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	477	3	)	)	PUNCT
ejpam-3889	477	4	)	)	PUNCT
ejpam-3889	477	5	is	be	AUX
ejpam-3889	477	6	quasi	quasi	ADJ
ejpam-3889	477	7	-	-	ADJ
ejpam-3889	477	8	injective	injective	ADJ
ejpam-3889	477	9	,	,	PUNCT
ejpam-3889	477	10	then	then	ADV
ejpam-3889	477	11	there	there	PRON
ejpam-3889	477	12	exists	exist	VERB
ejpam-3889	477	13	s−1(l(n	s−1(l(n	PROPN
ejpam-3889	477	14	)	)	PUNCT
ejpam-3889	477	15	)	)	PUNCT
ejpam-3889	477	16	a	a	DET
ejpam-3889	477	17	submodule	submodule	NOUN
ejpam-3889	477	18	of	of	ADP
ejpam-3889	477	19	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	477	20	)	)	PUNCT
ejpam-3889	477	21	)	)	PUNCT
ejpam-3889	477	22	such	such	ADJ
ejpam-3889	477	23	that	that	SCONJ
ejpam-3889	477	24	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	477	25	)	)	PUNCT
ejpam-3889	477	26	)	)	PUNCT
ejpam-3889	478	1	=	=	PUNCT
ejpam-3889	479	1	[	[	X
ejpam-3889	479	2	s−1(f(n))](s−1m(n))⊕	s−1(f(n))](s−1m(n))⊕	PROPN
ejpam-3889	479	3	s−1(l(n	s−1(l(n	PROPN
ejpam-3889	479	4	)	)	PUNCT
ejpam-3889	479	5	)	)	PUNCT
ejpam-3889	479	6	.	.	PUNCT
ejpam-3889	480	1	thus	thus	ADV
ejpam-3889	480	2	,	,	PUNCT
ejpam-3889	480	3	we	we	PRON
ejpam-3889	480	4	have	have	VERB
ejpam-3889	480	5	0	0	NUM
ejpam-3889	480	6	=	=	SYM
ejpam-3889	481	1	[	[	X
ejpam-3889	481	2	s−1(f(n))](s−1n)∩s−1(l(n	s−1(f(n))](s−1n)∩s−1(l(n	NOUN
ejpam-3889	481	3	)	)	PUNCT
ejpam-3889	481	4	)	)	PUNCT
ejpam-3889	482	1	=	=	SYM
ejpam-3889	482	2	s−1(n(n))∩s−1(l(n	s−1(n(n))∩s−1(l(n	PROPN
ejpam-3889	482	3	)	)	PUNCT
ejpam-3889	482	4	)	)	PUNCT
ejpam-3889	482	5	,	,	PUNCT
ejpam-3889	482	6	since	since	SCONJ
ejpam-3889	482	7	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	482	8	)	)	PUNCT
ejpam-3889	482	9	)	)	PUNCT
ejpam-3889	482	10	is	be	AUX
ejpam-3889	482	11	essential	essential	ADJ
ejpam-3889	482	12	,	,	PUNCT
ejpam-3889	482	13	then	then	ADV
ejpam-3889	482	14	s−1(l(n	s−1(l(n	PROPN
ejpam-3889	482	15	)	)	PUNCT
ejpam-3889	482	16	)	)	PUNCT
ejpam-3889	483	1	=	=	SYM
ejpam-3889	483	2	0	0	NUM
ejpam-3889	483	3	,	,	PUNCT
ejpam-3889	483	4	hence	hence	ADV
ejpam-3889	483	5	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	483	6	)	)	PUNCT
ejpam-3889	483	7	)	)	PUNCT
ejpam-3889	484	1	=	=	PUNCT
ejpam-3889	485	1	[	[	X
ejpam-3889	485	2	s−1(f(n))](s−1m(n	s−1(f(n))](s−1m(n	X
ejpam-3889	485	3	)	)	PUNCT
ejpam-3889	485	4	)	)	PUNCT
ejpam-3889	485	5	,	,	PUNCT
ejpam-3889	485	6	thus	thus	ADV
ejpam-3889	485	7	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	485	8	)	)	PUNCT
ejpam-3889	485	9	)	)	PUNCT
ejpam-3889	485	10	is	be	AUX
ejpam-3889	485	11	an	an	DET
ejpam-3889	485	12	epimorphism	epimorphism	NOUN
ejpam-3889	485	13	for	for	ADP
ejpam-3889	485	14	all	all	DET
ejpam-3889	485	15	n	n	PRON
ejpam-3889	485	16	∈	∈	NOUN
ejpam-3889	485	17	z	z	NOUN
ejpam-3889	485	18	=	=	NOUN
ejpam-3889	485	19	⇒	⇒	X
ejpam-3889	485	20	s−1(f∗	s−1(f∗	NOUN
ejpam-3889	485	21	)	)	PUNCT
ejpam-3889	485	22	is	be	AUX
ejpam-3889	485	23	an	an	DET
ejpam-3889	485	24	epimorphism	epimorphism	NOUN
ejpam-3889	485	25	of	of	ADP
ejpam-3889	485	26	chain	chain	NOUN
ejpam-3889	485	27	complex	complex	NOUN
ejpam-3889	485	28	,	,	PUNCT
ejpam-3889	485	29	so	so	ADV
ejpam-3889	485	30	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	485	31	)	)	PUNCT
ejpam-3889	485	32	is	be	AUX
ejpam-3889	485	33	cohopfian	cohopfian	ADJ
ejpam-3889	485	34	complex	complex	ADJ
ejpam-3889	485	35	sequence	sequence	NOUN
ejpam-3889	485	36	.	.	PUNCT
ejpam-3889	486	1	proposition	proposition	NOUN
ejpam-3889	486	2	2	2	NUM
ejpam-3889	486	3	.	.	PUNCT
ejpam-3889	487	1	let	let	VERB
ejpam-3889	487	2	m	m	PRON
ejpam-3889	487	3	be	be	AUX
ejpam-3889	487	4	a	a	DET
ejpam-3889	487	5	graded	grade	VERB
ejpam-3889	487	6	left	leave	VERB
ejpam-3889	487	7	a	a	DET
ejpam-3889	487	8	-	-	PUNCT
ejpam-3889	487	9	module	module	NOUN
ejpam-3889	487	10	and	and	CCONJ
ejpam-3889	487	11	s	s	VERB
ejpam-3889	487	12	a	a	DET
ejpam-3889	487	13	saturated	saturate	VERB
ejpam-3889	487	14	multiplicative	multiplicative	ADJ
ejpam-3889	487	15	part	part	NOUN
ejpam-3889	487	16	formed	form	VERB
ejpam-3889	487	17	by	by	ADP
ejpam-3889	487	18	the	the	DET
ejpam-3889	487	19	non	non	ADJ
ejpam-3889	487	20	-	-	ADJ
ejpam-3889	487	21	zero	zero	ADJ
ejpam-3889	487	22	homogeneous	homogeneous	ADJ
ejpam-3889	487	23	elements	element	NOUN
ejpam-3889	487	24	of	of	ADP
ejpam-3889	487	25	a	a	DET
ejpam-3889	487	26	verifying	verifying	NOUN
ejpam-3889	487	27	the	the	DET
ejpam-3889	487	28	left	left	ADJ
ejpam-3889	487	29	ore	ore	NOUN
ejpam-3889	487	30	conditions	condition	NOUN
ejpam-3889	487	31	.	.	PUNCT
ejpam-3889	488	1	if	if	SCONJ
ejpam-3889	488	2	m∗	m∗	PROPN
ejpam-3889	488	3	is	be	AUX
ejpam-3889	488	4	a	a	DET
ejpam-3889	488	5	hopfian	hopfian	ADJ
ejpam-3889	488	6	,	,	PUNCT
ejpam-3889	488	7	noetherian	noetherian	ADJ
ejpam-3889	488	8	and	and	CCONJ
ejpam-3889	488	9	quasi	quasi	ADJ
ejpam-3889	488	10	-	-	ADJ
ejpam-3889	488	11	injective	injective	ADJ
ejpam-3889	488	12	complex	complex	ADJ
ejpam-3889	488	13	sequence	sequence	NOUN
ejpam-3889	488	14	associated	associate	VERB
ejpam-3889	488	15	with	with	ADP
ejpam-3889	488	16	m	m	PROPN
ejpam-3889	488	17	,	,	PUNCT
ejpam-3889	488	18	then	then	ADV
ejpam-3889	488	19	the	the	DET
ejpam-3889	488	20	complex	complex	ADJ
ejpam-3889	488	21	sequence	sequence	NOUN
ejpam-3889	488	22	of	of	ADP
ejpam-3889	488	23	morphisms	morphism	NOUN
ejpam-3889	488	24	of	of	ADP
ejpam-3889	488	25	left	left	ADJ
ejpam-3889	488	26	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	488	27	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	488	28	)	)	PUNCT
ejpam-3889	488	29	has	have	VERB
ejpam-3889	488	30	the	the	DET
ejpam-3889	488	31	following	follow	VERB
ejpam-3889	488	32	property	property	NOUN
ejpam-3889	488	33	:	:	PUNCT
ejpam-3889	488	34	�	�	VERB
ejpam-3889	488	35	any	any	DET
ejpam-3889	488	36	epimorphism	epimorphism	NOUN
ejpam-3889	488	37	of	of	ADP
ejpam-3889	488	38	subcomplex	subcomplex	PROPN
ejpam-3889	488	39	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	488	40	)	)	PUNCT
ejpam-3889	488	41	of	of	ADP
ejpam-3889	488	42	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	488	43	)	)	PUNCT
ejpam-3889	488	44	is	be	AUX
ejpam-3889	488	45	an	an	DET
ejpam-3889	488	46	isomorphism	isomorphism	NOUN
ejpam-3889	488	47	�	�	NOUN
ejpam-3889	488	48	.	.	PUNCT
ejpam-3889	489	1	proof	proof	NOUN
ejpam-3889	489	2	.	.	PUNCT
ejpam-3889	490	1	let	let	VERB
ejpam-3889	490	2	n∗	n∗	PROPN
ejpam-3889	490	3	be	be	AUX
ejpam-3889	490	4	a	a	DET
ejpam-3889	490	5	subcomplex	subcomplex	NOUN
ejpam-3889	490	6	of	of	ADP
ejpam-3889	490	7	m∗	m∗	NOUN
ejpam-3889	490	8	and	and	CCONJ
ejpam-3889	490	9	s−1(f∗	s−1(f∗	NOUN
ejpam-3889	490	10	)	)	PUNCT
ejpam-3889	490	11	:	:	PUNCT
ejpam-3889	491	1	s−1n∗	s−1n∗	PROPN
ejpam-3889	491	2	−→	−→	NOUN
ejpam-3889	491	3	s−1m∗	s−1m∗	PROPN
ejpam-3889	491	4	an	an	DET
ejpam-3889	491	5	epimorphism	epimorphism	NOUN
ejpam-3889	491	6	of	of	ADP
ejpam-3889	491	7	chain	chain	NOUN
ejpam-3889	491	8	complex	complex	NOUN
ejpam-3889	491	9	.	.	PUNCT
ejpam-3889	492	1	since	since	SCONJ
ejpam-3889	492	2	m∗	m∗	NOUN
ejpam-3889	492	3	is	be	AUX
ejpam-3889	492	4	quasi	quasi	ADJ
ejpam-3889	492	5	-	-	ADJ
ejpam-3889	492	6	injective	injective	ADJ
ejpam-3889	492	7	,	,	PUNCT
ejpam-3889	492	8	then	then	ADV
ejpam-3889	492	9	for	for	SCONJ
ejpam-3889	492	10	all	all	DET
ejpam-3889	492	11	n	n	PRON
ejpam-3889	492	12	∈	∈	PROPN
ejpam-3889	492	13	z	z	PROPN
ejpam-3889	492	14	,	,	PUNCT
ejpam-3889	492	15	m(n	m(n	PROPN
ejpam-3889	492	16	)	)	PUNCT
ejpam-3889	492	17	is	be	AUX
ejpam-3889	492	18	quasi	quasi	ADJ
ejpam-3889	492	19	-	-	ADJ
ejpam-3889	492	20	injective	injective	ADJ
ejpam-3889	492	21	=	=	NOUN
