id sid tid token lemma pos cana-2413 1 1 communications communication NOUN cana-2413 1 2 on on ADP cana-2413 1 3 applied apply VERB cana-2413 1 4 nonlinear nonlinear ADJ cana-2413 1 5 analysis analysis NOUN cana-2413 1 6 issn issn NOUN cana-2413 1 7 : : PUNCT cana-2413 1 8 1074 1074 NUM cana-2413 1 9 - - PUNCT cana-2413 1 10 133x 133x NUM cana-2413 1 11 vol vol NOUN cana-2413 1 12 32 32 NUM cana-2413 1 13 no no NOUN cana-2413 1 14 . . PUNCT cana-2413 2 1 2s 2s NUM cana-2413 2 2 ( ( PUNCT cana-2413 2 3 2025 2025 NUM cana-2413 2 4 ) ) PUNCT cana-2413 2 5 404 404 NUM cana-2413 2 6 https://internationalpubls.com https://internationalpubls.com X cana-2413 2 7 zip zip NOUN cana-2413 2 8 property property NOUN cana-2413 2 9 of of ADP cana-2413 2 10 graded grade VERB cana-2413 2 11 and and CCONJ cana-2413 2 12 filtered filter VERB cana-2413 2 13 affine affine NOUN cana-2413 2 14 schemes scheme NOUN cana-2413 2 15 nawal nawal PROPN cana-2413 2 16 m. m. PROPN cana-2413 2 17 noureldeen1,3 noureldeen1,3 PROPN cana-2413 2 18 , , PUNCT cana-2413 2 19 * * PUNCT cana-2413 2 20 , , PUNCT cana-2413 2 21 helmy helmy ADJ cana-2413 2 22 a. a. NOUN cana-2413 2 23 a.2 a.2 PROPN cana-2413 2 24 , , PUNCT cana-2413 2 25 a. a. PROPN cana-2413 2 26 e. e. PROPN cana-2413 2 27 radwan radwan VERB cana-2413 2 28 2 2 NUM cana-2413 2 29 1department 1department NUM cana-2413 2 30 of of ADP cana-2413 2 31 mathematics mathematic NOUN cana-2413 2 32 , , PUNCT cana-2413 2 33 college college NOUN cana-2413 2 34 of of ADP cana-2413 2 35 science science PROPN cana-2413 2 36 , , PUNCT cana-2413 2 37 taibah taibah PROPN cana-2413 2 38 university university PROPN cana-2413 2 39 , , PUNCT cana-2413 2 40 madinah madinah PROPN cana-2413 2 41 , , PUNCT cana-2413 2 42 kingdom kingdom NOUN cana-2413 2 43 of of ADP cana-2413 2 44 saudi saudi PROPN cana-2413 2 45 arabia arabia PROPN cana-2413 2 46 . . PUNCT cana-2413 3 1 neldeen@taibahu.edu.sa neldeen@taibahu.edu.sa PROPN cana-2413 3 2 2department 2department NUM cana-2413 3 3 of of ADP cana-2413 3 4 mathematics mathematic NOUN cana-2413 3 5 , , PUNCT cana-2413 3 6 faculty faculty NOUN cana-2413 3 7 of of ADP cana-2413 3 8 science science NOUN cana-2413 3 9 , , PUNCT cana-2413 3 10 ain ain PROPN cana-2413 3 11 shams shams PROPN cana-2413 3 12 university university PROPN cana-2413 3 13 , , PUNCT cana-2413 3 14 cairo cairo PROPN cana-2413 3 15 , , PUNCT cana-2413 3 16 egypt egypt PROPN cana-2413 3 17 . . PUNCT cana-2413 4 1 helmy5962@hotmail.com helmy5962@hotmail.com X cana-2413 4 2 , , PUNCT cana-2413 4 3 zezorawan@yahoo.com zezorawan@yahoo.com X cana-2413 5 1 3department 3department NUM cana-2413 5 2 of of ADP cana-2413 5 3 mathematics mathematic NOUN cana-2413 5 4 , , PUNCT cana-2413 5 5 women woman NOUN cana-2413 5 6 ’s ’s PART cana-2413 5 7 college college PROPN cana-2413 5 8 of of ADP cana-2413 5 9 arts art NOUN cana-2413 5 10 , , PUNCT cana-2413 5 11 science science NOUN cana-2413 5 12 and and CCONJ cana-2413 5 13 education education NOUN cana-2413 5 14 , , PUNCT cana-2413 5 15 ain ain PROPN cana-2413 5 16 shams shams PROPN cana-2413 5 17 university university PROPN cana-2413 5 18 , , PUNCT cana-2413 5 19 egypt egypt PROPN cana-2413 5 20 . . PUNCT cana-2413 6 1 * * PUNCT cana-2413 6 2 corresponding correspond VERB cana-2413 6 3 author author NOUN cana-2413 6 4 . . PUNCT cana-2413 7 1 article article NOUN cana-2413 7 2 history history NOUN cana-2413 7 3 : : PUNCT cana-2413 7 4 received receive VERB cana-2413 7 5 : : PUNCT cana-2413 7 6 18 18 NUM cana-2413 7 7 - - SYM cana-2413 7 8 09 09 NUM cana-2413 7 9 - - PUNCT cana-2413 7 10 2024 2024 NUM cana-2413 7 11 revised revise VERB cana-2413 7 12 : : PUNCT cana-2413 7 13 26 26 NUM cana-2413 7 14 - - SYM cana-2413 7 15 10 10 NUM cana-2413 7 16 - - PUNCT cana-2413 7 17 2024 2024 NUM cana-2413 7 18 accepted accept VERB cana-2413 7 19 : : PUNCT cana-2413 7 20 06 06 NUM cana-2413 7 21 - - SYM cana-2413 7 22 11 11 NUM cana-2413 7 23 - - PUNCT cana-2413 7 24 2024 2024 NUM cana-2413 7 25 abstract abstract NOUN cana-2413 7 26 : : PUNCT cana-2413 7 27 in in ADP cana-2413 7 28 this this DET cana-2413 7 29 paper paper NOUN cana-2413 7 30 we we PRON cana-2413 7 31 study study VERB cana-2413 7 32 the the DET cana-2413 7 33 transfer transfer NOUN cana-2413 7 34 of of ADP cana-2413 7 35 zip zip NOUN cana-2413 7 36 property property NOUN cana-2413 7 37 between between ADP cana-2413 7 38 filtered filter VERB cana-2413 7 39 ( ( PUNCT cana-2413 7 40 graded grade VERB cana-2413 7 41 ) ) PUNCT cana-2413 7 42 rings ring NOUN cana-2413 7 43 and and CCONJ cana-2413 7 44 affine affine NOUN cana-2413 7 45 graded grade VERB cana-2413 7 46 ( ( PUNCT cana-2413 7 47 filtered filter VERB cana-2413 7 48 ) ) PUNCT cana-2413 7 49 structure structure NOUN cana-2413 7 50 schemes scheme NOUN cana-2413 7 51 . . PUNCT cana-2413 8 1 under under ADP cana-2413 8 2 some some DET cana-2413 8 3 conditions condition NOUN cana-2413 8 4 , , PUNCT cana-2413 8 5 the the DET cana-2413 8 6 zip zip NOUN cana-2413 8 7 property property NOUN cana-2413 8 8 of of ADP cana-2413 8 9 filtered filter VERB cana-2413 8 10 ( ( PUNCT cana-2413 8 11 graded grade VERB cana-2413 8 12 ) ) PUNCT cana-2413 8 13 rings ring NOUN cana-2413 8 14 is be AUX cana-2413 8 15 preserved preserve VERB cana-2413 8 16 under under ADP cana-2413 8 17 their their PRON cana-2413 8 18 graded grade VERB cana-2413 8 19 and and CCONJ cana-2413 8 20 filtered filter VERB cana-2413 8 21 affine affine NOUN cana-2413 8 22 schemes scheme NOUN cana-2413 8 23 . . PUNCT cana-2413 9 1 one one PRON cana-2413 9 2 may may AUX cana-2413 9 3 apply apply VERB cana-2413 9 4 these these DET cana-2413 9 5 results result NOUN cana-2413 9 6 up up ADP cana-2413 9 7 to to ADP cana-2413 9 8 the the DET cana-2413 9 9 formal formal ADJ cana-2413 9 10 level level NOUN cana-2413 9 11 as as ADP cana-2413 9 12 in in ADP cana-2413 9 13 [ [ X cana-2413 9 14 8 8 NUM cana-2413 9 15 ] ] PUNCT cana-2413 9 16 . . PUNCT cana-2413 10 1 introduction introduction NOUN cana-2413 10 2 : : PUNCT cana-2413 10 3 consider consider VERB cana-2413 10 4 a a DET cana-2413 10 5 zariskian zariskian NOUN cana-2413 10 6 filtered filter VERB cana-2413 10 7 ring ring NOUN cana-2413 10 8 𝑆 𝑆 PROPN cana-2413 10 9 such such ADJ cana-2413 10 10 that that SCONJ cana-2413 10 11 the the DET cana-2413 10 12 associated associate VERB cana-2413 10 13 graded grade VERB cana-2413 10 14 ring ring NOUN cana-2413 10 15 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 10 16 ) ) PUNCT cana-2413 10 17 = = PROPN cana-2413 10 18 βŠ• βŠ• PROPN cana-2413 10 19 𝐹𝑛𝑆 𝐹𝑛𝑆 X cana-2413 10 20 πΉπ‘›βˆ’1𝑆 πΉπ‘›βˆ’1𝑆 X