ejpam-3889	492	22	⇒	⇒	NOUN
ejpam-3889	492	23	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	492	24	)	)	PUNCT
ejpam-3889	492	25	)	)	PUNCT
ejpam-3889	493	1	is	be	AUX
ejpam-3889	493	2	quasi	quasi	ADJ
ejpam-3889	493	3	-	-	ADJ
ejpam-3889	493	4	injective	injective	ADJ
ejpam-3889	493	5	for	for	ADP
ejpam-3889	493	6	all	all	DET
ejpam-3889	493	7	n	n	PRON
ejpam-3889	493	8	∈	∈	PROPN
ejpam-3889	493	9	z	z	NOUN
ejpam-3889	493	10	,	,	PUNCT
ejpam-3889	493	11	so	so	ADV
ejpam-3889	493	12	,	,	PUNCT
ejpam-3889	493	13	there	there	PRON
ejpam-3889	493	14	exists	exist	VERB
ejpam-3889	493	15	s−1(f̃(n	s−1(f̃(n	NOUN
ejpam-3889	493	16	)	)	PUNCT
ejpam-3889	493	17	)	)	PUNCT
ejpam-3889	494	1	∈	∈	PROPN
ejpam-3889	494	2	end(s−1(m(n	end(s−1(m(n	NOUN
ejpam-3889	494	3	)	)	PUNCT
ejpam-3889	494	4	)	)	PUNCT
ejpam-3889	494	5	)	)	PUNCT
ejpam-3889	495	1	such	such	ADJ
ejpam-3889	495	2	that	that	DET
ejpam-3889	495	3	s−1(f̃(n))|s−1(n(n	s−1(f̃(n))|s−1(n(n	NOUN
ejpam-3889	495	4	)	)	PUNCT
ejpam-3889	495	5	)	)	PUNCT
ejpam-3889	496	1	=	=	SYM
ejpam-3889	496	2	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	496	3	)	)	PUNCT
ejpam-3889	496	4	)	)	PUNCT
ejpam-3889	496	5	.	.	PUNCT
ejpam-3889	497	1	or	or	CCONJ
ejpam-3889	497	2	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	497	3	)	)	PUNCT
ejpam-3889	497	4	)	)	PUNCT
ejpam-3889	497	5	is	be	AUX
ejpam-3889	497	6	surjective	surjective	ADJ
ejpam-3889	497	7	,	,	PUNCT
ejpam-3889	497	8	then	then	ADV
ejpam-3889	497	9	for	for	ADP
ejpam-3889	497	10	all	all	PRON
ejpam-3889	497	11	x	x	PRON
ejpam-3889	497	12	s	s	PROPN
ejpam-3889	497	13	∈	∈	PROPN
ejpam-3889	497	14	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	497	15	)	)	PUNCT
ejpam-3889	497	16	)	)	PUNCT
ejpam-3889	497	17	,	,	PUNCT
ejpam-3889	497	18	there	there	PRON
ejpam-3889	497	19	exists	exist	VERB
ejpam-3889	497	20	y	y	PROPN
ejpam-3889	497	21	t	t	PROPN
ejpam-3889	497	22	∈	∈	PROPN
ejpam-3889	497	23	s	s	PART
ejpam-3889	497	24	−1(n(n	−1(n(n	NOUN
ejpam-3889	497	25	)	)	PUNCT
ejpam-3889	497	26	)	)	PUNCT
ejpam-3889	498	1	such	such	ADJ
ejpam-3889	498	2	that	that	SCONJ
ejpam-3889	498	3	x	x	PRON
ejpam-3889	498	4	s	s	X
ejpam-3889	498	5	=	=	X
ejpam-3889	499	1	[	[	X
ejpam-3889	499	2	s−1(f(n))](yt	s−1(f(n))](yt	NUM
ejpam-3889	499	3	)	)	PUNCT
ejpam-3889	499	4	,	,	PUNCT
ejpam-3889	499	5	hence	hence	ADV
ejpam-3889	499	6	s−1(f̃(n	s−1(f̃(n	NOUN
ejpam-3889	499	7	)	)	PUNCT
ejpam-3889	499	8	)	)	PUNCT
ejpam-3889	499	9	is	be	AUX
ejpam-3889	499	10	surjective	surjective	ADJ
ejpam-3889	499	11	,	,	PUNCT
ejpam-3889	499	12	as	as	ADP
ejpam-3889	499	13	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	499	14	)	)	PUNCT
ejpam-3889	499	15	)	)	PUNCT
ejpam-3889	500	1	is	be	AUX
ejpam-3889	500	2	hopfian	hopfian	ADJ
ejpam-3889	500	3	,	,	PUNCT
ejpam-3889	500	4	thus	thus	ADV
ejpam-3889	500	5	˜s−1(f)(n	˜s−1(f)(n	NUM
ejpam-3889	500	6	)	)	PUNCT
ejpam-3889	500	7	∈	∈	PROPN
ejpam-3889	500	8	aut(s−1(n(n	aut(s−1(n(n	PROPN
ejpam-3889	500	9	)	)	PUNCT
ejpam-3889	500	10	)	)	PUNCT
ejpam-3889	500	11	)	)	PUNCT
ejpam-3889	501	1	for	for	ADP
ejpam-3889	501	2	all	all	DET
ejpam-3889	501	3	n	n	PRON
ejpam-3889	501	4	∈	∈	NOUN
ejpam-3889	501	5	z.	z.	NOUN
ejpam-3889	501	6	we	we	PRON
ejpam-3889	501	7	deduce	deduce	VERB
ejpam-3889	501	8	that	that	DET
ejpam-3889	501	9	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	501	10	)	)	PUNCT
ejpam-3889	501	11	)	)	PUNCT
ejpam-3889	501	12	is	be	AUX
ejpam-3889	501	13	a	a	DET
ejpam-3889	501	14	monomorphism	monomorphism	NOUN
ejpam-3889	501	15	of	of	ADP
ejpam-3889	501	16	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	501	17	)	)	PUNCT
ejpam-3889	501	18	)	)	PUNCT
ejpam-3889	501	19	into	into	ADP
ejpam-3889	501	20	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	501	21	)	)	PUNCT
ejpam-3889	501	22	)	)	PUNCT
ejpam-3889	501	23	,	,	PUNCT
ejpam-3889	501	24	for	for	ADP
ejpam-3889	501	25	all	all	DET
ejpam-3889	501	26	n	n	PRON
ejpam-3889	501	27	∈	∈	PROPN
ejpam-3889	501	28	z.	z.	PROPN
ejpam-3889	501	29	consequentely	consequentely	ADV
ejpam-3889	501	30	,	,	PUNCT
ejpam-3889	501	31	s−1(f∗	s−1(f∗	PROPN
ejpam-3889	501	32	)	)	PUNCT
ejpam-3889	501	33	is	be	AUX
ejpam-3889	501	34	an	an	DET
ejpam-3889	501	35	isomorphism	isomorphism	NOUN
ejpam-3889	501	36	of	of	ADP
ejpam-3889	501	37	chain	chain	NOUN
ejpam-3889	501	38	complex	complex	NOUN
ejpam-3889	501	39	.	.	PUNCT
ejpam-3889	502	1	theorem	theorem	VERB
ejpam-3889	502	2	17	17	NUM
ejpam-3889	502	3	.	.	PUNCT
ejpam-3889	503	1	let	let	VERB
ejpam-3889	503	2	m	m	PRON
ejpam-3889	503	3	be	be	AUX
ejpam-3889	503	4	a	a	DET
ejpam-3889	503	5	graded	grade	VERB
ejpam-3889	503	6	left	leave	VERB
ejpam-3889	503	7	a	a	DET
ejpam-3889	503	8	-	-	PUNCT
ejpam-3889	503	9	module	module	NOUN
ejpam-3889	503	10	,	,	PUNCT
ejpam-3889	503	11	n	n	CCONJ
ejpam-3889	503	12	a	a	DET
ejpam-3889	503	13	graded	grade	VERB
ejpam-3889	503	14	submodule	submodule	NOUN
ejpam-3889	503	15	of	of	ADP
ejpam-3889	503	16	m	m	PROPN
ejpam-3889	503	17	,	,	PUNCT
ejpam-3889	503	18	s	s	VERB
ejpam-3889	503	19	a	a	DET
ejpam-3889	503	20	saturated	saturate	VERB
ejpam-3889	503	21	multiplicative	multiplicative	ADJ
ejpam-3889	503	22	part	part	NOUN
ejpam-3889	503	23	formed	form	VERB
ejpam-3889	503	24	by	by	ADP
ejpam-3889	503	25	the	the	DET
ejpam-3889	503	26	non	non	ADJ
ejpam-3889	503	27	-	-	ADJ
ejpam-3889	503	28	zero	zero	ADJ
ejpam-3889	503	29	homogeneous	homogeneous	ADJ
ejpam-3889	503	30	elements	element	NOUN
ejpam-3889	503	31	of	of	ADP
ejpam-3889	503	32	a	a	DET
ejpam-3889	503	33	verifying	verifying	NOUN
ejpam-3889	503	34	the	the	DET
ejpam-3889	503	35	left	left	ADJ
ejpam-3889	503	36	ore	ore	NOUN
ejpam-3889	503	37	conditions	condition	NOUN
ejpam-3889	503	38	.	.	PUNCT
ejpam-3889	504	1	m∗	m∗	VERB
ejpam-3889	504	2	the	the	DET
ejpam-3889	504	3	quasi	quasi	ADJ
ejpam-3889	504	4	-	-	ADJ
ejpam-3889	504	5	projective	projective	ADJ
ejpam-3889	504	6	complex	complex	ADJ
ejpam-3889	504	7	sequence	sequence	NOUN
ejpam-3889	504	8	associated	associate	VERB
ejpam-3889	504	9	with	with	ADP
ejpam-3889	504	10	m	m	PROPN
ejpam-3889	504	11	and	and	CCONJ
ejpam-3889	504	12	n∗	n∗	VERB
ejpam-3889	504	13	a	a	DET
ejpam-3889	504	14	superfluous	superfluous	ADJ
ejpam-3889	504	15	and	and	CCONJ
ejpam-3889	504	16	completely	completely	ADV
ejpam-3889	504	17	invariant	invariant	ADJ
ejpam-3889	504	18	complex	complex	ADJ
ejpam-3889	504	19	sub	sub	NOUN
ejpam-3889	504	20	-	-	NOUN
ejpam-3889	504	21	sequence	sequence	NOUN
ejpam-3889	504	22	of	of	ADP
ejpam-3889	504	23	m∗.	m∗.	PROPN
ejpam-3889	504	24	then	then	ADV
ejpam-3889	504	25	the	the	DET
ejpam-3889	504	26	complex	complex	ADJ
ejpam-3889	504	27	morphism	morphism	NOUN
ejpam-3889	504	28	sequence	sequence	NOUN
ejpam-3889	504	29	of	of	ADP
ejpam-3889	504	30	left	left	ADJ
ejpam-3889	504	31	s−1(a)-modules	s−1(a)-module	NOUN
ejpam-3889	504	32	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	504	33	)	)	PUNCT
ejpam-3889	504	34	is	be	AUX
ejpam-3889	504	35	hopfian	hopfian	ADJ
ejpam-3889	504	36	if	if	SCONJ
ejpam-3889	504	37	,	,	PUNCT
ejpam-3889	504	38	and	and	CCONJ
ejpam-3889	504	39	only	only	ADV
ejpam-3889	504	40	if	if	SCONJ
ejpam-3889	504	41	,	,	PUNCT
ejpam-3889	504	42	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	504	43	)	)	PUNCT
ejpam-3889	504	44	the	the	DET
ejpam-3889	504	45	complex	complex	ADJ
ejpam-3889	504	46	sequence	sequence	NOUN
ejpam-3889	504	47	associated	associate	VERB
ejpam-3889	504	48	with	with	ADP
ejpam-3889	504	49	s−1(m	s−1(m	PROPN
ejpam-3889	504	50	/	/	SYM
ejpam-3889	504	51	n	n	CCONJ
ejpam-3889	504	52	)	)	PUNCT
ejpam-3889	504	53	is	be	AUX
ejpam-3889	504	54	hopfian	hopfian	ADJ
ejpam-3889	504	55	.	.	PUNCT
ejpam-3889	505	1	proof	proof	NOUN
ejpam-3889	505	2	.	.	PUNCT
ejpam-3889	506	1	suppose	suppose	VERB
ejpam-3889	506	2	that	that	SCONJ
ejpam-3889	506	3	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	506	4	)	)	PUNCT
ejpam-3889	506	5	is	be	AUX
ejpam-3889	506	6	hopfian	hopfian	ADJ
ejpam-3889	506	7	and	and	CCONJ
ejpam-3889	506	8	let	let	VERB
ejpam-3889	506	9	f∗	f∗	NOUN
ejpam-3889	506	10	:	:	PUNCT
ejpam-3889	506	11	m∗	m∗	VERB
ejpam-3889	506	12	−→m∗	−→m∗	PROPN
ejpam-3889	506	13	an	an	DET
ejpam-3889	506	14	epimorphism	epimorphism	NOUN
ejpam-3889	506	15	.	.	PUNCT
ejpam-3889	507	1	as	as	ADP
ejpam-3889	507	2	s−1(n∗	s−1(n∗	PROPN
ejpam-3889	507	3	)	)	PUNCT
ejpam-3889	507	4	is	be	AUX
ejpam-3889	507	5	completely	completely	ADV
ejpam-3889	507	6	invariant	invariant	ADJ
ejpam-3889	507	7	,	,	PUNCT
ejpam-3889	507	8	then	then	ADV
ejpam-3889	507	9	∀n	∀n	NUM
ejpam-3889	507	10	∈	∈	PROPN
ejpam-3889	507	11	z	z	NOUN
ejpam-3889	507	12	,	,	PUNCT
ejpam-3889	508	1	[	[	X
ejpam-3889	508	2	s−1(f(n))](s−1(n(n	s−1(f(n))](s−1(n(n	ADJ
ejpam-3889	508	3	)	)	PUNCT
ejpam-3889	508	4	)	)	PUNCT
ejpam-3889	508	5	)	)	PUNCT
ejpam-3889	509	1	⊂	⊂	PROPN
ejpam-3889	509	2	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	509	3	)	)	PUNCT
ejpam-3889	509	4	)	)	PUNCT
ejpam-3889	509	5	,	,	PUNCT
ejpam-3889	509	6	implies	imply	VERB
ejpam-3889	509	7	that	that	SCONJ
ejpam-3889	509	8	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	509	9	)	)	PUNCT
ejpam-3889	509	10	)	)	PUNCT
ejpam-3889	509	11	induces	induce	VERB
ejpam-3889	509	12	an	an	DET
ejpam-3889	509	13	epimorphism	epimorphism	NOUN
ejpam-3889	509	14	s−1(f)(n	s−1(f)(n	PROPN
ejpam-3889	509	15	)	)	PUNCT
ejpam-3889	509	16	:	:	PUNCT
ejpam-3889	509	17	s−1(m(n)/n(n	s−1(m(n)/n(n	PROPN
ejpam-3889	509	18	)	)	PUNCT
ejpam-3889	509	19	)	)	PUNCT
ejpam-3889	510	1	−→	−→	PROPN
ejpam-3889	510	2	s−1(m(n)/n(n	s−1(m(n)/n(n	PROPN
ejpam-3889	510	3	)	)	PUNCT
ejpam-3889	510	4	)	)	PUNCT
ejpam-3889	510	5	,	,	PUNCT
ejpam-3889	510	6	since	since	SCONJ
ejpam-3889	510	7	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	510	8	)	)	PUNCT
ejpam-3889	510	9	is	be	AUX
ejpam-3889	510	10	hopfian	hopfian	ADJ
ejpam-3889	510	11	,	,	PUNCT
ejpam-3889	510	12	then	then	ADV
ejpam-3889	510	13	s−1(f)(n	s−1(f)(n	PROPN
ejpam-3889	510	14	)	)	PUNCT
ejpam-3889	510	15	is	be	AUX
ejpam-3889	510	16	an	an	DET
ejpam-3889	510	17	automorphism	automorphism	NOUN
ejpam-3889	510	18	.	.	PUNCT
ejpam-3889	511	1	put	put	VERB
ejpam-3889	511	2	s−1(k(n	s−1(k(n	NOUN
ejpam-3889	511	3	)	)	PUNCT
ejpam-3889	511	4	)	)	PUNCT
ejpam-3889	512	1	=	=	SYM
ejpam-3889	512	2	ker(s−1f(n	ker(s−1f(n	PROPN
ejpam-3889	512	3	)	)	PUNCT
ejpam-3889	512	4	)	)	PUNCT
ejpam-3889	512	5	and	and	CCONJ
ejpam-3889	512	6	s−1(π(n	s−1(π(n	NOUN
ejpam-3889	512	7	)	)	PUNCT
ejpam-3889	512	8	)	)	PUNCT
ejpam-3889	512	9	:	:	PUNCT
ejpam-3889	512	10	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	512	11	)	)	PUNCT
ejpam-3889	512	12	)	)	PUNCT
ejpam-3889	513	1	−→	−→	PROPN
ejpam-3889	513	2	s−1(m(n)/n(n	s−1(m(n)/n(n	PROPN
ejpam-3889	513	3	)	)	PUNCT
ejpam-3889	513	4	)	)	PUNCT
ejpam-3889	514	1	the	the	DET
ejpam-3889	514	2	canonical	canonical	ADJ
ejpam-3889	514	3	projection	projection	NOUN
ejpam-3889	514	4	,	,	PUNCT
ejpam-3889	514	5	we	we	PRON
ejpam-3889	514	6	have	have	VERB
ejpam-3889	514	7	:	:	PUNCT
ejpam-3889	514	8	s−1(f)(n	s−1(f)(n	NUM
ejpam-3889	514	9	)	)	PUNCT
ejpam-3889	514	10	◦	◦	NOUN
ejpam-3889	515	1	[	[	X
ejpam-3889	515	2	s−1(π(n))](s−1(k(n	s−1(π(n))](s−1(k(n	NOUN
ejpam-3889	515	3	)	)	PUNCT
ejpam-3889	515	4	)	)	PUNCT
ejpam-3889	515	5	)	)	PUNCT
ejpam-3889	516	1	=	=	PUNCT
ejpam-3889	517	1	[	[	X
ejpam-3889	517	2	s−1(π(n	s−1(π(n	ADJ
ejpam-3889	517	3	)	)	PUNCT
ejpam-3889	517	4	◦	◦	NOUN
ejpam-3889	517	5	f(n))](k	f(n))](k	NUM
ejpam-3889	517	6	)	)	PUNCT
ejpam-3889	517	7	=	=	SYM
ejpam-3889	517	8	0	0	NUM
ejpam-3889	517	9	s.	s.	PROPN
ejpam-3889	517	10	a.	a.	PROPN
ejpam-3889	517	11	balde	balde	PROPN
ejpam-3889	517	12	,	,	PUNCT
ejpam-3889	517	13	m.	m.	PROPN
ejpam-3889	517	14	b.	b.	PROPN
ejpam-3889	517	15	maaouia	maaouia	PROPN
ejpam-3889	517	16	,	,	PUNCT
ejpam-3889	517	17	a.	a.	NOUN
ejpam-3889	517	18	o.	o.	NOUN
ejpam-3889	517	19	chbih	chbih	PROPN
ejpam-3889	517	20	/	/	SYM
ejpam-3889	517	21	eur	eur	PROPN
ejpam-3889	517	22	.	.	PUNCT
ejpam-3889	518	1	j.	j.	PROPN
ejpam-3889	518	2	pure	pure	PROPN
ejpam-3889	518	3	appl	appl	PROPN
ejpam-3889	518	4	.	.	PROPN
ejpam-3889	518	5	math	math	PROPN
ejpam-3889	518	6	,	,	PUNCT
ejpam-3889	518	7	14	14	NUM
ejpam-3889	518	8	(	(	PUNCT
ejpam-3889	518	9	2	2	NUM
ejpam-3889	518	10	)	)	PUNCT
ejpam-3889	518	11	(	(	PUNCT
ejpam-3889	518	12	2021	2021	NUM
ejpam-3889	518	13	)	)	PUNCT
ejpam-3889	518	14	,	,	PUNCT
ejpam-3889	518	15	404	404	NUM
ejpam-3889	518	16	-	-	SYM
ejpam-3889	518	17	422	422	NUM
ejpam-3889	518	18	420	420	NUM
ejpam-3889	518	19	indeed	indeed	ADV
ejpam-3889	518	20	,	,	PUNCT
ejpam-3889	518	21	∀xs	∀xs	PROPN
ejpam-3889	518	22	∈	∈	PROPN
ejpam-3889	518	23	s	s	PART
ejpam-3889	518	24	−1(k(n	−1(k(n	NOUN
ejpam-3889	518	25	)	)	PUNCT
ejpam-3889	518	26	)	)	PUNCT
ejpam-3889	519	1	,	,	PUNCT
ejpam-3889	519	2	we	we	PRON
ejpam-3889	519	3	have	have	VERB
ejpam-3889	519	4	:	:	PUNCT
ejpam-3889	519	5	[	[	X
ejpam-3889	519	6	s−1(f(n	s−1(f(n	X
ejpam-3889	519	7	)	)	PUNCT
ejpam-3889	519	8	◦	◦	NOUN
ejpam-3889	519	9	π(n))](xs	π(n))](xs	ADP
ejpam-3889	519	10	)	)	PUNCT
ejpam-3889	519	11	=	=	PUNCT
ejpam-3889	520	1	[	[	X
ejpam-3889	520	2	s−1(π(n	s−1(π(n	NOUN
ejpam-3889	520	3	)	)	PUNCT
ejpam-3889	520	4	◦	◦	NOUN
ejpam-3889	520	5	f(n))](xs	f(n))](xs	PROPN
ejpam-3889	520	6	)	)	PUNCT
ejpam-3889	520	7	,	,	PUNCT
ejpam-3889	520	8	so	so	CCONJ
ejpam-3889	520	9	[	[	X
ejpam-3889	520	10	s−1(π(n	s−1(π(n	ADJ
ejpam-3889	520	11	)	)	PUNCT
ejpam-3889	520	12	◦	◦	NOUN
ejpam-3889	520	13	f(n))](xs	f(n))](x	NOUN
ejpam-3889	520	14	)	)	PUNCT
ejpam-3889	521	1	=	=	PUNCT
ejpam-3889	522	1	[	[	X
ejpam-3889	522	2	s−1(π(n))](f(n)(x)s	s−1(π(n))](f(n)(x)s	NOUN
ejpam-3889	522	3	)	)	PUNCT
ejpam-3889	522	4	=	=	PUNCT
ejpam-3889	523	1	[	[	X
ejpam-3889	523	2	s−1(π(n))](0s	s−1(π(n))](0s	NOUN
ejpam-3889	523	3	)	)	PUNCT
ejpam-3889	523	4	=	=	SYM
ejpam-3889	523	5	0	0	NUM
ejpam-3889	523	6	,	,	PUNCT
ejpam-3889	523	7	thus	thus	ADV
ejpam-3889	523	8	[	[	X
ejpam-3889	523	9	s−1(s−1(f)(n	s−1(s−1(f)(n	NOUN
ejpam-3889	523	10	)	)	PUNCT
ejpam-3889	523	11	◦	◦	NOUN
ejpam-3889	523	12	π(n))](k	π(n))](k	PUNCT
ejpam-3889	523	13	)	)	PUNCT
ejpam-3889	523	14	=	=	SYM
ejpam-3889	524	1	0	0	X
ejpam-3889	524	2	.	.	PUNCT
ejpam-3889	525	1	we	we	PRON
ejpam-3889	525	2	have	have	VERB
ejpam-3889	525	3	:	:	PUNCT
ejpam-3889	526	1	[	[	X
ejpam-3889	526	2	s−1(f(n)	s−1(f(n)	NUM
ejpam-3889	526	3	◦	◦	ADJ
ejpam-3889	526	4	π(n))](k(n	π(n))](k(n	NOUN
ejpam-3889	526	5	)	)	PUNCT
ejpam-3889	526	6	)	)	PUNCT
ejpam-3889	527	1	=	=	PUNCT
ejpam-3889	528	1	[	[	X
ejpam-3889	528	2	s−1(π(n)	s−1(π(n)	NUM
ejpam-3889	528	3	◦	◦	NOUN
ejpam-3889	528	4	f(n))](k(n	f(n))](k(n	NOUN
ejpam-3889	528	5	)	)	PUNCT
ejpam-3889	528	6	)	)	PUNCT
ejpam-3889	529	1	=	=	PUNCT
ejpam-3889	529	2	0	0	PUNCT
ejpam-3889	530	1	=	=	AUX
ejpam-3889	530	2	⇒	⇒	X
ejpam-3889	530	3	[	[	X
ejpam-3889	530	4	s−1(f(n))](π(n)(k(n	s−1(f(n))](π(n)(k(n	X
ejpam-3889	530	5	)	)	PUNCT
ejpam-3889	530	6	)	)	PUNCT
ejpam-3889	530	7	)	)	PUNCT
ejpam-3889	531	1	=	=	PUNCT
ejpam-3889	531	2	0	0	PUNCT
ejpam-3889	532	1	=	=	AUX
ejpam-3889	532	2	⇒	⇒	X
ejpam-3889	532	3	[	[	X
ejpam-3889	532	4	s−1(π(n))](k(n	s−1(π(n))](k(n	X
ejpam-3889	532	5	)	)	PUNCT
ejpam-3889	532	6	)	)	PUNCT
ejpam-3889	533	1	⊂	⊂	PROPN
ejpam-3889	533	2	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	533	3	)	)	PUNCT
ejpam-3889	533	4	)	)	PUNCT
ejpam-3889	534	1	=	=	VERB
ejpam-3889	534	2	⇒	⇒	NOUN