cana-2413 10 21 β‰… β‰… PROPN cana-2413 10 22 οΏ½ οΏ½ PROPN cana-2413 10 23 Μƒ Μƒ PROPN cana-2413 10 24 οΏ½ οΏ½ PROPN cana-2413 10 25 𝑋 𝑋 PROPN cana-2413 10 26 οΏ½ οΏ½ PROPN cana-2413 10 27 Μƒ Μƒ PROPN cana-2413 10 28 οΏ½ οΏ½ PROPN cana-2413 10 29 is be AUX cana-2413 10 30 commutative commutative ADJ cana-2413 10 31 noetherian noetherian ADJ cana-2413 10 32 domain domain NOUN cana-2413 10 33 ; ; PUNCT cana-2413 10 34 [ [ X cana-2413 10 35 10 10 NUM cana-2413 10 36 ] ] PUNCT cana-2413 10 37 . . PUNCT cana-2413 11 1 this this PRON cana-2413 11 2 includes include VERB cana-2413 11 3 many many ADJ cana-2413 11 4 more more ADJ cana-2413 11 5 geometric geometric ADJ cana-2413 11 6 applications application NOUN cana-2413 11 7 , , PUNCT cana-2413 11 8 i.e. i.e. X cana-2413 11 9 this this DET cana-2413 11 10 situation situation NOUN cana-2413 11 11 is be AUX cana-2413 11 12 general general ADJ cana-2413 11 13 in in ADP cana-2413 11 14 the the DET cana-2413 11 15 sense sense NOUN cana-2413 11 16 that that SCONJ cana-2413 11 17 it it PRON cana-2413 11 18 allows allow VERB cana-2413 11 19 application application NOUN cana-2413 11 20 of of ADP cana-2413 11 21 the the DET cana-2413 11 22 results result NOUN cana-2413 11 23 to to ADP cana-2413 11 24 most most ADJ cana-2413 11 25 of of ADP cana-2413 11 26 the the DET cana-2413 11 27 important important ADJ cana-2413 11 28 examples example NOUN cana-2413 11 29 . . PUNCT cana-2413 12 1 the the DET cana-2413 12 2 topological topological ADJ cana-2413 12 3 base base NOUN cana-2413 12 4 space space NOUN cana-2413 12 5 𝑇 𝑇 PROPN cana-2413 12 6 will will AUX cana-2413 12 7 be be AUX cana-2413 12 8 𝑆𝑝𝑒𝑐𝑔 𝑆𝑝𝑒𝑐𝑔 PROPN cana-2413 12 9 of of ADP cana-2413 12 10 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 12 11 ) ) PUNCT cana-2413 12 12 . . PUNCT cana-2413 13 1 the the DET cana-2413 13 2 canonical canonical ADJ cana-2413 13 3 element element NOUN cana-2413 13 4 of of ADP cana-2413 13 5 degree degree NOUN cana-2413 13 6 one one NUM cana-2413 13 7 in in ADP cana-2413 13 8 οΏ½ οΏ½ PROPN cana-2413 13 9 Μƒ Μƒ PROPN cana-2413 13 10 οΏ½ οΏ½ PROPN cana-2413 13 11 = = PROPN cana-2413 13 12 βŠ• βŠ• PROPN cana-2413 13 13 𝐹𝑛𝑆 𝐹𝑛𝑆 PROPN cana-2413 13 14 β‰… β‰… NUM cana-2413 13 15 βˆ‘ βˆ‘ PROPN cana-2413 13 16 πΉπ‘›π‘›βˆˆπ›§ πΉπ‘›π‘›βˆˆπ›§ PROPN cana-2413 13 17 𝑆𝑋𝑛 𝑆𝑋𝑛 PROPN cana-2413 13 18 ≀ ≀ PROPN cana-2413 13 19 𝑆[𝑋 𝑆[𝑋 NOUN cana-2413 13 20 , , PUNCT cana-2413 13 21 π‘‹βˆ’1 π‘‹βˆ’1 X cana-2413 13 22 ] ] PUNCT cana-2413 13 23 is be AUX cana-2413 13 24 the the DET cana-2413 13 25 1 1 NUM cana-2413 13 26 ∈ ∈ NOUN cana-2413 13 27 𝐹1𝑆 𝐹1𝑆 NOUN cana-2413 13 28 in in ADP cana-2413 13 29 𝑆 𝑆 PROPN cana-2413 13 30 , , PUNCT cana-2413 13 31 we we PRON cana-2413 13 32 write write VERB cana-2413 13 33 it it PRON cana-2413 13 34 as as ADP cana-2413 13 35 𝑋. 𝑋. PROPN cana-2413 13 36 for for ADP cana-2413 13 37 moment moment NOUN cana-2413 13 38 let let VERB cana-2413 13 39 𝑆 𝑆 PROPN cana-2413 13 40 be be AUX cana-2413 13 41 a a DET cana-2413 13 42 graded grade VERB cana-2413 13 43 ring ring NOUN cana-2413 13 44 . . PUNCT cana-2413 14 1 for for ADP cana-2413 14 2 a a DET cana-2413 14 3 homogenous homogenous ADJ cana-2413 14 4 element element NOUN cana-2413 14 5 π‘Ž π‘Ž PROPN cana-2413 14 6 ∈ ∈ PROPN cana-2413 14 7 𝑆 𝑆 PROPN cana-2413 14 8 , , PUNCT cana-2413 14 9 the the DET cana-2413 14 10 annihilator annihilator PROPN cana-2413 14 11 ideal ideal PROPN cana-2413 14 12 π‘Žπ‘›π‘›π‘”(π‘Ž π‘Žπ‘›π‘›π‘”(π‘Ž PROPN cana-2413 14 13 ) ) PUNCT cana-2413 14 14 = = PRON cana-2413 14 15 { { PUNCT cana-2413 14 16 𝑠 𝑠 PROPN cana-2413 14 17 ∈ ∈ PROPN cana-2413 14 18 𝑆 𝑆 PROPN cana-2413 14 19 : : PUNCT cana-2413 14 20 π‘ π‘Ž π‘ π‘Ž NOUN cana-2413 14 21 = = NOUN cana-2413 14 22 0 0 NUM cana-2413 14 23 } } PUNCT cana-2413 14 24 is be AUX cana-2413 14 25 a a DET cana-2413 14 26 homogenous homogenous ADJ cana-2413 14 27 ideal ideal NOUN cana-2413 14 28 , , PUNCT cana-2413 14 29 as as SCONJ cana-2413 14 30 is be AUX cana-2413 14 31 the the DET cana-2413 14 32 ideal ideal ADJ cana-2413 14 33 annihilator annihilator NOUN cana-2413 14 34 π‘Žπ‘›π‘›π‘”(𝐴 π‘Žπ‘›π‘›π‘”(𝐴 PROPN cana-2413 14 35 ) ) PUNCT cana-2413 15 1 = = PRON cana-2413 15 2 { { PUNCT cana-2413 15 3 𝑠 𝑠 PROPN cana-2413 15 4 ∈ ∈ PROPN cana-2413 15 5 𝑆 𝑆 PROPN cana-2413 15 6 : : PUNCT cana-2413 15 7 𝑠𝐴 𝑠𝐴 PROPN cana-2413 15 8 = = PUNCT cana-2413 15 9 0 0 NUM cana-2413 15 10 } } PUNCT cana-2413 15 11 ; ; PUNCT cana-2413 15 12 𝐴 𝐴 PROPN cana-2413 15 13 βŠ† βŠ† NUM cana-2413 15 14 𝑆 𝑆 PROPN cana-2413 15 15 a a DET cana-2413 15 16 set set NOUN cana-2413 15 17 of of ADP cana-2413 15 18 homogenous homogenous ADJ cana-2413 15 19 elements element NOUN cana-2413 15 20 and and CCONJ cana-2413 15 21 as as ADP cana-2413 15 22 is be AUX cana-2413 15 23 the the DET cana-2413 15 24 ideal ideal ADJ cana-2413 15 25 annihilator annihilator PROPN cana-2413 15 26 π‘Žπ‘›π‘›π‘”(𝐼 π‘Žπ‘›π‘›π‘”(𝐼 PROPN cana-2413 15 27 ) ) PUNCT cana-2413 16 1 = = PRON cana-2413 16 2 { { PUNCT cana-2413 16 3 𝑠 𝑠 PROPN cana-2413 16 4 ∈ ∈ PROPN cana-2413 16 5 𝑆 𝑆 PROPN cana-2413 16 6 : : PUNCT cana-2413 16 7 𝑠𝐼 𝑠𝐼 NOUN cana-2413 16 8 = = SYM cana-2413 16 9 0 0 NUM cana-2413 16 10 } } PUNCT cana-2413 16 11 ; ; PUNCT cana-2413 16 12 𝐼 𝐼 ADP cana-2413 16 13 ⊲ ⊲ NOUN cana-2413 16 14     SPACE cana-2413 16 15 𝑆 𝑆 PROPN cana-2413 16 16 an an DET cana-2413 16 17 ideal ideal NOUN cana-2413 16 18 of of ADP cana-2413 16 19 homogenous homogenous ADJ cana-2413 16 20 elements element NOUN cana-2413 16 21 . . PUNCT cana-2413 17 1 a a DET cana-2413 17 2 graded grade VERB cana-2413 17 3 ring ring NOUN cana-2413 17 4 𝑆 𝑆 PROPN cana-2413 17 5 is be AUX cana-2413 17 6 said say VERB cana-2413 17 7 to to PART cana-2413 17 8 be be AUX cana-2413 17 9 zip zip NOUN cana-2413 17 10 if if SCONJ cana-2413 17 11 βˆ€ βˆ€ NOUN cana-2413 17 12     SPACE cana-2413 17 13 𝐴 𝐴 PROPN cana-2413 17 14 βŠ† βŠ† NUM cana-2413 17 15 𝑆 𝑆 PROPN cana-2413 17 16 : : PUNCT cana-2413 17 17       SPACE cana-2413 17 18 π‘Žπ‘›π‘›π‘”(𝐴 π‘Žπ‘›π‘›π‘”(𝐴 NOUN cana-2413 17 19 ) ) PUNCT cana-2413 17 20 = = SYM cana-2413 17 21 0 0 NUM cana-2413 17 22 β‡’ β‡’ PROPN cana-2413 17 23 βˆƒ βˆƒ PROPN cana-2413 17 24     SPACE cana-2413 17 25 𝐴0 𝐴0 PROPN cana-2413 17 26 βŠ† βŠ† NUM cana-2413 17 27 𝐴 𝐴 PROPN cana-2413 17 28 , , PUNCT cana-2413 17 29 finite finite PROPN cana-2413 17 30 subset subset NOUN cana-2413 17 31 of of ADP cana-2413 17 32 homogenous homogenous ADJ cana-2413 17 33 elements element NOUN cana-2413 17 34 : : PUNCT cana-2413 17 35 π‘Žπ‘›π‘›π‘”(𝐴0 π‘Žπ‘›π‘›π‘”(𝐴0 X cana-2413 17 36 ) ) PUNCT cana-2413 18 1 = = PUNCT cana-2413 18 2 0 0 X cana-2413 18 3 . . PUNCT cana-2413 19 1 in in ADP cana-2413 19 2 this this DET cana-2413 19 3 definition definition NOUN cana-2413 19 4 , , PUNCT cana-2413 19 5 we we PRON cana-2413 19 6 can can AUX cana-2413 19 7 equivalently equivalently ADV cana-2413 19 8 need need VERB cana-2413 19 9 to to PART cana-2413 19 10 use use VERB cana-2413 19 11 that that SCONJ cana-2413 19 12 𝐴 𝐴 PROPN cana-2413 19 13 is be AUX cana-2413 19 14 a a DET cana-2413 19 15 graded grade VERB cana-2413 19 16 ideal ideal NOUN cana-2413 19 17 of of ADP cana-2413 19 18 𝑆. 