ejpam-3889	534	3	s−1(k(n	s−1(k(n	NOUN
ejpam-3889	534	4	)	)	PUNCT
ejpam-3889	534	5	)	)	PUNCT
ejpam-3889	534	6	⊂	⊂	PROPN
ejpam-3889	534	7	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	534	8	)	)	PUNCT
ejpam-3889	534	9	)	)	PUNCT
ejpam-3889	534	10	since	since	SCONJ
ejpam-3889	534	11	m∗	m∗	NOUN
ejpam-3889	534	12	is	be	AUX
ejpam-3889	534	13	quasiprojective	quasiprojective	ADJ
ejpam-3889	534	14	=	=	NOUN
ejpam-3889	534	15	⇒	⇒	NOUN
ejpam-3889	534	16	s−1m∗	s−1m∗	NOUN
ejpam-3889	534	17	quasi	quasi	NOUN
ejpam-3889	534	18	-	-	NOUN
ejpam-3889	534	19	projective	projective	ADJ
ejpam-3889	534	20	,	,	PUNCT
ejpam-3889	534	21	there	there	PRON
ejpam-3889	534	22	exists	exist	VERB
ejpam-3889	534	23	an	an	DET
ejpam-3889	534	24	endomorphism	endomorphism	NOUN
ejpam-3889	534	25	s−(s(n	s−(s(n	NOUN
ejpam-3889	534	26	)	)	PUNCT
ejpam-3889	534	27	)	)	PUNCT
ejpam-3889	534	28	:	:	PUNCT
ejpam-3889	534	29	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	534	30	)	)	PUNCT
ejpam-3889	534	31	)	)	PUNCT
ejpam-3889	534	32	−→	−→	NOUN
ejpam-3889	534	33	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	534	34	)	)	PUNCT
ejpam-3889	534	35	)	)	PUNCT
ejpam-3889	534	36	such	such	ADJ
ejpam-3889	534	37	that	that	DET
ejpam-3889	534	38	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	534	39	)	)	PUNCT
ejpam-3889	534	40	◦	◦	NOUN
ejpam-3889	534	41	s(n	s(n	PROPN
ejpam-3889	534	42	)	)	PUNCT
ejpam-3889	534	43	)	)	PUNCT
ejpam-3889	535	1	=	=	SYM
ejpam-3889	536	1	s−1(id(n)s−1(m(n	s−1(id(n)s−1(m(n	NOUN
ejpam-3889	536	2	)	)	PUNCT
ejpam-3889	536	3	)	)	PUNCT
ejpam-3889	536	4	)	)	PUNCT
ejpam-3889	537	1	,	,	PUNCT
ejpam-3889	537	2	this	this	PRON
ejpam-3889	537	3	implies	imply	VERB
ejpam-3889	537	4	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	537	5	)	)	PUNCT
ejpam-3889	537	6	)	)	PUNCT
ejpam-3889	538	1	=	=	SYM
ejpam-3889	538	2	s−1(k(n	s−1(k(n	X
ejpam-3889	538	3	)	)	PUNCT
ejpam-3889	538	4	⊕	⊕	PROPN
ejpam-3889	538	5	ims(n	ims(n	PROPN
ejpam-3889	538	6	)	)	PUNCT
ejpam-3889	538	7	)	)	PUNCT
ejpam-3889	538	8	,	,	PUNCT
ejpam-3889	538	9	or	or	CCONJ
ejpam-3889	538	10	k(n	k(n	X
ejpam-3889	538	11	)	)	PUNCT
ejpam-3889	538	12	=	=	SYM
ejpam-3889	538	13	n(n	n(n	NOUN
ejpam-3889	538	14	)	)	PUNCT
ejpam-3889	538	15	and	and	CCONJ
ejpam-3889	538	16	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	538	17	)	)	PUNCT
ejpam-3889	538	18	)	)	PUNCT
ejpam-3889	538	19	is	be	AUX
ejpam-3889	538	20	superfluous	superfluous	ADJ
ejpam-3889	538	21	in	in	ADP
ejpam-3889	538	22	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	538	23	)	)	PUNCT
ejpam-3889	538	24	)	)	PUNCT
ejpam-3889	538	25	,	,	PUNCT
ejpam-3889	538	26	∀n	∀n	X
ejpam-3889	538	27	∈	∈	PROPN
ejpam-3889	539	1	z	z	NOUN
ejpam-3889	539	2	,	,	PUNCT
ejpam-3889	539	3	then	then	ADV
ejpam-3889	539	4	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	539	5	)	)	PUNCT
ejpam-3889	539	6	)	)	PUNCT
ejpam-3889	540	1	=	=	PUNCT
ejpam-3889	540	2	s−1(ims(n	s−1(ims(n	NOUN
ejpam-3889	540	3	)	)	PUNCT
ejpam-3889	540	4	)	)	PUNCT
ejpam-3889	540	5	,	,	PUNCT
ejpam-3889	540	6	so	so	ADV
ejpam-3889	540	7	s−1(k(n	s−1(k(n	X
ejpam-3889	540	8	)	)	PUNCT
ejpam-3889	540	9	)	)	PUNCT
ejpam-3889	541	1	=	=	SYM
ejpam-3889	541	2	ker(s−1(f)(n	ker(s−1(f)(n	PROPN
ejpam-3889	541	3	)	)	PUNCT
ejpam-3889	541	4	)	)	PUNCT
ejpam-3889	542	1	=	=	SYM
ejpam-3889	542	2	0,∀n	0,∀n	PUNCT
ejpam-3889	542	3	∈	∈	PROPN
ejpam-3889	542	4	z	z	NOUN
ejpam-3889	542	5	,	,	PUNCT
ejpam-3889	542	6	thus	thus	ADV
ejpam-3889	542	7	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	542	8	)	)	PUNCT
ejpam-3889	542	9	)	)	PUNCT
ejpam-3889	542	10	is	be	AUX
ejpam-3889	542	11	a	a	DET
ejpam-3889	542	12	monomorphism	monomorphism	NOUN
ejpam-3889	542	13	,	,	PUNCT
ejpam-3889	542	14	∀n	∀n	SYM
ejpam-3889	542	15	∈	∈	PROPN
ejpam-3889	542	16	z	z	NOUN
ejpam-3889	542	17	finally	finally	ADV
ejpam-3889	542	18	,	,	PUNCT
ejpam-3889	542	19	s−1(f∗	s−1(f∗	PROPN
ejpam-3889	542	20	)	)	PUNCT
ejpam-3889	542	21	is	be	AUX
ejpam-3889	542	22	a	a	DET
ejpam-3889	542	23	monomorphism	monomorphism	NOUN
ejpam-3889	542	24	,	,	PUNCT
ejpam-3889	542	25	so	so	ADV
ejpam-3889	542	26	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	542	27	)	)	PUNCT
ejpam-3889	542	28	is	be	AUX
ejpam-3889	542	29	hopfian	hopfian	ADJ
ejpam-3889	542	30	.	.	PUNCT
ejpam-3889	543	1	reciprocally	reciprocally	PROPN
ejpam-3889	543	2	,	,	PUNCT
ejpam-3889	543	3	if	if	SCONJ
ejpam-3889	543	4	s−1(m∗	s−1(m∗	PROPN
ejpam-3889	543	5	)	)	PUNCT
ejpam-3889	543	6	is	be	AUX
ejpam-3889	543	7	hopfian	hopfian	ADJ
ejpam-3889	543	8	,	,	PUNCT
ejpam-3889	543	9	show	show	VERB
ejpam-3889	543	10	that	that	SCONJ
ejpam-3889	543	11	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	543	12	)	)	PUNCT
ejpam-3889	543	13	is	be	AUX
ejpam-3889	543	14	hopfian	hopfian	ADJ
ejpam-3889	543	15	.	.	PUNCT
ejpam-3889	544	1	let	let	VERB
ejpam-3889	544	2	s−1(ϕ(n	s−1(ϕ(n	NUM
ejpam-3889	544	3	)	)	PUNCT
ejpam-3889	544	4	)	)	PUNCT
ejpam-3889	544	5	:	:	PUNCT
ejpam-3889	545	1	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	545	2	)	)	PUNCT
ejpam-3889	545	3	−→	−→	PROPN
ejpam-3889	545	4	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	545	5	)	)	PUNCT
ejpam-3889	545	6	an	an	DET
ejpam-3889	545	7	epimorphism	epimorphism	NOUN
ejpam-3889	545	8	of	of	ADP
ejpam-3889	545	9	chain	chain	NOUN
ejpam-3889	545	10	complex	complex	NOUN
ejpam-3889	545	11	,	,	PUNCT
ejpam-3889	545	12	as	as	SCONJ
ejpam-3889	545	13	s−1(m∗	s−1(m∗	NOUN
ejpam-3889	545	14	)	)	PUNCT
ejpam-3889	545	15	is	be	AUX
ejpam-3889	545	16	quasi	quasi	ADJ
ejpam-3889	545	17	-	-	ADJ
ejpam-3889	545	18	projective	projective	ADJ
ejpam-3889	545	19	,	,	PUNCT
ejpam-3889	545	20	then	then	ADV
ejpam-3889	545	21	∀n	∀n	NUM
ejpam-3889	545	22	∈	∈	PROPN
ejpam-3889	545	23	z	z	PROPN
ejpam-3889	545	24	,	,	PUNCT
ejpam-3889	545	25	s−(m(n	s−(m(n	PROPN
ejpam-3889	545	26	)	)	PUNCT
ejpam-3889	545	27	)	)	PUNCT
ejpam-3889	545	28	is	be	AUX
ejpam-3889	545	29	quasi	quasi	ADJ
ejpam-3889	545	30	-	-	ADJ
ejpam-3889	545	31	projective	projective	ADJ
ejpam-3889	545	32	.	.	PUNCT
ejpam-3889	546	1	consider	consider	VERB
ejpam-3889	546	2	s−1(π(n	s−1(π(n	NOUN
ejpam-3889	546	3	)	)	PUNCT
ejpam-3889	546	4	)	)	PUNCT
ejpam-3889	546	5	:	:	PUNCT
ejpam-3889	546	6	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	546	7	)	)	PUNCT
ejpam-3889	546	8	)	)	PUNCT
ejpam-3889	547	1	−→	−→	PROPN
ejpam-3889	547	2	s−1(m(n)/n(n	s−1(m(n)/n(n	PROPN
ejpam-3889	547	3	)	)	PUNCT
ejpam-3889	547	4	)	)	PUNCT
ejpam-3889	547	5	,	,	PUNCT
ejpam-3889	547	6	then	then	ADV
ejpam-3889	547	7	there	there	PRON
ejpam-3889	547	8	exists	exist	VERB
ejpam-3889	547	9	s−1(f(n	s−1(f(n	NOUN
ejpam-3889	547	10	)	)	PUNCT
ejpam-3889	547	11	)	)	PUNCT
ejpam-3889	548	1	∈	∈	PROPN
ejpam-3889	548	2	end(s−1(m(n	end(s−1(m(n	NOUN
ejpam-3889	548	3	)	)	PUNCT
ejpam-3889	548	4	)	)	PUNCT
ejpam-3889	548	5	)	)	PUNCT
ejpam-3889	549	1	such	such	ADJ
ejpam-3889	549	2	that	that	DET
ejpam-3889	549	3	s−1(π(n	s−1(π(n	NOUN
ejpam-3889	549	4	)	)	PUNCT
ejpam-3889	549	5	◦	◦	PROPN
ejpam-3889	549	6	f(n	f(n	PROPN
ejpam-3889	549	7	)	)	PUNCT
ejpam-3889	549	8	)	)	PUNCT
ejpam-3889	550	1	=	=	SYM
ejpam-3889	550	2	s−1(ϕ(n	s−1(ϕ(n	X
ejpam-3889	550	3	)	)	PUNCT
ejpam-3889	550	4	◦	◦	NOUN
ejpam-3889	550	5	π(n	π(n	PROPN
ejpam-3889	550	6	)	)	PUNCT
ejpam-3889	550	7	.	.	PUNCT
ejpam-3889	551	1	since	since	SCONJ
ejpam-3889	551	2	s−1(ϕ(n	s−1(ϕ(n	NUM
ejpam-3889	551	3	)	)	PUNCT
ejpam-3889	551	4	)	)	PUNCT
ejpam-3889	551	5	is	be	AUX
ejpam-3889	551	6	an	an	DET
ejpam-3889	551	7	epimorphism	epimorphism	NOUN
ejpam-3889	551	8	,	,	PUNCT
ejpam-3889	551	9	∀xs	∀xs	PROPN
ejpam-3889	551	10	∈	∈	PROPN
ejpam-3889	551	11	s−1(m(n)/n(n)),∃(yt	s−1(m(n)/n(n)),∃(yt	PROPN
ejpam-3889	551	12	)	)	PUNCT