𝑆. PROPN cana-2413 19 19 we we PRON cana-2413 19 20 need need VERB cana-2413 19 21 only only ADV cana-2413 19 22 zip zip NOUN cana-2413 19 23 expression expression NOUN cana-2413 19 24 of of ADP cana-2413 19 25 commutative commutative ADJ cana-2413 19 26 case case NOUN cana-2413 19 27 . . PUNCT cana-2413 20 1 for for ADP cana-2413 20 2 elementary elementary ADJ cana-2413 20 3 notions notion NOUN cana-2413 20 4 , , PUNCT cana-2413 20 5 conventions convention NOUN cana-2413 20 6 and and CCONJ cana-2413 20 7 generalities generality NOUN cana-2413 20 8 , , PUNCT cana-2413 20 9 which which PRON cana-2413 20 10 we we PRON cana-2413 20 11 need need VERB cana-2413 20 12 here here ADV cana-2413 20 13 in in ADP cana-2413 20 14 this this DET cana-2413 20 15 paper paper NOUN cana-2413 20 16 we we PRON cana-2413 20 17 refer refer VERB cana-2413 20 18 to to ADP cana-2413 20 19 the the DET cana-2413 20 20 list list NOUN cana-2413 20 21 of of ADP cana-2413 20 22 references reference NOUN cana-2413 20 23 . . PUNCT cana-2413 21 1 objectives objective NOUN cana-2413 21 2 : : PUNCT cana-2413 21 3 in in ADP cana-2413 21 4 this this DET cana-2413 21 5 paper paper NOUN cana-2413 21 6 , , PUNCT cana-2413 21 7 we we PRON cana-2413 21 8 study study VERB cana-2413 21 9 the the DET cana-2413 21 10 transfer transfer NOUN cana-2413 21 11 of of ADP cana-2413 21 12 zip zip NOUN cana-2413 21 13 property property NOUN cana-2413 21 14 from from ADP cana-2413 21 15 filtered filter VERB cana-2413 21 16 ( ( PUNCT cana-2413 21 17 graded grade VERB cana-2413 21 18 ) ) PUNCT cana-2413 21 19 rings ring NOUN cana-2413 21 20 to to ADP cana-2413 21 21 the the DET cana-2413 21 22 graded grade VERB cana-2413 21 23 and and CCONJ cana-2413 21 24 filtered filter VERB cana-2413 21 25 structure structure NOUN cana-2413 21 26 affine affine NOUN cana-2413 21 27 schemes scheme NOUN cana-2413 21 28 . . PUNCT cana-2413 22 1 results result NOUN cana-2413 22 2 : : PUNCT cana-2413 22 3 according accord VERB cana-2413 22 4 to to ADP cana-2413 22 5 the the DET cana-2413 22 6 work work NOUN cana-2413 22 7 of of ADP cana-2413 22 8 leroy leroy PROPN cana-2413 22 9 and and CCONJ cana-2413 22 10 matczuk matczuk ADJ cana-2413 22 11 ( ( PUNCT cana-2413 22 12 [ [ X cana-2413 22 13 4 4 NUM cana-2413 22 14 ] ] PUNCT cana-2413 22 15 , , PUNCT cana-2413 22 16 theorem theorem VERB cana-2413 22 17 3.2(1 3.2(1 NUM cana-2413 22 18 ) ) PUNCT cana-2413 22 19 ) ) PUNCT cana-2413 22 20 , , PUNCT cana-2413 22 21 who who PRON cana-2413 22 22 investigated investigate VERB cana-2413 22 23 the the DET cana-2413 22 24 behavior behavior NOUN cana-2413 22 25 of of ADP cana-2413 22 26 the the DET cana-2413 22 27 zip zip NOUN cana-2413 22 28 property property NOUN cana-2413 22 29 for for ADP cana-2413 22 30 a a DET cana-2413 22 31 localization localization NOUN cana-2413 22 32 of of ADP cana-2413 22 33 a a DET cana-2413 22 34 ring ring NOUN cana-2413 22 35 , , PUNCT cana-2413 22 36 we we PRON cana-2413 22 37 extend extend VERB cana-2413 22 38 this this DET cana-2413 22 39 result result NOUN cana-2413 22 40 for for ADP cana-2413 22 41 graded grade VERB cana-2413 22 42 and and CCONJ cana-2413 22 43 filtered filter VERB cana-2413 22 44 affine affine NOUN cana-2413 22 45 schemes scheme NOUN cana-2413 22 46 . . PUNCT cana-2413 23 1 conclusion conclusion NOUN cana-2413 23 2 : : PUNCT cana-2413 23 3 in in ADP cana-2413 23 4 this this DET cana-2413 23 5 research research NOUN cana-2413 23 6 , , PUNCT cana-2413 23 7 we we PRON cana-2413 23 8 investigate investigate VERB cana-2413 23 9 the the DET cana-2413 23 10 zip zip NOUN cana-2413 23 11 property property NOUN cana-2413 23 12 of of ADP cana-2413 23 13 filtered filter VERB cana-2413 23 14 ( ( PUNCT cana-2413 23 15 graded grade VERB cana-2413 23 16 ) ) PUNCT cana-2413 23 17 rings ring NOUN cana-2413 23 18 is be AUX cana-2413 23 19 preserved preserve VERB cana-2413 23 20 under under ADP cana-2413 23 21 their their PRON cana-2413 23 22 graded grade VERB cana-2413 23 23 and and CCONJ cana-2413 23 24 filtered filter VERB cana-2413 23 25 affine affine NOUN cana-2413 23 26 schemes scheme NOUN cana-2413 23 27 . . PUNCT cana-2413 24 1 in in ADP cana-2413 24 2 the the DET cana-2413 24 3 forthcoming forthcoming ADJ cana-2413 24 4 work work NOUN cana-2413 24 5 , , PUNCT cana-2413 24 6 we we PRON cana-2413 24 7 hope hope VERB cana-2413 24 8 to to PART cana-2413 24 9 come come VERB cana-2413 24 10 back back ADV cana-2413 24 11 to to PART cana-2413 24 12 introduce introduce VERB cana-2413 24 13 the the DET cana-2413 24 14 same same ADJ cana-2413 24 15 results result NOUN cana-2413 24 16 on on ADP cana-2413 24 17 the the DET cana-2413 24 18 formal formal ADJ cana-2413 24 19 level level NOUN cana-2413 24 20 , , PUNCT cana-2413 24 21 one one PRON cana-2413 24 22 may may AUX cana-2413 24 23 make make VERB cana-2413 24 24 this this PRON cana-2413 24 25 by by ADP cana-2413 24 26 [ [ X cana-2413 24 27 8 8 NUM cana-2413 24 28 ] ] PUNCT cana-2413 24 29 . . PUNCT cana-2413 25 1 keywords keyword NOUN cana-2413 25 2 : : PUNCT cana-2413 25 3 graded grade VERB cana-2413 25 4 annihilator annihilator PROPN cana-2413 25 5 , , PUNCT cana-2413 25 6 affine affine NOUN cana-2413 25 7 schemes scheme NOUN cana-2413 25 8 , , PUNCT cana-2413 25 9 zip zip NOUN cana-2413 25 10 property property NOUN cana-2413 25 11 . . PUNCT cana-2413 26 1 mailto:helmy5962@hotmail.com mailto:helmy5962@hotmail.com X cana-2413 26 2 mailto:zezorawan@yahoo.com mailto:zezorawan@yahoo.com X cana-2413 26 3 communications communication NOUN cana-2413 26 4 on on ADP cana-2413 26 5 applied apply VERB cana-2413 26 6 nonlinear nonlinear ADJ cana-2413 26 7 analysis analysis NOUN cana-2413 26 8 issn issn NOUN cana-2413 26 9 : : PUNCT cana-2413 26 10 1074 1074 NUM cana-2413 26 11 - - PUNCT cana-2413 26 12 133x 133x NUM cana-2413 26 13 vol vol NOUN cana-2413 26 14 32 32 NUM cana-2413 26 15 no no NOUN cana-2413 26 16 . . PUNCT cana-2413 27 1 2s 2s NUM cana-2413 27 2 ( ( PUNCT cana-2413 27 3 2025 2025 NUM cana-2413 27 4 ) ) PUNCT cana-2413 27 5 405 405 NUM cana-2413 27 6 https://internationalpubls.com