ejpam-3889	551	13	∈	∈	PROPN
ejpam-3889	551	14	s−1(m(n)/n(n	s−1(m(n)/n(n	PROPN
ejpam-3889	551	15	)	)	PUNCT
ejpam-3889	551	16	)	)	PUNCT
ejpam-3889	551	17	such	such	ADJ
ejpam-3889	551	18	that	that	SCONJ
ejpam-3889	551	19	[	[	X
ejpam-3889	551	20	s−1(ϕ(n))](yt	s−1(ϕ(n))](yt	NUM
ejpam-3889	551	21	)	)	PUNCT
ejpam-3889	551	22	=	=	PUNCT
ejpam-3889	552	1	x	x	SYM
ejpam-3889	552	2	s	s	X
ejpam-3889	552	3	=	=	X
ejpam-3889	553	1	[	[	PUNCT
ejpam-3889	553	2	s−1(ϕ(n))](π(n)(y)t	s−1(ϕ(n))](π(n)(y)t	PROPN
ejpam-3889	553	3	)	)	PUNCT
ejpam-3889	554	1	[	[	X
ejpam-3889	554	2	s−1(π(n	s−1(π(n	ADJ
ejpam-3889	554	3	)	)	PUNCT
ejpam-3889	554	4	◦	◦	NOUN
ejpam-3889	554	5	f(n))](yt	f(n))](yt	NUM
ejpam-3889	554	6	)	)	PUNCT
ejpam-3889	555	1	=	=	PUNCT
ejpam-3889	556	1	[	[	X
ejpam-3889	556	2	s−1(ϕ(n	s−1(ϕ(n	NOUN
ejpam-3889	556	3	)	)	PUNCT
ejpam-3889	556	4	◦	◦	NOUN
ejpam-3889	556	5	π(n))](yt	π(n))](yt	X
ejpam-3889	556	6	)	)	PUNCT
ejpam-3889	557	1	[	[	X
ejpam-3889	557	2	s−1(π(n	s−1(π(n	NOUN
ejpam-3889	557	3	)	)	PUNCT
ejpam-3889	557	4	)	)	PUNCT
ejpam-3889	557	5	]	]	PUNCT
ejpam-3889	557	6	(	(	PUNCT
ejpam-3889	557	7	(	(	PUNCT
ejpam-3889	557	8	f(n)(x))s	f(n)(x))	NOUN
ejpam-3889	557	9	)	)	PUNCT
ejpam-3889	557	10	=	=	PUNCT
ejpam-3889	558	1	[	[	X
ejpam-3889	558	2	s−1(ϕ)](yt	s−1(ϕ)](yt	X
ejpam-3889	558	3	)	)	PUNCT
ejpam-3889	559	1	=	=	VERB
ejpam-3889	559	2	⇒	⇒	NOUN
ejpam-3889	559	3	[	[	X
ejpam-3889	559	4	s−1(ϕ(n))](yt	s−1(ϕ(n))](yt	NUM
ejpam-3889	559	5	)	)	PUNCT
ejpam-3889	559	6	=	=	PUNCT
ejpam-3889	560	1	[	[	X
ejpam-3889	560	2	s−1(f(n))](yt	s−1(f(n))](yt	NUM
ejpam-3889	560	3	)	)	PUNCT
ejpam-3889	560	4	=	=	PUNCT
ejpam-3889	561	1	x	x	SYM
ejpam-3889	561	2	s	s	NOUN
ejpam-3889	561	3	=	=	NOUN
ejpam-3889	561	4	⇒	⇒	NOUN
ejpam-3889	561	5	(	(	PUNCT
ejpam-3889	561	6	f(n))(y)t	f(n))(y)t	PROPN
ejpam-3889	561	7	−	−	VERB
ejpam-3889	561	8	x	x	SYM
ejpam-3889	561	9	s	s	NOUN
ejpam-3889	561	10	=	=	SYM
ejpam-3889	561	11	0	0	PUNCT
ejpam-3889	561	12	=	=	NOUN
ejpam-3889	561	13	⇒	⇒	NOUN
ejpam-3889	561	14	f(n)(y	f(n)(y	NUM
ejpam-3889	561	15	)	)	PUNCT
ejpam-3889	561	16	t	t	NOUN
ejpam-3889	561	17	−	−	NOUN
ejpam-3889	561	18	x	x	SYM
ejpam-3889	561	19	s	s	X
ejpam-3889	561	20	∈	∈	NOUN
ejpam-3889	561	21	s	s	PART
ejpam-3889	561	22	−1(n(n	−1(n(n	NOUN
ejpam-3889	561	23	)	)	PUNCT
ejpam-3889	561	24	)	)	PUNCT
ejpam-3889	561	25	,	,	PUNCT
ejpam-3889	561	26	then	then	ADV
ejpam-3889	561	27	s−1(m	s−1(m	PROPN
ejpam-3889	561	28	)	)	PUNCT
ejpam-3889	561	29	=	=	SYM
ejpam-3889	561	30	s−1(im(f)(n))+s−1(n(n	s−1(im(f)(n))+s−1(n(n	NOUN
ejpam-3889	561	31	)	)	PUNCT
ejpam-3889	561	32	,	,	PUNCT
ejpam-3889	561	33	as	as	ADP
ejpam-3889	561	34	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	561	35	)	)	PUNCT
ejpam-3889	561	36	)	)	PUNCT
ejpam-3889	561	37	is	be	AUX
ejpam-3889	561	38	superfluous	superfluous	ADJ
ejpam-3889	561	39	,	,	PUNCT
ejpam-3889	561	40	then	then	ADV
ejpam-3889	561	41	s−1(im(f)(n	s−1(im(f)(n	NOUN
ejpam-3889	561	42	)	)	PUNCT
ejpam-3889	561	43	)	)	PUNCT
ejpam-3889	562	1	=	=	SYM
ejpam-3889	562	2	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	562	3	)	)	PUNCT
ejpam-3889	562	4	)	)	PUNCT
ejpam-3889	563	1	=	=	SYM
ejpam-3889	563	2	⇒	⇒	X
ejpam-3889	563	3	s−(f(n	s−(f(n	PROPN
ejpam-3889	563	4	)	)	PUNCT
ejpam-3889	563	5	)	)	PUNCT
ejpam-3889	563	6	is	be	AUX
ejpam-3889	563	7	an	an	DET
ejpam-3889	563	8	epimorphism	epimorphism	NOUN
ejpam-3889	563	9	.	.	PUNCT
ejpam-3889	564	1	so	so	ADV
ejpam-3889	564	2	,	,	PUNCT
ejpam-3889	564	3	s−1f(n	s−1f(n	PROPN
ejpam-3889	564	4	)	)	PUNCT
ejpam-3889	564	5	is	be	AUX
ejpam-3889	564	6	an	an	DET
ejpam-3889	564	7	automorphism	automorphism	NOUN
ejpam-3889	564	8	,	,	PUNCT
ejpam-3889	564	9	because	because	SCONJ
ejpam-3889	564	10	s−1(m(n	s−1(m(n	NOUN
ejpam-3889	564	11	)	)	PUNCT
ejpam-3889	564	12	)	)	PUNCT
ejpam-3889	565	1	is	be	AUX
ejpam-3889	565	2	hopfian	hopfian	ADJ
ejpam-3889	565	3	for	for	ADP
ejpam-3889	565	4	all	all	DET
ejpam-3889	565	5	n	n	PRON
ejpam-3889	565	6	∈	∈	NOUN
ejpam-3889	565	7	z.	z.	PROPN
ejpam-3889	566	1	so	so	ADV
ejpam-3889	566	2	the	the	DET
ejpam-3889	566	3	restriction	restriction	NOUN
ejpam-3889	566	4	of	of	ADP
ejpam-3889	566	5	s−1(f(n	s−1(f(n	PROPN
ejpam-3889	566	6	)	)	PUNCT
ejpam-3889	566	7	)	)	PUNCT
ejpam-3889	566	8	over	over	ADP
ejpam-3889	566	9	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	566	10	)	)	PUNCT
ejpam-3889	566	11	)	)	PUNCT
ejpam-3889	566	12	is	be	AUX
ejpam-3889	566	13	an	an	DET
ejpam-3889	566	14	automorphisme	automorphisme	NOUN
ejpam-3889	566	15	of	of	ADP
ejpam-3889	566	16	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	566	17	)	)	PUNCT
ejpam-3889	566	18	)	)	PUNCT
ejpam-3889	566	19	.	.	PUNCT
ejpam-3889	567	1	if	if	SCONJ
ejpam-3889	567	2	[	[	X
ejpam-3889	567	3	s−1(ϕ)(n)](xs	s−1(ϕ)(n)](xs	INTJ
ejpam-3889	567	4	)	)	PUNCT
ejpam-3889	567	5	=	=	PUNCT
ejpam-3889	568	1	[	[	X
ejpam-3889	568	2	s−1(f(n))](xs	s−1(f(n))](xs	NUM
ejpam-3889	568	3	)	)	PUNCT
ejpam-3889	568	4	=	=	SYM
ejpam-3889	568	5	0	0	NUM
ejpam-3889	568	6	,	,	PUNCT
ejpam-3889	568	7	then	then	ADV
ejpam-3889	568	8	s−1[(f(n))](xs	s−1[(f(n))](xs	ADJ
ejpam-3889	568	9	)	)	PUNCT
ejpam-3889	568	10	∈	∈	PROPN
ejpam-3889	568	11	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	568	12	)	)	PUNCT
ejpam-3889	568	13	)	)	PUNCT
ejpam-3889	568	14	,	,	PUNCT
ejpam-3889	568	15	or	or	CCONJ
ejpam-3889	568	16	s−1(n(n	s−1(n(n	PROPN
ejpam-3889	568	17	)	)	PUNCT
ejpam-3889	568	18	)	)	PUNCT
ejpam-3889	568	19	is	be	AUX
ejpam-3889	568	20	completely	completely	ADV
ejpam-3889	568	21	invariant	invariant	ADJ
ejpam-3889	568	22	,	,	PUNCT
ejpam-3889	568	23	then	then	ADV
ejpam-3889	568	24	x	x	X
ejpam-3889	568	25	s	s	PROPN
ejpam-3889	568	26	∈	∈	PROPN
ejpam-3889	568	27	s	s	PART
ejpam-3889	568	28	−1(n(n	−1(n(n	NOUN
ejpam-3889	568	29	)	)	PUNCT
ejpam-3889	568	30	)	)	PUNCT
ejpam-3889	568	31	,	,	PUNCT
ejpam-3889	568	32	so	so	CCONJ
ejpam-3889	568	33	x	x	X
ejpam-3889	568	34	s	s	NOUN
ejpam-3889	568	35	=	=	SYM
ejpam-3889	568	36	0	0	PUNCT
ejpam-3889	569	1	=	=	NOUN
ejpam-3889	569	2	⇒	⇒	NOUN
ejpam-3889	569	3	ker(s−1(ϕ(n	ker(s−1(ϕ(n	PROPN
ejpam-3889	569	4	)	)	PUNCT
ejpam-3889	569	5	)	)	PUNCT
ejpam-3889	569	6	)	)	PUNCT
ejpam-3889	570	1	=	=	PUNCT
ejpam-3889	570	2	s−1(n(n	s−1(n(n	NOUN
ejpam-3889	570	3	)	)	PUNCT
ejpam-3889	570	4	)	)	PUNCT
ejpam-3889	571	1	=	=	PUNCT
ejpam-3889	571	2	0	0	PUNCT
ejpam-3889	572	1	=	=	AUX
ejpam-3889	572	2	⇒	⇒	NOUN
ejpam-3889	572	3	s−1(ϕ(n	s−1(ϕ(n	NUM
ejpam-3889	572	4	)	)	PUNCT
ejpam-3889	572	5	)	)	PUNCT
ejpam-3889	572	6	is	be	AUX
ejpam-3889	572	7	a	a	DET
ejpam-3889	572	8	monomorphism	monomorphism	NOUN
ejpam-3889	572	9	=	=	PRON
ejpam-3889	572	10	⇒	⇒	NOUN
ejpam-3889	572	11	ϕ(n	ϕ(n	PROPN
ejpam-3889	572	12	)	)	PUNCT
ejpam-3889	572	13	is	be	AUX
ejpam-3889	572	14	an	an	DET
ejpam-3889	572	15	automorphism	automorphism	NOUN
ejpam-3889	572	16	for	for	ADP
ejpam-3889	572	17	all	all	DET
ejpam-3889	572	18	n	n	PRON
ejpam-3889	572	19	∈	∈	PROPN
ejpam-3889	572	20	z	z	NOUN
ejpam-3889	572	21	,	,	PUNCT
ejpam-3889	572	22	lastly	lastly	ADV
ejpam-3889	572	23	s−1(m(n)/n(n	s−1(m(n)/n(n	PROPN
ejpam-3889	572	24	)	)	PUNCT
ejpam-3889	572	25	)	)	PUNCT
ejpam-3889	573	1	is	be	AUX
ejpam-3889	573	2	hopfian	hopfian	ADJ
ejpam-3889	573	3	for	for	ADP
ejpam-3889	573	4	all	all	DET
ejpam-3889	573	5	n	n	PRON
ejpam-3889	573	6	∈	∈	PROPN
ejpam-3889	573	7	z	z	NOUN
ejpam-3889	573	8	,	,	PUNCT
ejpam-3889	573	9	hence	hence	ADV
ejpam-3889	573	10	s−1(m∗/n∗	s−1(m∗/n∗	PROPN
ejpam-3889	573	11	)	)	PUNCT
ejpam-3889	573	12	is	be	AUX
ejpam-3889	573	13	a	a	DET
ejpam-3889	573	14	hopfian	hopfian	ADJ
ejpam-3889	573	15	complex	complex	ADJ
ejpam-3889	573	16	sequence	sequence	NOUN
ejpam-3889	573	17	.	.	PUNCT
ejpam-3889	574	1	acknowledgements	acknowledgement	NOUN
ejpam-3889	574	2	i	i	PRON
ejpam-3889	574	3	am	be	AUX
ejpam-3889	574	4	grateful	grateful	ADJ
ejpam-3889	574	5	to	to	ADP
ejpam-3889	574	6	professor	professor	PROPN
ejpam-3889	574	7	mohamed	mohamed	PROPN
ejpam-3889	574	8	ben	ben	PROPN
ejpam-3889	574	9	maaouia	maaouia	PROPN
ejpam-3889	574	10	,	,	PUNCT
ejpam-3889	574	11	director	director	NOUN
ejpam-3889	574	12	of	of	ADP
ejpam-3889	574	13	laboratory	laboratory	NOUN
ejpam-3889	574	14	of	of	ADP
ejpam-3889	574	15	algebra	algebra	PROPN
ejpam-3889	574	16	,	,	PUNCT
ejpam-3889	574	17	cryptography	cryptography	NOUN
ejpam-3889	574	18	,	,	PUNCT
ejpam-3889	574	19	codes	code	NOUN
ejpam-3889	574	20	and	and	CCONJ
ejpam-3889	574	21	applications	application	NOUN
ejpam-3889	574	22	(	(	PUNCT
ejpam-3889	574	23	lacca	lacca	VERB
ejpam-3889	574	24	)	)	PUNCT
ejpam-3889	574	25	ufr	ufr	PROPN
ejpam-3889	574	26	-	-	PUNCT
ejpam-3889	574	27	sat	sit	VERB
ejpam-3889	574	28	,	,	PUNCT
ejpam-3889	574	29	for	for	ADP
ejpam-3889	574	30	helpful	helpful	ADJ
ejpam-3889	574	31	discussions	discussion	NOUN
ejpam-3889	574	32	references	reference	VERB
ejpam-3889	574	33	421	421	NUM
ejpam-3889	574	34	references	reference	NOUN
ejpam-3889	574	35	[	[	X
ejpam-3889	574	36	1	1	NUM
ejpam-3889	574	37	]	]	PUNCT
ejpam-3889	574	38	a.	a.	NOUN
ejpam-3889	574	39	o.	o.	PROPN
ejpam-3889	574	40	chbih	chbih	PROPN
ejpam-3889	574	41	,	,	PUNCT
ejpam-3889	574	42	thèse	thèse	PROPN
ejpam-3889	574	43	de	de	X
ejpam-3889	574	44	doctorat	doctorat	PROPN
ejpam-3889	574	45	unique	unique	ADJ
ejpam-3889	574	46	,	,	PUNCT
ejpam-3889	574	47	ufr	ufr	PROPN
ejpam-3889	574	48	sat	sit	VERB
ejpam-3889	574	49	,	,	PUNCT
ejpam-3889	574	50	ugb	ugb	PROPN
ejpam-3889	574	51	,	,	PUNCT
ejpam-3889	574	52	saint	saint	NOUN
ejpam-3889	574	53	-	-	PUNCT
ejpam-3889	574	54	louis	louis	NOUN
ejpam-3889	574	55	,	,	PUNCT
ejpam-3889	574	56	avril	avril	PROPN
ejpam-3889	574	57	,	,	PUNCT
ejpam-3889	574	58	2016	2016	NUM
ejpam-3889	574	59	[	[	X
ejpam-3889	574	60	2	2	X
ejpam-3889	574	61	]	]	X
ejpam-3889	574	62	a.o	a.o	PROPN
ejpam-3889	574	63	chbih	chbih	PROPN
ejpam-3889	574	64	,	,	PUNCT
ejpam-3889	574	65	m.	m.	NOUN
ejpam-3889	574	66	ben	ben	PROPN
ejpam-3889	574	67	maaouia	maaouia	PROPN
ejpam-3889	574	68	and	and	CCONJ
ejpam-3889	574	69	m.	m.	NOUN
ejpam-3889	574	70	sanghare	sanghare	PROPN
ejpam-3889	574	71	graduation	graduation	NOUN
ejpam-3889	574	72	of	of	ADP
ejpam-3889	574	73	module	module	NOUN
ejpam-3889	574	74	of	of	ADP
ejpam-3889	574	75	fraction	fraction	NOUN
ejpam-3889	574	76	on	on	ADP
ejpam-3889	574	77	a	a	DET
ejpam-3889	574	78	graded	grade	VERB
ejpam-3889	574	79	domain	domain	NOUN
ejpam-3889	574	80	ring	ring	NOUN
ejpam-3889	574	81	not	not	PART
ejpam-3889	574	82	necessarily	necessarily	ADV
ejpam-3889	574	83	commutative	commutative	ADJ
ejpam-3889	574	84	,	,	PUNCT
ejpam-3889	574	85	international	international	ADJ
ejpam-3889	574	86	journal	journal	NOUN
ejpam-3889	574	87	of	of	ADP
ejpam-3889	574	88	algebra	algebra	PROPN
ejpam-3889	574	89	,	,	PUNCT
ejpam-3889	574	90	vol	vol	NOUN
ejpam-3889	574	91	.	.	NOUN
ejpam-3889	574	92	9	9	NUM
ejpam-3889	574	93	,	,	PUNCT
ejpam-3889	574	94	2015	2015	NUM
ejpam-3889	574	95	,	,	PUNCT
ejpam-3889	574	96	no	no	INTJ
ejpam-3889	574	97	.	.	NOUN
ejpam-3889	574	98	10	10	NUM
ejpam-3889	574	99	,	,	PUNCT
ejpam-3889	574	100	457	457	NUM
ejpam-3889	574	101	474	474	NUM
ejpam-3889	574	102	.	.	PUNCT
ejpam-3889	575	1	[	[	X
ejpam-3889	575	2	3	3	X
ejpam-3889	575	3	]	]	X
ejpam-3889	575	4	a.o	a.o	PROPN
ejpam-3889	575	5	chbih	chbih	PROPN
ejpam-3889	575	6	,	,	PUNCT
ejpam-3889	575	7	m.	m.	NOUN
ejpam-3889	575	8	ben	ben	PROPN
ejpam-3889	575	9	maaouia	maaouia	PROPN
ejpam-3889	575	10	and	and	CCONJ
ejpam-3889	575	11	m.	m.	NOUN
ejpam-3889	575	12	sanghare	sanghare	NOUN
ejpam-3889	575	13	factorization	factorization	NOUN
ejpam-3889	575	14	of	of	ADP
ejpam-3889	575	15	graded	grade	VERB
ejpam-3889	575	16	modules	module	NOUN
ejpam-3889	575	17	of	of	ADP
ejpam-3889	575	18	fractions	fraction	NOUN
ejpam-3889	575	19	,	,	PUNCT
ejpam-3889	575	20	international	international	PROPN
ejpam-3889	575	21	mathematical	mathematical	ADJ
ejpam-3889	575	22	forum	forum	PROPN
ejpam-3889	575	23	,	,	PUNCT
ejpam-3889	575	24	vol	vol	NOUN
ejpam-3889	575	25	.	.	PROPN
ejpam-3889	575	26	11	11	NUM
ejpam-3889	575	27	,	,	PUNCT
ejpam-3889	575	28	2016	2016	NUM
ejpam-3889	575	29	,	,	PUNCT
ejpam-3889	575	30	no	no	INTJ
ejpam-3889	575	31	.	.	NOUN
ejpam-3889	575	32	22	22	NUM
ejpam-3889	575	33	,	,	PUNCT
ejpam-3889	575	34	1067	1067	NUM
ejpam-3889	575	35	1088	1088	NUM
ejpam-3889	575	36	.	.	PUNCT
ejpam-3889	576	1	[	[	X
ejpam-3889	576	2	4	4	NUM
ejpam-3889	576	3	]	]	PUNCT
ejpam-3889	576	4	a.	a.	NOUN
ejpam-3889	576	5	hmainou	hmainou	PROPN
ejpam-3889	576	6	,	,	PUNCT
ejpam-3889	576	7	a.	a.	NOUN
ejpam-3889	576	8	kaidi	kaidi	PROPN
ejpam-3889	576	9	and	and	CCONJ
ejpam-3889	576	10	e.	e.	PROPN
ejpam-3889	576	11	c.	c.	PROPN
ejpam-3889	576	12	sanchez	sanchez	PROPN
ejpam-3889	576	13	generalising	generalise	VERB
ejpam-3889	576	14	fitting	fitting	ADJ
ejpam-3889	576	15	modules	module	NOUN
ejpam-3889	576	16	and	and	CCONJ
ejpam-3889	576	17	rings	ring	NOUN
ejpam-3889	576	18	,	,	PUNCT
ejpam-3889	576	19	journal	journal	NOUN
ejpam-3889	576	20	of	of	ADP
ejpam-3889	576	21	algebra	algebra	PROPN
ejpam-3889	576	22	.	.	PUNCT
ejpam-3889	577	1	(	(	PUNCT
ejpam-3889	577	2	2006	2006	NUM
ejpam-3889	577	3	)	)	PUNCT
ejpam-3889	577	4	.	.	PUNCT
ejpam-3889	578	1	[	[	X
ejpam-3889	578	2	5	5	X
ejpam-3889	578	3	]	]	PUNCT
ejpam-3889	578	4	c.	c.	PROPN
ejpam-3889	578	5	nastasescu	nastasescu	PROPN
ejpam-3889	578	6	and	and	CCONJ
ejpam-3889	578	7	van	van	PROPN
ejpam-3889	578	8	oystaeyen	oystaeyen	PROPN
ejpam-3889	578	9	f.	f.	PROPN
ejpam-3889	578	10	graded	grade	VERB
ejpam-3889	578	11	ring	ring	NOUN
ejpam-3889	578	12	theory	theory	NOUN
ejpam-3889	578	13	,	,	PUNCT
ejpam-3889	578	14	mathematical	mathematical	ADJ
ejpam-3889	578	15	library	library	NOUN
ejpam-3889	578	16	28	28	NUM
ejpam-3889	578	17	,	,	PUNCT
ejpam-3889	578	18	north	north	NOUN
ejpam-3889	578	19	holland	holland	PROPN
ejpam-3889	578	20	,	,	PUNCT
ejpam-3889	578	21	amesterdam	amesterdam	PROPN
ejpam-3889	578	22	.	.	PUNCT
ejpam-3889	579	1	(	(	PUNCT
ejpam-3889	579	2	1982	1982	NUM
ejpam-3889	579	3	)	)	PUNCT
ejpam-3889	580	1	[	[	X
ejpam-3889	580	2	6	6	NUM
ejpam-3889	580	3	]	]	PUNCT
ejpam-3889	580	4	e.	e.	PROPN
ejpam-3889	580	5	c.	c.	PROPN
ejpam-3889	580	6	dade	dade	PROPN
ejpam-3889	580	7	,	,	PUNCT
ejpam-3889	580	8	group	group	NOUN
ejpam-3889	580	9	graded	grade	VERB
ejpam-3889	580	10	rings	ring	NOUN
ejpam-3889	580	11	and	and	CCONJ
ejpam-3889	580	12	modules	module	NOUN
ejpam-3889	580	13	,	,	PUNCT
ejpam-3889	580	14	math	math	NOUN
ejpam-3889	580	15	.	.	PUNCT
ejpam-3889	581	1	z.	z.	PROPN
ejpam-3889	581	2	,	,	PUNCT
ejpam-3889	581	3	174	174	NUM
ejpam-3889	581	4	(	(	PUNCT
ejpam-3889	581	5	1980	1980	NUM
ejpam-3889	581	6	)	)	PUNCT
ejpam-3889	581	7	,	,	PUNCT
ejpam-3889	581	8	241	241	PROPN
ejpam-3889	581	9	-	-	SYM
ejpam-3889	581	10	262	262	NUM
ejpam-3889	581	11	.	.	PUNCT
ejpam-3889	582	1	[	[	X
ejpam-3889	582	2	7	7	X
ejpam-3889	582	3	]	]	X
ejpam-3889	582	4	e.	e.	PROPN
ejpam-3889	582	5	o.	o.	PROPN
ejpam-3889	582	6	diallo	diallo	PROPN
ejpam-3889	582	7	,	,	PUNCT
ejpam-3889	582	8	thèse	thèse	PROPN
ejpam-3889	582	9	de	de	X
ejpam-3889	582	10	doctorat	doctorat	PROPN
ejpam-3889	582	11	unique	unique	ADJ
ejpam-3889	582	12	,	,	PUNCT
ejpam-3889	582	13	faculté	faculté	NOUN
ejpam-3889	582	14	des	des	PROPN
ejpam-3889	582	15	sciences	sciences	PROPN
ejpam-3889	582	16	et	et	PROPN
ejpam-3889	582	17	techniques	technique	NOUN
ejpam-3889	582	18	,	,	PUNCT