https://internationalpubls.com X cana-2413 27 7 1 1 X cana-2413 27 8 . . PUNCT cana-2413 27 9 introduction introduction NOUN cana-2413 27 10 throughout throughout ADP cana-2413 27 11 the the DET cana-2413 27 12 paper paper NOUN cana-2413 27 13 𝑆 𝑆 PROPN cana-2413 27 14 will will AUX cana-2413 27 15 denote denote VERB cana-2413 27 16 a a DET cana-2413 27 17 zariskian zariskian NOUN cana-2413 27 18 filtered filter VERB cana-2413 27 19 ring ring NOUN cana-2413 27 20 such such ADJ cana-2413 27 21 that that SCONJ cana-2413 27 22 the the DET cana-2413 27 23 associated associate VERB cana-2413 27 24 graded grade VERB cana-2413 27 25 ring ring NOUN cana-2413 27 26 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 27 27 ) ) PUNCT cana-2413 27 28 = = SYM cana-2413 27 29 βŠ• βŠ• PROPN cana-2413 27 30 𝐹𝑛𝑆 𝐹𝑛𝑆 PROPN cana-2413 27 31 πΉπ‘›βˆ’1𝑆 πΉπ‘›βˆ’1𝑆 X cana-2413 27 32 β‰… β‰… PROPN cana-2413 27 33 οΏ½ οΏ½ PROPN cana-2413 27 34 Μƒ Μƒ PROPN cana-2413 27 35 οΏ½ οΏ½ PROPN cana-2413 27 36 𝑋 𝑋 PROPN cana-2413 27 37 οΏ½ οΏ½ PROPN cana-2413 27 38 Μƒ Μƒ PROPN cana-2413 27 39 οΏ½ οΏ½ PROPN cana-2413 27 40 is be AUX cana-2413 27 41 commutative commutative ADJ cana-2413 27 42 noetherian noetherian ADJ cana-2413 27 43 domain domain NOUN cana-2413 27 44 ; ; PUNCT cana-2413 27 45 [ [ X cana-2413 27 46 10 10 NUM cana-2413 27 47 ] ] PUNCT cana-2413 27 48 . . PUNCT cana-2413 28 1 this this PRON cana-2413 28 2 includes include VERB cana-2413 28 3 many many ADJ cana-2413 28 4 more more ADJ cana-2413 28 5 geometric geometric ADJ cana-2413 28 6 applications application NOUN cana-2413 28 7 , , PUNCT cana-2413 28 8 i.e. i.e. X cana-2413 28 9 this this DET cana-2413 28 10 situation situation NOUN cana-2413 28 11 is be AUX cana-2413 28 12 general general ADJ cana-2413 28 13 in in ADP cana-2413 28 14 the the DET cana-2413 28 15 sense sense NOUN cana-2413 28 16 that that SCONJ cana-2413 28 17 it it PRON cana-2413 28 18 allows allow VERB cana-2413 28 19 application application NOUN cana-2413 28 20 of of ADP cana-2413 28 21 the the DET cana-2413 28 22 results result NOUN cana-2413 28 23 to to ADP cana-2413 28 24 most most ADJ cana-2413 28 25 of of ADP cana-2413 28 26 the the DET cana-2413 28 27 important important ADJ cana-2413 28 28 examples example NOUN cana-2413 28 29 . . PUNCT cana-2413 29 1 the the DET cana-2413 29 2 topological topological ADJ cana-2413 29 3 base base NOUN cana-2413 29 4 space space NOUN cana-2413 29 5 𝑇 𝑇 PROPN cana-2413 29 6 will will AUX cana-2413 29 7 be be AUX cana-2413 29 8 𝑆𝑝𝑒𝑐𝑔 𝑆𝑝𝑒𝑐𝑔 PROPN cana-2413 29 9 of of ADP cana-2413 29 10 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 29 11 ) ) PUNCT cana-2413 29 12 . . PUNCT cana-2413 30 1 the the DET cana-2413 30 2 canonical canonical ADJ cana-2413 30 3 element element NOUN cana-2413 30 4 of of ADP cana-2413 30 5 degree degree NOUN cana-2413 30 6 one one NUM cana-2413 30 7 in in ADP cana-2413 30 8 οΏ½ οΏ½ PROPN cana-2413 30 9 Μƒ Μƒ PROPN cana-2413 30 10 οΏ½ οΏ½ PROPN cana-2413 30 11 = = PROPN cana-2413 30 12 βŠ• βŠ• PROPN cana-2413 30 13 𝐹𝑛𝑆 𝐹𝑛𝑆 PROPN cana-2413 30 14 β‰… β‰… NUM cana-2413 30 15 βˆ‘ βˆ‘ PROPN cana-2413 30 16 πΉπ‘›π‘›βˆˆπ›§ πΉπ‘›π‘›βˆˆπ›§ PROPN cana-2413 30 17 𝑆𝑋𝑛 𝑆𝑋𝑛 PROPN cana-2413 30 18 ≀ ≀ PROPN cana-2413 30 19 𝑆[𝑋 𝑆[𝑋 NOUN cana-2413 30 20 , , PUNCT cana-2413 30 21 π‘‹βˆ’1 π‘‹βˆ’1 X cana-2413 30 22 ] ] PUNCT cana-2413 30 23 is be AUX cana-2413 30 24 the the DET cana-2413 30 25 1 1 NUM cana-2413 30 26 ∈ ∈ NOUN cana-2413 30 27 𝐹1𝑆 𝐹1𝑆 NOUN cana-2413 30 28 in in ADP cana-2413 30 29 𝑆 𝑆 PROPN cana-2413 30 30 , , PUNCT cana-2413 30 31 we we PRON cana-2413 30 32 write write VERB cana-2413 30 33 it it PRON cana-2413 30 34 as as ADP cana-2413 30 35 𝑋. 𝑋. PROPN cana-2413 30 36 for for ADP cana-2413 30 37 moment moment NOUN cana-2413 30 38 let let VERB cana-2413 30 39 𝑆 𝑆 PROPN cana-2413 30 40 be be AUX cana-2413 30 41 a a DET cana-2413 30 42 graded grade VERB cana-2413 30 43 ring ring NOUN cana-2413 30 44 . . PUNCT cana-2413 31 1 for for ADP cana-2413 31 2 a a DET cana-2413 31 3 homogenous homogenous ADJ cana-2413 31 4 element element NOUN cana-2413 31 5 π‘Ž π‘Ž PROPN cana-2413 31 6 ∈ ∈ PROPN cana-2413 31 7 𝑆 𝑆 PROPN cana-2413 31 8 , , PUNCT cana-2413 31 9 the the DET cana-2413 31 10 annihilator annihilator PROPN cana-2413 31 11 ideal ideal PROPN cana-2413 31 12 π‘Žπ‘›π‘›π‘”(π‘Ž π‘Žπ‘›π‘›π‘”(π‘Ž PROPN cana-2413 31 13 ) ) PUNCT cana-2413 31 14 = = PRON cana-2413 31 15 { { PUNCT cana-2413 31 16 𝑠 𝑠 PROPN cana-2413 31 17 ∈ ∈ PROPN cana-2413 31 18 𝑆 𝑆 PROPN cana-2413 31 19 : : PUNCT cana-2413 31 20 π‘ π‘Ž π‘ π‘Ž NOUN cana-2413 31 21 = = NOUN cana-2413 31 22 0 0 NUM cana-2413 31 23 } } PUNCT cana-2413 31 24 is be AUX cana-2413 31 25 a a DET cana-2413 31 26 homogenous homogenous ADJ cana-2413 31 27 ideal ideal NOUN cana-2413 31 28 , , PUNCT cana-2413 31 29 as as SCONJ cana-2413 31 30 is be AUX cana-2413 31 31 the the DET cana-2413 31 32 ideal ideal ADJ cana-2413 31 33 annihilator annihilator NOUN cana-2413 31 34 π‘Žπ‘›π‘›π‘”(𝐴 π‘Žπ‘›π‘›π‘”(𝐴 PROPN cana-2413 31 35 ) ) PUNCT cana-2413 32 1 = = PRON cana-2413 32 2 { { PUNCT cana-2413 32 3 𝑠 𝑠 PROPN cana-2413 32 4 ∈ ∈ PROPN cana-2413 32 5 𝑆 𝑆 PROPN cana-2413 32 6 : : PUNCT cana-2413 32 7 𝑠𝐴 𝑠𝐴 PROPN cana-2413 32 8 = = PUNCT cana-2413 32 9 0 0 NUM cana-2413 32 10 } } PUNCT cana-2413 32 11 ; ; PUNCT cana-2413 32 12 𝐴 𝐴 PROPN cana-2413 32 13 βŠ† βŠ† NUM cana-2413 32 14 𝑆 𝑆 PROPN cana-2413 32 15 a a DET cana-2413 32 16 set set NOUN cana-2413 32 17 of of ADP cana-2413 32 18 homogenous homogenous ADJ cana-2413 32 19 elements element NOUN cana-2413 32 20 and and CCONJ cana-2413 32 21 as as ADP cana-2413 32 22 is be AUX cana-2413 32 23 the the DET cana-2413 32 24 ideal ideal ADJ cana-2413 32 25 annihilator annihilator PROPN cana-2413 32 26 π‘Žπ‘›π‘›π‘”(𝐼 π‘Žπ‘›π‘›π‘”(𝐼 PROPN cana-2413 32 27 ) ) PUNCT cana-2413 33 1 = = PRON cana-2413 33 2 { { PUNCT cana-2413 33 3 𝑠 𝑠 PROPN cana-2413 33 4 ∈ ∈ PROPN cana-2413 33 5 𝑆 𝑆 PROPN cana-2413 33 6 : : PUNCT cana-2413 33 7 𝑠𝐼 𝑠𝐼 NOUN cana-2413 33 8 = = SYM cana-2413 33 9 0 0 NUM cana-2413 33 10 } } PUNCT cana-2413 33 11 ; ; PUNCT cana-2413 33 12 𝐼 𝐼 ADP cana-2413 33 13 ⊲ ⊲ NOUN cana-2413 33 14     SPACE cana-2413 33 15 𝑆 𝑆 PROPN cana-2413 33 16 an an DET cana-2413 33 17 ideal ideal NOUN cana-2413 33 18 of of ADP cana-2413 33 19 homogenous homogenous ADJ cana-2413 33 20 elements element NOUN cana-2413 33 21 . . PUNCT cana-2413 34 1 a a DET cana-2413 34 2 graded grade VERB cana-2413 34 3 ring ring NOUN cana-2413 34 4 𝑆 𝑆 PROPN cana-2413 34 5 is be AUX cana-2413 34 6 said say VERB cana-2413 34 7 to to PART cana-2413 34 8 be be AUX cana-2413 34 9 zip zip NOUN cana-2413 