ejpam-3889	582	19	ucad	ucad	ADJ
ejpam-3889	582	20	,	,	PUNCT
ejpam-3889	582	21	dakar	dakar	NOUN
ejpam-3889	582	22	,	,	PUNCT
ejpam-3889	582	23	octobre	octobre	PROPN
ejpam-3889	582	24	,	,	PUNCT
ejpam-3889	582	25	2013	2013	NUM
ejpam-3889	583	1	[	[	X
ejpam-3889	583	2	8	8	NUM
ejpam-3889	583	3	]	]	PUNCT
ejpam-3889	583	4	e.	e.	PROPN
ejpam-3889	583	5	enoch	enoch	PROPN
ejpam-3889	583	6	,	,	PUNCT
ejpam-3889	583	7	o.	o.	PROPN
ejpam-3889	583	8	m.g	m.g	PROPN
ejpam-3889	583	9	.	.	PROPN
ejpam-3889	583	10	jenda	jenda	PROPN
ejpam-3889	583	11	,	,	PUNCT
ejpam-3889	583	12	relative	relative	ADJ
ejpam-3889	583	13	homological	homological	ADJ
ejpam-3889	583	14	algebra	algebra	NOUN
ejpam-3889	583	15	,	,	PUNCT
ejpam-3889	583	16	mathematics	mathematic	NOUN
ejpam-3889	583	17	subject	subject	ADJ
ejpam-3889	583	18	classification	classification	NOUN
ejpam-3889	583	19	,	,	PUNCT
ejpam-3889	583	20	2010	2010	NUM
ejpam-3889	584	1	.	.	PUNCT
ejpam-3889	585	1	[	[	X
ejpam-3889	585	2	9	9	NUM
ejpam-3889	585	3	]	]	PUNCT
ejpam-3889	585	4	f.	f.	PROPN
ejpam-3889	585	5	w.	w.	PROPN
ejpam-3889	585	6	anderson	anderson	PROPN
ejpam-3889	585	7	and	and	CCONJ
ejpam-3889	585	8	k.	k.	PROPN
ejpam-3889	585	9	r.	r.	PROPN
ejpam-3889	585	10	fuller	fuller	PROPN
ejpam-3889	585	11	,	,	PUNCT
ejpam-3889	585	12	rings	ring	NOUN
ejpam-3889	585	13	and	and	CCONJ
ejpam-3889	585	14	categories	category	NOUN
ejpam-3889	585	15	of	of	ADP
ejpam-3889	585	16	modules	module	NOUN
ejpam-3889	585	17	,	,	PUNCT
ejpam-3889	585	18	springer	springer	NOUN
ejpam-3889	585	19	-	-	PUNCT
ejpam-3889	585	20	verlag	verlag	NOUN
ejpam-3889	586	1	[	[	X
ejpam-3889	586	2	10	10	NUM
ejpam-3889	586	3	]	]	X
ejpam-3889	586	4	g.	g.	PROPN
ejpam-3889	586	5	renault	renault	PROPN
ejpam-3889	586	6	,	,	PUNCT
ejpam-3889	586	7	algèbre	algèbre	PROPN
ejpam-3889	586	8	non	non	X
ejpam-3889	586	9	commutative	commutative	ADJ
ejpam-3889	586	10	,	,	PUNCT
ejpam-3889	586	11	gauthier	gauthier	NOUN
ejpam-3889	586	12	-	-	PUNCT
ejpam-3889	586	13	villars	villar	NOUN
ejpam-3889	586	14	,	,	PUNCT
ejpam-3889	586	15	paris	paris	PROPN
ejpam-3889	586	16	-	-	PUNCT
ejpam-3889	586	17	bruxelles	bruxelles	PROPN
ejpam-3889	586	18	-	-	PUNCT
ejpam-3889	586	19	montréal	montréal	PROPN
ejpam-3889	586	20	(	(	PUNCT
ejpam-3889	586	21	1975	1975	NUM
ejpam-3889	586	22	)	)	PUNCT
ejpam-3889	586	23	.	.	PUNCT
ejpam-3889	587	1	[	[	X
ejpam-3889	587	2	11	11	NUM
ejpam-3889	587	3	]	]	PUNCT
ejpam-3889	587	4	j.	j.	PROPN
ejpam-3889	587	5	rotman	rotman	PROPN
ejpam-3889	587	6	,	,	PUNCT
ejpam-3889	587	7	notes	note	NOUN
ejpam-3889	587	8	on	on	ADP
ejpam-3889	587	9	homological	homological	ADJ
ejpam-3889	587	10	algebra	algebra	NOUN
ejpam-3889	587	11	,	,	PUNCT
ejpam-3889	587	12	university	university	NOUN
ejpam-3889	587	13	of	of	ADP
ejpam-3889	587	14	illinois	illinois	PROPN
ejpam-3889	587	15	,	,	PUNCT
ejpam-3889	587	16	uraba	uraba	PROPN
ejpam-3889	587	17	(	(	PUNCT
ejpam-3889	587	18	1968	1968	NUM
ejpam-3889	587	19	)	)	PUNCT
ejpam-3889	587	20	.	.	PUNCT
ejpam-3889	588	1	[	[	X
ejpam-3889	588	2	12	12	NUM
ejpam-3889	588	3	]	]	PUNCT
ejpam-3889	588	4	k.	k.	PROPN
ejpam-3889	588	5	divaani	divaani	PROPN
ejpam-3889	588	6	-	-	PUNCT
ejpam-3889	588	7	aazar1	aazar1	PROPN
ejpam-3889	588	8	and	and	CCONJ
ejpam-3889	588	9	a.	a.	NOUN
ejpam-3889	588	10	mafi	mafi	PROPN
ejpam-3889	588	11	hopfian	hopfian	PROPN
ejpam-3889	588	12	and	and	CCONJ
ejpam-3889	588	13	co	co	ADJ
ejpam-3889	588	14	-	-	ADJ
ejpam-3889	588	15	hopfian	hopfian	ADJ
ejpam-3889	588	16	modules	module	NOUN
ejpam-3889	588	17	over	over	ADP
ejpam-3889	588	18	commutative	commutative	ADJ
ejpam-3889	588	19	rings	ring	NOUN
ejpam-3889	588	20	,	,	PUNCT
ejpam-3889	588	21	vietnam	vietnam	PROPN
ejpam-3889	588	22	journal	journal	NOUN
ejpam-3889	588	23	of	of	ADP
ejpam-3889	588	24	mathematics35:3	mathematics35:3	PROPN
ejpam-3889	588	25	275–283	275–283	NUM
ejpam-3889	588	26	(	(	PUNCT
ejpam-3889	588	27	2007	2007	NUM
ejpam-3889	588	28	)	)	PUNCT
ejpam-3889	589	1	[	[	X
ejpam-3889	589	2	13	13	NUM
ejpam-3889	589	3	]	]	PUNCT
ejpam-3889	589	4	k.	k.	NOUN
ejpam-3889	589	5	varadarajan	varadarajan	PROPN
ejpam-3889	589	6	hopfian	hopfian	PROPN
ejpam-3889	589	7	and	and	CCONJ
ejpam-3889	589	8	cohopfian	cohopfian	ADJ
ejpam-3889	589	9	objects	object	NOUN
ejpam-3889	589	10	,	,	PUNCT
ejpam-3889	589	11	publicacions	publicacion	NOUN
ejpam-3889	589	12	mathemàtiques	mathemàtique	NOUN
ejpam-3889	589	13	research	research	NOUN
ejpam-3889	589	14	;	;	PUNCT
ejpam-3889	589	15	vol	vol	NOUN
ejpam-3889	589	16	.	.	PROPN
ejpam-3889	589	17	36	36	NUM
ejpam-3889	589	18	,	,	PUNCT
ejpam-3889	589	19	(	(	PUNCT
ejpam-3889	589	20	1992	1992	NUM
ejpam-3889	589	21	)	)	PUNCT
ejpam-3889	589	22	,	,	PUNCT
ejpam-3889	589	23	293	293	NUM
ejpam-3889	589	24	-	-	SYM
ejpam-3889	589	25	317	317	NUM
ejpam-3889	589	26	.	.	PUNCT
ejpam-3889	590	1	[	[	X
ejpam-3889	590	2	14	14	NUM
ejpam-3889	590	3	]	]	X
ejpam-3889	590	4	l.	l.	PROPN
ejpam-3889	590	5	faouzi	faouzi	PROPN
ejpam-3889	590	6	,	,	PUNCT
ejpam-3889	590	7	mémoire	mémoire	PROPN
ejpam-3889	590	8	de	de	X
ejpam-3889	590	9	fin	fin	NOUN
ejpam-3889	590	10	d’études	d’étude	NOUN
ejpam-3889	590	11	,	,	PUNCT
ejpam-3889	590	12	faculté	faculté	NOUN
ejpam-3889	590	13	des	des	PROPN
ejpam-3889	590	14	sciences	sciences	PROPN
ejpam-3889	590	15	et	et	PROPN
ejpam-3889	590	16	techniques	technique	NOUN
ejpam-3889	590	17	,	,	PUNCT
ejpam-3889	590	18	tetouan	tetouan	PROPN
ejpam-3889	590	19	,	,	PUNCT
ejpam-3889	590	20	maroc	maroc	PROPN
ejpam-3889	590	21	,	,	PUNCT
ejpam-3889	590	22	21	21	NUM
ejpam-3889	590	23	février	février	PROPN
ejpam-3889	590	24	,	,	PUNCT
ejpam-3889	590	25	2003	2003	NUM
ejpam-3889	590	26	.	.	PUNCT
ejpam-3889	591	1	[	[	X
ejpam-3889	591	2	15	15	NUM
ejpam-3889	591	3	]	]	X
ejpam-3889	591	4	m.f.atiyah	m.f.atiyah	NOUN
ejpam-3889	591	5	and	and	CCONJ
ejpam-3889	591	6	i.g.macdonald	i.g.macdonald	NOUN
ejpam-3889	591	7	,	,	PUNCT
ejpam-3889	591	8	introduction	introduction	NOUN
ejpam-3889	591	9	to	to	ADP
ejpam-3889	591	10	commutative	commutative	ADJ
ejpam-3889	591	11	algebra	algebra	NOUN
ejpam-3889	591	12	,	,	PUNCT
ejpam-3889	591	13	addisonwesley	addisonwesley	ADJ
ejpam-3889	591	14	publishing	publishing	NOUN
ejpam-3889	591	15	company	company	NOUN
ejpam-3889	591	16	,	,	PUNCT
ejpam-3889	591	17	university	university	NOUN
ejpam-3889	591	18	of	of	ADP
ejpam-3889	591	19	oxford	oxford	PROPN
ejpam-3889	591	20	.	.	PUNCT
ejpam-3889	592	1	[	[	X
ejpam-3889	592	2	16	16	NUM
ejpam-3889	592	3	]	]	PUNCT
ejpam-3889	592	4	m.	m.	PROPN
ejpam-3889	592	5	f.	f.	PROPN
ejpam-3889	592	6	maaouia	maaouia	PROPN
ejpam-3889	592	7	,	,	PUNCT
ejpam-3889	592	8	thèse	thèse	SCONJ
ejpam-3889	592	9	d’état	d’état	PROPN
ejpam-3889	592	10	,	,	PUNCT
ejpam-3889	592	11	faculté	faculté	NOUN
ejpam-3889	592	12	des	des	PROPN
ejpam-3889	592	13	sciences	sciences	PROPN
ejpam-3889	592	14	et	et	PROPN
ejpam-3889	592	15	techniques	technique	NOUN
ejpam-3889	592	16	,	,	PUNCT
ejpam-3889	592	17	ucad	ucad	ADJ
ejpam-3889	592	18	,	,	PUNCT
ejpam-3889	592	19	dakar	dakar	NOUN
ejpam-3889	592	20	,	,	PUNCT
ejpam-3889	592	21	juillet	juillet	PROPN
ejpam-3889	592	22	,	,	PUNCT
ejpam-3889	592	23	2009	2009	NUM
ejpam-3889	592	24	[	[	X
ejpam-3889	592	25	17	17	NUM
ejpam-3889	592	26	]	]	PUNCT
ejpam-3889	592	27	m.f.maaouia	m.f.maaouia	NOUN
ejpam-3889	592	28	,	,	PUNCT
ejpam-3889	592	29	doctorat	doctorat	X
ejpam-3889	592	30	3ème	3ème	NUM
ejpam-3889	592	31	cycle	cycle	NOUN
ejpam-3889	592	32	,	,	PUNCT
ejpam-3889	592	33	faculté	faculté	NOUN
ejpam-3889	592	34	des	des	PROPN
ejpam-3889	592	35	sciences	sciences	PROPN
ejpam-3889	592	36	et	et	PROPN
ejpam-3889	592	37	techniques	technique	NOUN
ejpam-3889	592	38	,	,	PUNCT
ejpam-3889	592	39	ucad	ucad	ADJ
ejpam-3889	592	40	,	,	PUNCT
ejpam-3889	592	41	dakar	dakar	NOUN
ejpam-3889	592	42	,	,	PUNCT
ejpam-3889	592	43	juillet	juillet	PROPN
ejpam-3889	592	44	,	,	PUNCT
ejpam-3889	592	45	2003	2003	NUM
ejpam-3889	592	46	.	.	PUNCT
ejpam-3889	593	1	references	reference	NOUN
ejpam-3889	593	2	422	422	NUM
ejpam-3889	593	3	[	[	X
ejpam-3889	593	4	18	18	NUM
ejpam-3889	593	5	]	]	PUNCT
ejpam-3889	593	6	m.	m.	PROPN
ejpam-3889	593	7	f.	f.	PROPN
ejpam-3889	593	8	maaouia	maaouia	PROPN
ejpam-3889	593	9	and	and	CCONJ
ejpam-3889	593	10	m.	m.	NOUN
ejpam-3889	593	11	sanghare	sanghare	PROPN
ejpam-3889	593	12	,	,	PUNCT
ejpam-3889	593	13	module	module	NOUN
ejpam-3889	593	14	de	de	NOUN
ejpam-3889	593	15	fractions	fraction	NOUN
ejpam-3889	593	16	,	,	PUNCT
ejpam-3889	593	17	sous	sous	ADJ
ejpam-3889	593	18	-	-	PUNCT
ejpam-3889	593	19	modules	module	NOUN
ejpam-3889	593	20	s−saturée	s−saturée	PROPN
ejpam-3889	593	21	et	et	NOUN
ejpam-3889	593	22	foncteur	foncteur	PROPN
ejpam-3889	593	23	s−1,international	s−1,international	PROPN
ejpam-3889	593	24	journal	journal	NOUN
ejpam-3889	593	25	of	of	ADP
ejpam-3889	593	26	algebra	algebra	PROPN
ejpam-3889	593	27	,	,	PUNCT
ejpam-3889	593	28	issn	issn	PROPN
ejpam-3889	593	29	0973	0973	NUM
ejpam-3889	593	30	-	-	SYM
ejpam-3889	593	31	1768	1768	NUM
ejpam-3889	593	32	,	,	PUNCT
ejpam-3889	593	33	volume	volume	NOUN
ejpam-3889	593	34	6	6	NUM
ejpam-3889	593	35	,	,	PUNCT
ejpam-3889	593	36	number	number	NOUN
ejpam-3889	593	37	16	16	NUM
ejpam-3889	593	38	(	(	PUNCT
ejpam-3889	593	39	2012	2012	NUM
ejpam-3889	593	40	)	)	PUNCT
ejpam-3889	593	41	.	.	PUNCT
ejpam-3889	594	1	[	[	X
ejpam-3889	594	2	19	19	NUM
ejpam-3889	594	3	]	]	PUNCT
ejpam-3889	594	4	m.	m.	PROPN
ejpam-3889	594	5	mendelson	mendelson	PROPN
ejpam-3889	594	6	graded	grade	VERB
ejpam-3889	594	7	rings	ring	NOUN
ejpam-3889	594	8	,	,	PUNCT
ejpam-3889	594	9	modules	module	NOUN
ejpam-3889	594	10	and	and	CCONJ
ejpam-3889	594	11	algebras	algebra	NOUN
ejpam-3889	594	12	,	,	PUNCT
ejpam-3889	594	13	m.i.t	m.i.t	NOUN
ejpam-3889	594	14	.	.	PUNCT
ejpam-3889	595	1	(	(	PUNCT
ejpam-3889	595	2	1970	1970	NUM
ejpam-3889	595	3	)	)	PUNCT
ejpam-3889	595	4	.	.	PUNCT
ejpam-3889	596	1	[	[	X
ejpam-3889	596	2	20	20	NUM
ejpam-3889	596	3	]	]	SYM
ejpam-3889	596	4	m.sanghare	m.sanghare	NOUN
ejpam-3889	596	5	,	,	PUNCT
ejpam-3889	596	6	doctorat	doctorat	PROPN
ejpam-3889	596	7	d’état	d’état	PROPN
ejpam-3889	596	8	,	,	PUNCT
ejpam-3889	596	9	faculté	faculté	NOUN
ejpam-3889	596	10	des	des	PROPN
ejpam-3889	596	11	sciences	sciences	PROPN
ejpam-3889	596	12	et	et	PROPN
ejpam-3889	596	13	techniques	technique	NOUN
ejpam-3889	596	14	,	,	PUNCT
ejpam-3889	596	15	ucad	ucad	ADJ
ejpam-3889	596	16	,	,	PUNCT
ejpam-3889	596	17	dakar	dakar	NOUN
ejpam-3889	596	18	,	,	PUNCT
ejpam-3889	596	19	17	17	NUM
ejpam-3889	596	20	décembre	décembre	PROPN
ejpam-3889	596	21	,	,	PUNCT
ejpam-3889	596	22	1993	1993	NUM
ejpam-3889	596	23	.	.	PUNCT
ejpam-3889	597	1	[	[	X
ejpam-3889	597	2	21	21	NUM
ejpam-3889	597	3	]	]	X
ejpam-3889	597	4	r.	r.	PROPN
ejpam-3889	597	5	abu	abu	PROPN
ejpam-3889	597	6	-	-	PUNCT
ejpam-3889	597	7	dawwas	dawwa	VERB
ejpam-3889	597	8	more	more	ADJ
ejpam-3889	597	9	on	on	ADP
ejpam-3889	597	10	crossed	cross	VERB
ejpam-3889	597	11	product	product	NOUN
ejpam-3889	597	12	over	over	ADP
ejpam-3889	597	13	the	the	DET
ejpam-3889	597	14	support	support	NOUN
ejpam-3889	597	15	of	of	ADP
ejpam-3889	597	16	graded	grade	VERB
ejpam-3889	597	17	rings	ring	NOUN
ejpam-3889	597	18	,	,	PUNCT
ejpam-3889	597	19	international	international	PROPN
ejpam-3889	597	20	mathematical	mathematical	ADJ
ejpam-3889	597	21	forum	forum	PROPN
ejpam-3889	597	22	,	,	PUNCT
ejpam-3889	597	23	5	5	NUM
ejpam-3889	597	24	(	(	PUNCT
ejpam-3889	597	25	63	63	NUM
ejpam-3889	597	26	)	)	PUNCT
ejpam-3889	597	27	,	,	PUNCT
ejpam-3889	597	28	3121	3121	NUM
ejpam-3889	597	29	-	-	SYM
ejpam-3889	597	30	3126	3126	NUM
ejpam-3889	597	31	.	.	PUNCT
ejpam-3889	598	1	(	(	PUNCT
ejpam-3889	598	2	2010	2010	NUM
ejpam-3889	598	3	)	)	PUNCT
ejpam-3889	599	1	[	[	X
ejpam-3889	599	2	22	22	NUM
ejpam-3889	599	3	]	]	PUNCT
ejpam-3889	599	4	r.	r.	PROPN
ejpam-3889	599	5	abu	abu	PROPN
ejpam-3889	599	6	-	-	PUNCT
ejpam-3889	599	7	dawwas	dawwas	PROPN
ejpam-3889	599	8	and	and	CCONJ
ejpam-3889	599	9	m.	m.	NOUN
ejpam-3889	599	10	refai	refai	NOUN
ejpam-3889	599	11	further	further	ADJ
ejpam-3889	599	12	results	result	NOUN
ejpam-3889	599	13	on	on	ADP
ejpam-3889	599	14	graded	grade	VERB
ejpam-3889	599	15	prime	prime	ADJ
ejpam-3889	599	16	submodules	submodule	NOUN
ejpam-3889	599	17	,	,	PUNCT
ejpam-3889	599	18	international	international	ADJ
ejpam-3889	599	19	journal	journal	NOUN
ejpam-3889	599	20	of	of	ADP
ejpam-3889	599	21	algebra	algebra	PROPN
ejpam-3889	599	22	,	,	PUNCT
ejpam-3889	599	23	4	4	NUM
ejpam-3889	599	24	,	,	PUNCT
ejpam-3889	599	25	no	no	INTJ
ejpam-3889	599	26	.	.	NOUN
ejpam-3889	599	27	28	28	NUM
ejpam-3889	599	28	,	,	PUNCT
ejpam-3889	599	29	1413	1413	NUM
ejpam-3889	599	30	-	-	SYM
ejpam-3889	599	31	1419	1419	NUM
ejpam-3889	599	32	.	.	PUNCT
ejpam-3889	600	1	(	(	PUNCT
ejpam-3889	600	2	2010	2010	NUM
ejpam-3889	600	3	)	)	PUNCT
ejpam-3889	601	1	[	[	X
ejpam-3889	601	2	23	23	NUM
ejpam-3889	601	3	]	]	X
ejpam-3889	601	4	s.a	s.a	PROPN
ejpam-3889	601	5	.	.	PROPN
ejpam-3889	601	6	balde	balde	PROPN
ejpam-3889	601	7	,	,	PUNCT
ejpam-3889	601	8	m.	m.	NOUN
ejpam-3889	601	9	ben	ben	PROPN
ejpam-3889	601	10	maaouia	maaouia	PROPN
ejpam-3889	601	11	and	and	CCONJ
ejpam-3889	601	12	a.	a.	NOUN
ejpam-3889	601	13	o.	o.	PROPN
ejpam-3889	601	14	schbih	schbih	PROPN
ejpam-3889	601	15	,	,	PUNCT
ejpam-3889	601	16	hopfian	hopfian	ADJ
ejpam-3889	601	17	and	and	CCONJ
ejpam-3889	601	18	cohopfian	cohopfian	ADJ
ejpam-3889	601	19	objects	object	NOUN
ejpam-3889	601	20	in	in	ADP
ejpam-3889	601	21	the	the	DET
ejpam-3889	601	22	categories	category	NOUN
ejpam-3889	601	23	of	of	ADP
ejpam-3889	601	24	gr(a−mod	gr(a−mod	NOUN
ejpam-3889	601	25	)	)	PUNCT
ejpam-3889	601	26	and	and	CCONJ
ejpam-3889	601	27	comp	comp	NOUN
ejpam-3889	601	28	(	(	PUNCT
ejpam-3889	601	29	gr(a−mod	gr(a−mod	NOUN
ejpam-3889	601	30	)	)	PUNCT
ejpam-3889	601	31	)	)	PUNCT
ejpam-3889	601	32	,	,	PUNCT
ejpam-3889	601	33	journal	journal	NOUN
ejpam-3889	601	34	of	of	ADP
ejpam-3889	601	35	mathematics	mathematics	PROPN
ejpam-3889	601	36	research	research	NOUN
ejpam-3889	601	37	;	;	PUNCT
ejpam-3889	601	38	vol	vol	NOUN
ejpam-3889	601	39	.	.	PROPN
ejpam-3889	601	40	12	12	NUM
ejpam-3889	601	41	,	,	PUNCT
ejpam-3889	601	42	no	no	INTJ
ejpam-3889	601	43	.	.	NOUN
ejpam-3889	601	44	2	2	NUM
ejpam-3889	601	45	;	;	PUNCT
ejpam-3889	601	46	april	april	PROPN
ejpam-3889	601	47	2020	2020	NUM
ejpam-3889	601	48	.	.	PUNCT
ejpam-3889	602	1	[	[	X
ejpam-3889	602	2	24	24	NUM
ejpam-3889	602	3	]	]	PUNCT
ejpam-3889	602	4	v.	v.	ADP
ejpam-3889	602	5	a.	a.	NOUN
ejpam-3889	602	6	hirmath	hirmath	PROPN
ejpam-3889	602	7	,	,	PUNCT
ejpam-3889	602	8	hopfian	hopfian	ADJ
ejpam-3889	602	9	rings	ring	NOUN
ejpam-3889	602	10	and	and	CCONJ
ejpam-3889	602	11	hopfian	hopfian	ADJ
ejpam-3889	602	12	modules	module	NOUN
ejpam-3889	602	13	,	,	PUNCT
ejpam-3889	602	14	indian	indian	PROPN
ejpam-3889	602	15	.	.	PUNCT
ejpam-3889	603	1	j.	j.	PROPN
ejpam-3889	603	2	pure	pure	ADJ
ejpam-3889	603	3	and	and	CCONJ
ejpam-3889	603	4	applied	applied	ADJ
ejpam-3889	603	5	mathematics	mathematic	NOUN
ejpam-3889	603	6	.	.	PUNCT
ejpam-3889	604	1	17	17	NUM
ejpam-3889	604	2	,	,	PUNCT
ejpam-3889	604	3	(	(	PUNCT
ejpam-3889	604	4	1986	1986	NUM
ejpam-3889	604	5	)	)	PUNCT
ejpam-3889	604	6	,	,	PUNCT
ejpam-3889	604	7	895	895	NUM
ejpam-3889	604	8	-	-	SYM
ejpam-3889	604	9	900	900	NUM
ejpam-3889	604	10	.	.	PUNCT