34 10 if if SCONJ cana-2413 34 11 βˆ€ βˆ€ NOUN cana-2413 34 12     SPACE cana-2413 34 13 𝐴 𝐴 PROPN cana-2413 34 14 βŠ† βŠ† NUM cana-2413 34 15 𝑆 𝑆 PROPN cana-2413 34 16 : : PUNCT cana-2413 34 17       SPACE cana-2413 34 18 π‘Žπ‘›π‘›π‘”(𝐴 π‘Žπ‘›π‘›π‘”(𝐴 NOUN cana-2413 34 19 ) ) PUNCT cana-2413 34 20 = = SYM cana-2413 34 21 0 0 NUM cana-2413 34 22 β‡’ β‡’ PROPN cana-2413 34 23 βˆƒ βˆƒ PROPN cana-2413 34 24     SPACE cana-2413 34 25 𝐴0 𝐴0 PROPN cana-2413 34 26 βŠ† βŠ† NUM cana-2413 34 27 𝐴 𝐴 PROPN cana-2413 34 28 , , PUNCT cana-2413 34 29 finite finite PROPN cana-2413 34 30 subset subset NOUN cana-2413 34 31 of of ADP cana-2413 34 32 homogenous homogenous ADJ cana-2413 34 33 elements element NOUN cana-2413 34 34 : : PUNCT cana-2413 34 35 π‘Žπ‘›π‘›π‘”(𝐴0 π‘Žπ‘›π‘›π‘”(𝐴0 X cana-2413 34 36 ) ) PUNCT cana-2413 35 1 = = PUNCT cana-2413 35 2 0 0 X cana-2413 35 3 . . PUNCT cana-2413 36 1 in in ADP cana-2413 36 2 this this DET cana-2413 36 3 definition definition NOUN cana-2413 36 4 , , PUNCT cana-2413 36 5 we we PRON cana-2413 36 6 can can AUX cana-2413 36 7 equivalently equivalently ADV cana-2413 36 8 need need VERB cana-2413 36 9 to to PART cana-2413 36 10 use use VERB cana-2413 36 11 that that SCONJ cana-2413 36 12 𝐴 𝐴 PROPN cana-2413 36 13 is be AUX cana-2413 36 14 a a DET cana-2413 36 15 graded grade VERB cana-2413 36 16 ideal ideal NOUN cana-2413 36 17 of of ADP cana-2413 36 18 𝑆. 𝑆. PROPN cana-2413 36 19 we we PRON cana-2413 36 20 need need VERB cana-2413 36 21 only only ADV cana-2413 36 22 zip zip NOUN cana-2413 36 23 expression expression NOUN cana-2413 36 24 of of ADP cana-2413 36 25 commutative commutative ADJ cana-2413 36 26 case case NOUN cana-2413 36 27 . . PUNCT cana-2413 37 1 2 2 X cana-2413 37 2 . . X cana-2413 37 3 zip zip NOUN cana-2413 37 4 graded grade VERB cana-2413 37 5 affine affine NOUN cana-2413 37 6 schemes scheme NOUN cana-2413 37 7 a a DET cana-2413 37 8 graded grade VERB cana-2413 37 9 sheaf sheaf NOUN cana-2413 37 10 𝑂𝑇 𝑂𝑇 PROPN cana-2413 37 11 𝑔 𝑔 PROPN cana-2413 37 12 of of ADP cana-2413 37 13 graded grade VERB cana-2413 37 14 rings ring NOUN cana-2413 37 15 , , PUNCT cana-2413 37 16 over over ADP cana-2413 37 17 a a DET cana-2413 37 18 topological topological ADJ cana-2413 37 19 space space NOUN cana-2413 37 20 𝑇 𝑇 PROPN cana-2413 37 21 , , PUNCT cana-2413 37 22 is be AUX cana-2413 37 23 zip zip NOUN cana-2413 37 24 graded grade VERB cana-2413 37 25 sheaf sheaf NOUN cana-2413 37 26 over over ADP cana-2413 37 27 𝑇 𝑇 PROPN cana-2413 37 28 if if SCONJ cana-2413 37 29 locally locally ADV cana-2413 37 30 is be AUX cana-2413 37 31 zip zip NOUN cana-2413 37 32 sheaf sheaf NOUN cana-2413 37 33 i.e. i.e. X cana-2413 37 34 βˆ€ βˆ€ X cana-2413 37 35     SPACE cana-2413 37 36 𝑃 𝑃 NOUN cana-2413 37 37 ∈ ∈ PROPN cana-2413 37 38 𝑇 𝑇 PROPN cana-2413 37 39 β‡’ β‡’ NOUN cana-2413 37 40 𝑂𝑇,𝑃 𝑂𝑇,𝑃 PROPN cana-2413 37 41 𝑔 𝑔 PROPN cana-2413 37 42 is be AUX cana-2413 37 43 zip zip NOUN cana-2413 37 44 graded grade VERB cana-2413 37 45 ring ring NOUN cana-2413 37 46 . . PUNCT cana-2413 38 1 as as ADP cana-2413 38 2 in in ADP cana-2413 38 3 section section NOUN cana-2413 38 4 one one NUM cana-2413 38 5 , , PUNCT cana-2413 38 6 we we PRON cana-2413 38 7 consider consider VERB cana-2413 38 8 𝑇 𝑇 PROPN cana-2413 38 9 = = PUNCT cana-2413 38 10 𝑆𝑝𝑒𝑐𝑔(𝐺(𝑆 𝑆𝑝𝑒𝑐𝑔(𝐺(𝑆 PROPN cana-2413 38 11 ) ) PUNCT cana-2413 38 12 ) ) PUNCT cana-2413 38 13 , , PUNCT cana-2413 38 14 the the DET cana-2413 38 15 graded grade VERB cana-2413 38 16 prime prime ADJ cana-2413 38 17 spectrum spectrum NOUN cana-2413 38 18 of of ADP cana-2413 38 19 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 38 20 ) ) PUNCT cana-2413 38 21 . . PUNCT cana-2413 39 1 write write VERB cana-2413 39 2 𝛽 𝛽 PROPN cana-2413 39 3 for for ADP cana-2413 39 4 the the DET cana-2413 39 5 basis basis NOUN cana-2413 39 6 of of ADP cana-2413 39 7 the the DET cana-2413 39 8 zariski zariski ADJ cana-2413 39 9 topology topology NOUN cana-2413 39 10 on on ADP cana-2413 39 11 𝑇 𝑇 PROPN cana-2413 39 12 consisting consist VERB cana-2413 39 13 of of ADP cana-2413 39 14 the the DET cana-2413 39 15 basic basic ADJ cana-2413 39 16 open open ADJ cana-2413 39 17 sets set NOUN cana-2413 39 18 𝑇(𝑓 𝑇(𝑓 PRON cana-2413 39 19 ) ) PUNCT cana-2413 39 20 = = PRON cana-2413 39 21 { { PUNCT cana-2413 39 22 𝑃 𝑃 PROPN cana-2413 39 23 ∈ ∈ PROPN cana-2413 39 24 𝑇 𝑇 PROPN cana-2413 39 25 ; ; PUNCT cana-2413 39 26 𝑓 𝑓 DET cana-2413 39 27 βˆ‰ βˆ‰ PROPN cana-2413 39 28 𝑃 𝑃 PROPN cana-2413 39 29 } } PUNCT cana-2413 39 30 ; ; PUNCT cana-2413 39 31 𝑓 𝑓 DET cana-2413 39 32 homogenous homogenous ADJ cana-2413 39 33 element element NOUN cana-2413 39 34 . . PUNCT cana-2413 40 1 we we PRON cana-2413 40 2 may may AUX cana-2413 40 3 define define VERB cana-2413 40 4 graded grade VERB cana-2413 40 5 structure structure NOUN cana-2413 40 6 sheaf sheaf NOUN cana-2413 40 7 on on ADP cana-2413 40 8 𝑇 𝑇 PROPN cana-2413 40 9 : : PUNCT cana-2413 40 10 we we PRON cana-2413 40 11 may may AUX cana-2413 40 12 associate associate VERB cana-2413 40 13 to to ADP cana-2413 40 14 𝑇(𝑓 𝑇(𝑓 X cana-2413 40 15 ) ) PUNCT cana-2413 40 16 the the DET cana-2413 40 17 graded grade VERB cana-2413 40 18 ring ring NOUN cana-2413 40 19 𝑄𝑓 𝑄𝑓 PROPN cana-2413 40 20 𝑔(𝐺(𝑆 𝑔(𝐺(𝑆 PROPN cana-2413 40 21 ) ) PUNCT cana-2413 40 22 ) ) PUNCT cana-2413 41 1 = = PUNCT cana-2413 42 1 𝑆𝑓 𝑆𝑓 NOUN cana-2413 42 2 βˆ’1 βˆ’1 NOUN cana-2413 42 3 β‹… β‹… NOUN cana-2413 42 4 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 42 5 ) ) PUNCT cana-2413 43 1 ; ; PUNCT cana-2413 43 2 just just ADV cana-2413 43 3 , , PUNCT cana-2413 43 4 by by ADP cana-2413 43 5 inverting invert VERB cana-2413 43 6 the the DET cana-2413 43 7 homogenous homogenous ADJ cana-2413 43 8 set set NOUN cana-2413 44 1 𝑆𝑓 𝑆𝑓 PROPN cana-2413 44 2 = = PUNCT cana-2413 44 3 { { PUNCT cana-2413 44 4 1 1 NUM cana-2413 44 5 , , PUNCT cana-2413 44 6 𝑓 𝑓 DET cana-2413 44 7 , , PUNCT cana-2413 44 8 𝑓2 𝑓2 NOUN cana-2413 44 9 , , PUNCT cana-2413 44 10 β‹― β‹― VERB cana-2413 44 11 } } PUNCT cana-2413 44 12 of of ADP cana-2413 44 13 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 44 14 ) ) PUNCT cana-2413 44 15 in in ADP cana-2413 44 16 the the DET cana-2413 44 17 classical classical ADJ cana-2413 44 18 way way NOUN cana-2413 44 19 and and CCONJ cana-2413 44 20 we we PRON cana-2413 44 21 obtain obtain VERB cana-2413 44 22 the the DET cana-2413 44 23 graded grade VERB cana-2413 44 24 structure structure NOUN cana-2413 44 25 sheaf sheaf NOUN cana-2413 44 26 𝑄𝑇 𝑄𝑇 PROPN cana-2413 44 27 𝑔 𝑔 PROPN cana-2413 44 28 on on ADP cana-2413 44 29 𝑇 𝑇 PROPN cana-2413 44 30 having have VERB cana-2413 44 31 as as ADP cana-2413 44 32 the the DET cana-2413 44 33 stalk stalk NOUN cana-2413 44 34 at at ADP cana-2413 44 35 𝑃 𝑃 PROPN cana-2413 44 36 ∈ ∈ PROPN cana-2413 44 37 𝑇 𝑇 PROPN cana-2413 44 38 the the DET cana-2413 44 39 graded grade VERB cana-2413 44 40 local local ADJ cana-2413 44 41 ring ring NOUN cana-2413 44 42 𝑄𝑃 𝑄𝑃 ADJ cana-2413 44 43 𝑔(𝐺(𝑆 𝑔(𝐺(𝑆 NOUN cana-2413 44 44 ) ) PUNCT cana-2413 44 45 ) ) PUNCT cana-2413 45 1 = = SYM cana-2413 45 2 𝑆𝑃 𝑆𝑃 PROPN cana-2413 45 3 βˆ’1 βˆ’1 NOUN cana-2413 45 4 β‹… β‹… NOUN cana-2413 45 5 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 45 6 ) ) PUNCT cana-2413 45 7 ; ; PUNCT cana-2413 45 8 𝑆𝑃 𝑆𝑃 PROPN cana-2413 45 9 = = SYM cana-2413 45 10 β„Ž(𝐺(𝑆 β„Ž(𝐺(𝑆 PROPN cana-2413 45 11 ) ) PUNCT cana-2413 45 12 βˆ’ βˆ’ PROPN cana-2413 45 13 𝑃 𝑃 NOUN cana-2413 45 14 ) ) PUNCT cana-2413 45 15 . . PUNCT cana-2413 46 1 proposition proposition NOUN cana-2413 46 2 1 1 NUM cana-2413 46 3 . . PUNCT cana-2413 46 4 with with ADP cana-2413 46 5 the the DET cana-2413 46 6 same same ADJ cana-2413 46 7 consideration consideration NOUN cana-2413 46 8 if if SCONJ cana-2413 46 9 𝑆 𝑆 PROPN cana-2413 46 10 is be AUX cana-2413 46 11 zip zip NOUN cana-2413 46 12 commutative commutative ADJ cana-2413 46 13 filtered filter VERB cana-2413 46 14 ring ring NOUN cana-2413 46 15 , , PUNCT cana-2413 46 16 then then ADV cana-2413 46 17 οΏ½ οΏ½ PROPN cana-2413 46 18 Μƒ Μƒ PROPN cana-2413 46 19 οΏ½ οΏ½ PROPN cana-2413 46 20 is be AUX cana-2413 46 21 zip zip NOUN cana-2413 46 22 commutative commutative ADJ cana-2413 46 23 graded grade VERB cana-2413 46 24 ring ring NOUN cana-2413 46 25 . . PUNCT cana-2413 47 1 proof proof NOUN cana-2413 47 2 : : PUNCT cana-2413 47 3 follows follow VERB cana-2413 47 4 from from ADP cana-2413 47 5 [ [ X cana-2413 47 6 4 4 NUM cana-2413 47 7 ] ] PUNCT cana-2413 47 8 , , PUNCT cana-2413 47 9 [ [ X cana-2413 47 10 5 5 NUM cana-2413 47 11 ] ] PUNCT cana-2413 47 12 just just ADV cana-2413 47 13 at at ADP cana-2413 47 14 the the DET cana-2413 47 15 filtered filter VERB cana-2413 47 16 ( ( PUNCT cana-2413 47 17 graded grade VERB cana-2413 47 18 ) ) PUNCT cana-2413 47 19 level level NOUN cana-2413 47 20 . . PUNCT cana-2413 48 1 proposition proposition NOUN cana-2413 48 2 2 2 NUM cana-2413 48 3 . . PUNCT cana-2413 49 1 under under ADP cana-2413 49 2 the the DET cana-2413 49 3 assumption assumption NOUN cana-2413 49 4 and and CCONJ cana-2413 49 5 notation notation NOUN cana-2413 49 6 mentioned mention VERB cana-2413 49 7 above above ADV cana-2413 49 8 we we PRON cana-2413 49 9 have have VERB cana-2413 49 10 : : PUNCT cana-2413 49 11 i. i. NOUN cana-2413 49 12 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 49 13 ) ) PUNCT cana-2413 49 14 and and CCONJ cana-2413 49 15 οΏ½ οΏ½ PROPN cana-2413 49 16 Μƒ Μƒ PROPN cana-2413 49 17 οΏ½ οΏ½ PROPN cana-2413 49 18 ( ( PUNCT cana-2413 49 19 1 1 NUM cana-2413 49 20 βˆ’ βˆ’ PROPN cana-2413 49 21 𝑋) 𝑋) PROPN cana-2413 49 22 οΏ½ οΏ½ PROPN cana-2413 49 23 Μƒ Μƒ PROPN cana-2413 49 24 οΏ½ οΏ½ PROPN cana-2413 49 25 ⁄ ⁄ PROPN cana-2413 49 26 are be AUX cana-2413 49 27 zip zip NOUN cana-2413 49 28 graded grade VERB cana-2413 49 29 rings ring NOUN cana-2413 49 30 . . PUNCT cana-2413 50 1 ii ii PROPN cana-2413 50 2 . . PUNCT cana-2413 51 1 𝑄𝑓 𝑄𝑓 PROPN cana-2413 51 2 𝑔(𝐺(𝑆 𝑔(𝐺(𝑆 PROPN cana-2413 51 3 ) ) PUNCT cana-2413 51 4 ) ) PUNCT cana-2413 51 5 is be AUX cana-2413 51 6 zip zip NOUN cana-2413 51 7 graded grade VERB cana-2413 51 8 ring ring NOUN cana-2413 51 9 ; ; PUNCT cana-2413 51 10 𝑓 𝑓 DET cana-2413 51 11 ∈ ∈ NOUN cana-2413 51 12 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 51 13 ) ) PUNCT cana-2413 51 14 . . PUNCT cana-2413 52 1 iii iii X cana-2413 52 2 . . PUNCT cana-2413 52 3 𝑄𝑃 𝑄𝑃 PROPN cana-2413 52 4 𝑔(𝐺(𝑆 𝑔(𝐺(𝑆 PROPN cana-2413 52 5 ) ) PUNCT cana-2413 52 6 ) ) PUNCT cana-2413 52 7 is be AUX cana-2413 52 8 zip zip NOUN cana-2413 52 9 graded grade VERB cana-2413 52 10 ring ring NOUN cana-2413 52 11 ; ; PUNCT cana-2413 52 12 𝑃 𝑃 PROPN cana-2413 52 13 ∈ ∈ PROPN cana-2413 52 14 𝑇. 𝑇. PROPN cana-2413 52 15 proof proof NOUN cana-2413 52 16 : : PUNCT cana-2413 52 17 i. i. NOUN cana-2413 52 18 by by ADP cana-2413 52 19 assumption assumption NOUN cana-2413 52 20 , , PUNCT cana-2413 52 21 since since SCONJ cana-2413 52 22 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 52 23 ) ) PUNCT cana-2413 52 24 and and CCONJ cana-2413 52 25 οΏ½ οΏ½ PROPN cana-2413 52 26 Μƒ Μƒ PROPN cana-2413 52 27 οΏ½ οΏ½ PROPN cana-2413 52 28 ( ( PUNCT cana-2413 52 29 1 1 NUM cana-2413 52 30 βˆ’ βˆ’ PROPN cana-2413 52 31 𝑋) 𝑋) PROPN cana-2413 52 32 οΏ½ οΏ½ PROPN cana-2413 52 33 Μƒ Μƒ PROPN cana-2413 52 34 οΏ½ οΏ½ PROPN cana-2413 52 35 ⁄ ⁄ PROPN cana-2413 52 36 are be AUX cana-2413 52 37 commutative commutative ADJ cana-2413 52 38 noetherian noetherian ADJ cana-2413 52 39 domains domain NOUN cana-2413 52 40 . . PUNCT cana-2413 52 41 ii ii PROPN cana-2413 52 42 . . PROPN cana-2413 52 43 or or CCONJ cana-2413 52 44 iii iii X cana-2413 52 45 . . PUNCT cana-2413 53 1 this this PRON cana-2413 53 2 is be AUX cana-2413 53 3 an an DET cana-2413 53 4 adaptation adaptation NOUN cana-2413 53 5 of of ADP cana-2413 53 6 theorem theorem NOUN cana-2413 53 7 3.2 3.2 NUM cana-2413 53 8 . . PUNCT cana-2413 54 1 ( ( PUNCT cana-2413 54 2 1 1 NUM cana-2413 54 3 ) ) PUNCT cana-2413 54 4 of of ADP cana-2413 54 5 [ [ X cana-2413 54 6 4 4 NUM cana-2413 54 7 ] ] PUNCT cana-2413 54 8 ; ; PUNCT cana-2413 54 9 indeed indeed ADV cana-2413 54 10 , , PUNCT cana-2413 54 11 this this DET cana-2413 54 12 theorem theorem NOUN cana-2413 54 13 is be AUX cana-2413 54 14 phrased phrase VERB cana-2413 54 15 for for ADP cana-2413 54 16 ring ring NOUN cana-2413 54 17 theory theory NOUN cana-2413 54 18 but but CCONJ cana-2413 54 19 the the DET cana-2413 54 20 fact fact NOUN cana-2413 54 21 that that SCONJ cana-2413 54 22 we we PRON cana-2413 54 23 work work VERB cana-2413 54 24 with with ADP cana-2413 54 25 graded grade VERB cana-2413 54 26 objects object NOUN cana-2413 54 27 . . PUNCT cana-2413 55 1 the the DET cana-2413 55 2 proof proof NOUN cana-2413 55 3 of of ADP cana-2413 55 4 theorem theorem ADJ cana-2413 55 5 3.2 3.2 NUM cana-2413 55 6 . . PUNCT cana-2413 56 1 ( ( PUNCT cana-2413 56 2 1 1 NUM cana-2413 56 3 ) ) PUNCT cana-2413 56 4 of of ADP cana-2413 56 5 [ [ X cana-2413 56 6 4 4 X cana-2413 56 7 ] ] PUNCT cana-2413 56 8 then then ADV cana-2413 56 9 carries carry VERB cana-2413 56 10 over over ADP cana-2413 56 11 after after ADP cana-2413 56 12 the the DET cana-2413 56 13 common common ADJ cana-2413 56 14 modifications modification NOUN cana-2413 56 15 of of ADP cana-2413 56 16 graded grade VERB cana-2413 56 17 nature nature NOUN cana-2413 56 18 . . PUNCT cana-2413 57 1 communications communication NOUN cana-2413 57 2 on on ADP cana-2413 57 3 applied apply VERB cana-2413 57 4 nonlinear nonlinear ADJ cana-2413 57 5 analysis analysis NOUN cana-2413 57 6 issn issn NOUN cana-2413 57 7 : : PUNCT cana-2413 57 8 1074 1074 NUM cana-2413 57 9 - - PUNCT cana-2413 57 10 133x 133x NUM cana-2413 57 11 vol vol NOUN cana-2413 57 12 32 32 NUM cana-2413 57 13 no no NOUN cana-2413 57 14 . . PUNCT cana-2413 58 1 2s 2s NUM cana-2413 58 2 ( ( PUNCT cana-2413 58 3 2025 2025 NUM cana-2413 58 4 ) ) PUNCT cana-2413 58 5 406 406 NUM cana-2413 58 6 https://internationalpubls.com https://internationalpubls.com X cana-2413 58 7 now now ADV cana-2413 58 8 , , PUNCT cana-2413 58 9 we we PRON cana-2413 58 10 come come VERB cana-2413 58 11 the the DET cana-2413 58 12 first first ADJ cana-2413 58 13 main main ADJ cana-2413 58 14 result result NOUN cana-2413 58 15 of of ADP cana-2413 58 16 this this DET cana-2413 58 17 paper paper NOUN cana-2413 58 18 which which PRON cana-2413 58 19 comes come VERB cana-2413 58 20 directly directly ADV cana-2413 58 21 from from ADP cana-2413 58 22 definition definition NOUN cana-2413 58 23 and and CCONJ cana-2413 58 24 the the DET cana-2413 58 25 above above ADJ cana-2413 58 26 proposition proposition NOUN cana-2413 58 27 . . PUNCT cana-2413 59 1 proposition proposition NOUN cana-2413 59 2 3 3 NUM cana-2413 59 3 . . PUNCT cana-2413 59 4 with with ADP cana-2413 59 5 conventions convention NOUN cana-2413 59 6 notation notation NOUN cana-2413 59 7 as as ADP cana-2413 59 8 before before ADV cana-2413 59 9 : : PUNCT cana-2413 59 10 ( ( PUNCT cana-2413 59 11 𝑇 𝑇 PROPN cana-2413 59 12 , , PUNCT cana-2413 59 13 𝑂𝑇 𝑂𝑇 PROPN cana-2413 59 14 𝑔 𝑔 PROPN cana-2413 59 15 ) ) PUNCT cana-2413 59 16 is be AUX cana-2413 59 17 zip zip NOUN cana-2413 59 18 graded grade VERB cana-2413 59 19 affine affine NOUN cana-2413 59 20 scheme scheme NOUN cana-2413 59 21 . . PUNCT cana-2413 60 1 also also ADV cana-2413 60 2 , , PUNCT cana-2413 60 3 we we PRON cana-2413 60 4 may may AUX cana-2413 60 5 define define VERB cana-2413 60 6 the the DET cana-2413 60 7 graded grade VERB cana-2413 60 8 structure structure NOUN cana-2413 60 9 sheaf sheaf NOUN cana-2413 60 10 𝑂𝑛,𝑇 𝑂𝑛,𝑇 PROPN cana-2413 60 11 𝑔 𝑔 PROPN cana-2413 60 12 on on ADP cana-2413 60 13 𝑇 𝑇 PROPN cana-2413 60 14 : : PUNCT cana-2413 60 15 we we PRON cana-2413 60 16 may may AUX cana-2413 60 17 associate associate VERB cana-2413 60 18 to to ADP cana-2413 60 19 𝑇(𝑓 𝑇(𝑓 NUM cana-2413 60 20 ) ) PUNCT cana-2413 60 21 ; ; PUNCT cana-2413 60 22 𝑓 𝑓 DET cana-2413 60 23 ∈ ∈ NOUN cana-2413 60 24 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 60 25 ) ) PUNCT cana-2413 60 26 homogenous homogenous ADJ cana-2413 60 27 element element NOUN cana-2413 60 28 , , PUNCT cana-2413 60 29 the the DET cana-2413 60 30 graded grade VERB cana-2413 60 31 ring ring NOUN cana-2413 60 32 𝑆 𝑆 PROPN cana-2413 60 33 οΏ½ οΏ½ PROPN cana-2413 60 34 Μƒ Μƒ PROPN cana-2413 60 35 οΏ½ οΏ½ NOUN cana-2413 60 36 (𝑛 (𝑛 NOUN cana-2413 60 37 ) ) PUNCT cana-2413 60 38 βˆ’1 βˆ’1 NOUN cana-2413 60 39 ( ( PUNCT cana-2413 60 40 οΏ½ οΏ½ PROPN cana-2413 60 41 Μƒ Μƒ PROPN cana-2413 60 42 οΏ½ οΏ½ PROPN cana-2413 60 43 𝑋𝑛 𝑋𝑛 PROPN cana-2413 60 44 οΏ½ οΏ½ PROPN cana-2413 60 45 Μƒ Μƒ PROPN cana-2413 60 46 οΏ½ οΏ½ PROPN cana-2413 60 47 ) ) PUNCT cana-2413 60 48 , , PUNCT cana-2413 60 49 where where SCONJ cana-2413 60 50 𝑆 𝑆 PROPN cana-2413 60 51 οΏ½ οΏ½ PROPN cana-2413 60 52 Μƒ Μƒ PROPN cana-2413 60 53 οΏ½ οΏ½ NOUN cana-2413 60 54 (𝑛 (𝑛 NOUN cana-2413 60 55 ) ) PUNCT cana-2413 60 56 is be AUX cana-2413 60 57 the the DET cana-2413 60 58 homogenous homogenous ADJ cana-2413 60 59 image image NOUN cana-2413 60 60 of of ADP cana-2413 60 61 𝑆 𝑆 PROPN cana-2413 60 62 οΏ½ οΏ½ PROPN cana-2413 60 63 Μƒ Μƒ PROPN cana-2413 60 64 οΏ½ οΏ½ PROPN cana-2413 60 65 in in ADP cana-2413 60 66 οΏ½ οΏ½ PROPN cana-2413 60 67 Μƒ Μƒ PROPN cana-2413 60 68 οΏ½ οΏ½ PROPN cana-2413 60 69 and and CCONJ cana-2413 60 70 we we PRON cana-2413 60 71 obtain obtain VERB cana-2413 60 72 𝑂𝑛,𝑇 𝑂𝑛,𝑇 PROPN cana-2413 60 73 𝑔 𝑔 PROPN cana-2413 60 74 on on ADP cana-2413 60 75 𝑇 𝑇 PROPN cana-2413 60 76 having have VERB cana-2413 60 77 as as ADP cana-2413 60 78 stalk stalk NOUN cana-2413 60 79 at at ADP cana-2413 60 80 𝑃 𝑃 PROPN cana-2413 60 81 ∈ ∈ PROPN cana-2413 60 82 𝑇 𝑇 PROPN cana-2413 60 83 , , PUNCT cana-2413 60 84 the the DET cana-2413 60 85 graded grade VERB cana-2413 60 86 local local ADJ cana-2413 60 87 ring ring NOUN cana-2413 60 88 𝑄𝑝(𝑛 𝑄𝑝(𝑛 NOUN cana-2413 60 89 ) ) PUNCT cana-2413 60 90 𝑔 𝑔 PROPN cana-2413 60 91 ( ( PUNCT cana-2413 60 92 οΏ½ οΏ½ PROPN cana-2413 60 93 Μƒ Μƒ PROPN cana-2413 60 94 οΏ½ οΏ½ PROPN cana-2413 60 95 𝑋𝑛 𝑋𝑛 PROPN cana-2413 60 96 οΏ½ οΏ½ PROPN cana-2413 60 97 Μƒ Μƒ PROPN cana-2413 60 98 οΏ½ οΏ½ PROPN cana-2413 60 99 ⁄ ⁄ PROPN cana-2413 60 100 ) ) PUNCT cana-2413 60 101 = = PUNCT cana-2413 60 102 𝑆 𝑆 PROPN cana-2413 60 103 οΏ½ οΏ½ PROPN cana-2413 60 104 Μƒ Μƒ PROPN cana-2413 60 105 οΏ½ οΏ½ NOUN cana-2413 60 106 (𝑛 (𝑛 NOUN cana-2413 60 107 ) ) PUNCT cana-2413 60 108 βˆ’1 βˆ’1 NOUN cana-2413 60 109 ( ( PUNCT cana-2413 60 110 οΏ½ οΏ½ PROPN cana-2413 60 111 Μƒ Μƒ NOUN cana-2413 60 112 οΏ½ οΏ½ PROPN cana-2413 60 113 𝑋𝑛 𝑋𝑛 PROPN cana-2413 60 114 οΏ½ οΏ½ PROPN cana-2413 60 115 Μƒ Μƒ PROPN cana-2413 60 116 οΏ½ οΏ½ PROPN cana-2413 60 117 ⁄ ⁄ PROPN cana-2413 60 118 ) ) PUNCT cana-2413 60 119 . . PUNCT cana-2413 61 1 proposition proposition NOUN cana-2413 61 2 4 4 NUM cana-2413 61 3 . . PUNCT cana-2413 62 1 under under ADP cana-2413 62 2 the the DET cana-2413 62 3 same same ADJ cana-2413 62 4 assumptions assumption NOUN cana-2413 62 5 : : PUNCT cana-2413 62 6 if if SCONJ cana-2413 62 7 οΏ½ οΏ½ PROPN cana-2413 62 8 Μƒ Μƒ PROPN cana-2413 62 9 οΏ½ οΏ½ PROPN cana-2413 62 10 is be AUX cana-2413 62 11 zip zip NOUN cana-2413 62 12 domain domain NOUN cana-2413 62 13 , , PUNCT cana-2413 62 14 then then ADV cana-2413 62 15 βˆ€ βˆ€ X cana-2413 62 16     SPACE cana-2413 62 17 𝑛 𝑛 PRON cana-2413 62 18 ∈ ∈ PROPN cana-2413 62 19 𝛧+ 𝛧+ NOUN cana-2413 62 20 , , PUNCT cana-2413 62 21 οΏ½ οΏ½ PROPN cana-2413 62 22 Μƒ Μƒ PROPN cana-2413 62 23 οΏ½ οΏ½ PROPN cana-2413 62 24 𝑋𝑛 𝑋𝑛 PROPN cana-2413 62 25 οΏ½ οΏ½ PROPN cana-2413 62 26 Μƒ Μƒ PROPN cana-2413 62 27 οΏ½ οΏ½ PROPN cana-2413 62 28 ⁄ ⁄ PROPN cana-2413 62 29 = = SYM cana-2413 62 30 οΏ½ οΏ½ PROPN cana-2413 62 31 Μ„Μƒ Μ„Μƒ NOUN cana-2413 62 32 οΏ½ οΏ½ PROPN cana-2413 62 33 (𝑛 (𝑛 NOUN cana-2413 62 34 ) ) PUNCT cana-2413 62 35 zip zip NOUN cana-2413 62 36 graded grade VERB cana-2413 62 37 ring ring NOUN cana-2413 62 38 . . PUNCT cana-2413 63 1 proof proof NOUN cana-2413 63 2 . . PUNCT cana-2413 64 1 it it PRON cana-2413 64 2 is be AUX cana-2413 64 3 easily easily ADV cana-2413 64 4 checked check VERB cana-2413 64 5 that that SCONJ cana-2413 64 6 , , PUNCT cana-2413 64 7 if if SCONJ cana-2413 64 8 𝐼(𝑛 𝐼(𝑛 ADP cana-2413 64 9 ) ) PUNCT cana-2413 64 10     SPACE cana-2413 64 11 ⊲ ⊲ NOUN cana-2413 64 12     SPACE cana-2413 64 13 𝐼 𝐼 PROPN cana-2413 64 14     SPACE cana-2413 64 15 οΏ½ οΏ½ PROPN cana-2413 64 16 Μ„Μƒ Μ„Μƒ NOUN cana-2413 64 17 οΏ½ οΏ½ NOUN cana-2413 64 18 (𝑛 (𝑛 NOUN cana-2413 64 19 ): ): PUNCT cana-2413 64 20     SPACE cana-2413 64 21 π‘Žπ‘›π‘›π‘” π‘Žπ‘›π‘›π‘” NOUN cana-2413 64 22 ( ( PUNCT cana-2413 64 23 𝐼(𝑛 𝐼(𝑛 NOUN cana-2413 64 24 ) ) PUNCT cana-2413 64 25 ) ) PUNCT cana-2413 64 26 = = SYM cana-2413 65 1 0 0 NUM cana-2413 65 2 ; ; PUNCT cana-2413 65 3 𝐼(𝑛 𝐼(𝑛 PRON cana-2413 65 4 ) ) PUNCT cana-2413 65 5 = = SYM cana-2413 65 6 𝐼/𝑋𝑛 𝐼/𝑋𝑛 PROPN cana-2413 65 7 οΏ½ οΏ½ PROPN cana-2413 65 8 Μƒ Μƒ PROPN cana-2413 65 9 οΏ½ οΏ½ PROPN cana-2413 65 10 ; ; PUNCT cana-2413 65 11 𝐼 𝐼 PROPN cana-2413 65 12     SPACE cana-2413 65 13 ⊲ ⊲ NOUN cana-2413 65 14     SPACE cana-2413 65 15 𝐼 𝐼 PROPN cana-2413 65 16     SPACE cana-2413 65 17 οΏ½ οΏ½ PROPN cana-2413 65 18 Μƒ Μƒ PROPN cana-2413 65 19 οΏ½ οΏ½ PROPN cana-2413 65 20 .then .then PUNCT cana-2413 65 21 π‘Žπ‘›π‘›π‘”(𝐼 π‘Žπ‘›π‘›π‘”(𝐼 PROPN cana-2413 65 22 ) ) PUNCT cana-2413 65 23 = = PUNCT cana-2413 66 1 0 0 NUM cana-2413 66 2 ; ; PUNCT cana-2413 66 3 . . PUNCT cana-2413 67 1 in in ADP cana-2413 67 2 𝑆 𝑆 PROPN cana-2413 67 3 . . PUNCT cana-2413 68 1 then then ADV cana-2413 68 2 there there PRON cana-2413 68 3 exists exist VERB cana-2413 68 4 𝐼0 𝐼0 ADJ cana-2413 68 5 ⊲ ⊲ NOUN cana-2413 68 6     SPACE cana-2413 68 7 𝐼(𝑓𝑖𝑛𝑖𝑑𝑒 𝐼(𝑓𝑖𝑛𝑖𝑑𝑒 PROPN cana-2413 68 8 ) ) PUNCT cana-2413 68 9 ∢ ∢ NOUN cana-2413 68 10 π‘Žπ‘›π‘›π‘”(𝐼0 π‘Žπ‘›π‘›π‘”(𝐼0 NOUN cana-2413 68 11 ) ) PUNCT cana-2413 69 1 = = PUNCT cana-2413 69 2 0 0 X cana-2413 69 3 . . PUNCT cana-2413 70 1 then then ADV cana-2413 70 2 there there PRON cana-2413 70 3 exists exist VERB cana-2413 70 4 𝐼0/𝑋𝑛 𝐼0/𝑋𝑛 PROPN cana-2413 70 5 οΏ½ οΏ½ PROPN cana-2413 70 6 Μƒ Μƒ PROPN cana-2413 70 7 οΏ½ οΏ½ NOUN cana-2413 70 8 = = SYM cana-2413 70 9     SPACE cana-2413 70 10 𝐼0 𝐼0 ADJ cana-2413 70 11     SPACE cana-2413 70 12 ⊲ ⊲ NOUN cana-2413 70 13     SPACE cana-2413 70 14 𝐼 𝐼 PROPN cana-2413 70 15 : : PUNCT cana-2413 70 16 π‘Žπ‘›π‘›π‘”(𝐼0 π‘Žπ‘›π‘›π‘”(𝐼0 NOUN cana-2413 70 17 ) ) PUNCT cana-2413 71 1 = = SYM cana-2413 71 2 0 0 PUNCT cana-2413 72 1 and and CCONJ cana-2413 72 2 we we PRON cana-2413 72 3 have have VERB cana-2413 72 4 οΏ½ οΏ½ PROPN cana-2413 72 5 Μ„Μƒ Μ„Μƒ NOUN cana-2413 72 6 οΏ½ οΏ½ NOUN cana-2413 72 7 (𝑛 (𝑛 NOUN cana-2413 72 8 ) ) PUNCT cana-2413 72 9 , , PUNCT cana-2413 72 10 βˆ€π‘› βˆ€π‘› NOUN cana-2413 72 11 , , PUNCT cana-2413 72 12 is be AUX cana-2413 72 13 zip zip NOUN cana-2413 72 14 . . PUNCT cana-2413 72 15 proposition proposition NOUN cana-2413 72 16 5 5 NUM cana-2413 72 17 . . PUNCT cana-2413 73 1 ( ( PUNCT cana-2413 73 2 graded grade VERB cana-2413 73 3 version version NOUN cana-2413 73 4 of of ADP cana-2413 73 5 theorem theorem NOUN cana-2413 73 6 3.2.(1 3.2.(1 NUM cana-2413 73 7 ) ) PUNCT cana-2413 73 8 in in ADP cana-2413 73 9 [ [ X cana-2413 73 10 4 4 NUM cana-2413 73 11 ] ] PUNCT cana-2413 73 12 ) ) PUNCT cana-2413 73 13 under under ADP cana-2413 73 14 the the DET cana-2413 73 15 same same ADJ cana-2413 73 16 consideration consideration NOUN cana-2413 73 17 : : PUNCT cana-2413 73 18 𝑄 𝑄 PROPN cana-2413 73 19 οΏ½ οΏ½ PROPN cana-2413 73 20 Μ„Μƒ Μ„Μƒ NOUN cana-2413 73 21 οΏ½ οΏ½ PROPN cana-2413 73 22 𝑔 𝑔 PROPN cana-2413 73 23 ( ( PUNCT cana-2413 73 24 οΏ½ οΏ½ PROPN cana-2413 73 25 Μ„Μƒ Μ„Μƒ NOUN cana-2413 73 26 οΏ½ οΏ½ NOUN cana-2413 73 27 ) ) PUNCT cana-2413 73 28 is be AUX cana-2413 73 29 zip zip NOUN cana-2413 73 30 graded grade VERB cana-2413 73 31 ring ring NOUN cana-2413 73 32 . . PUNCT cana-2413 74 1 again again ADV cana-2413 74 2 , , PUNCT cana-2413 74 3 we we PRON cana-2413 74 4 can can AUX cana-2413 74 5 mention mention VERB cana-2413 74 6 the the DET cana-2413 74 7 second second ADJ cana-2413 74 8 main main ADJ cana-2413 74 9 important important ADJ cana-2413 74 10 result result NOUN cana-2413 74 11 of of ADP cana-2413 74 12 this this DET cana-2413 74 13 paper paper NOUN cana-2413 74 14 which which PRON cana-2413 74 15 comes come VERB cana-2413 74 16 directly directly ADV cana-2413 74 17 from from ADP cana-2413 74 18 definition definition NOUN cana-2413 74 19 and and CCONJ cana-2413 74 20 the the DET cana-2413 74 21 above above ADJ cana-2413 74 22 proposition proposition NOUN cana-2413 74 23 . . PUNCT cana-2413 75 1 proposition proposition NOUN cana-2413 75 2 6 6 NUM cana-2413 75 3 . . PUNCT cana-2413 75 4 with with ADP cana-2413 75 5 conventions convention NOUN cana-2413 75 6 and and CCONJ cana-2413 75 7 notations notation NOUN cana-2413 75 8 as as ADP cana-2413 75 9 before before ADV cana-2413 75 10 : : PUNCT cana-2413 75 11 if if SCONJ cana-2413 75 12 οΏ½ οΏ½ PROPN cana-2413 75 13 Μƒ Μƒ PROPN cana-2413 75 14 οΏ½ οΏ½ PROPN cana-2413 75 15 is be AUX cana-2413 75 16 zip zip NOUN cana-2413 75 17 domain domain NOUN cana-2413 75 18 then then ADV cana-2413 75 19 the the DET cana-2413 75 20 graded grade VERB cana-2413 75 21 structure structure NOUN cana-2413 75 22 affine affine NOUN cana-2413 75 23 scheme scheme NOUN cana-2413 75 24 ( ( PUNCT cana-2413 75 25 𝑇 𝑇 PROPN cana-2413 75 26 , , PUNCT cana-2413 75 27 𝑂𝑛,𝑇 𝑂𝑛,𝑇 NOUN cana-2413 75 28 𝑔 𝑔 PROPN cana-2413 75 29 ) ) PUNCT cana-2413 75 30 , , PUNCT cana-2413 75 31 βˆ€ βˆ€ X cana-2413 75 32     SPACE cana-2413 75 33 𝑛 𝑛 PRON cana-2413 75 34 ∈ ∈ PROPN cana-2413 75 35 𝛧+ 𝛧+ NOUN cana-2413 75 36 , , PUNCT cana-2413 75 37 is be AUX cana-2413 75 38 zip zip NOUN cana-2413 75 39 graded grade VERB cana-2413 75 40 affine affine NOUN cana-2413 75 41 scheme scheme NOUN cana-2413 75 42 . . PUNCT cana-2413 76 1 a a DET cana-2413 76 2 similar similar ADJ cana-2413 76 3 result result NOUN cana-2413 76 4 holds hold VERB cana-2413 76 5 in in ADP cana-2413 76 6 case case NOUN cana-2413 76 7 of of ADP cana-2413 76 8 the the DET cana-2413 76 9 rees rees PROPN cana-2413 76 10 graded grade VERB cana-2413 76 11 micro micro NOUN cana-2413 76 12 - - NOUN cana-2413 76 13 localization localization ADJ cana-2413 76 14 rings ring NOUN cana-2413 76 15 οΏ½ οΏ½ PROPN cana-2413 76 16 Μƒ Μƒ PROPN cana-2413 76 17 οΏ½ οΏ½ PROPN cana-2413 76 18 οΏ½ οΏ½ PROPN cana-2413 76 19 Μƒ Μƒ PROPN cana-2413 76 20 οΏ½ οΏ½ PROPN cana-2413 76 21 πœ‡ πœ‡ X cana-2413 76 22 ( ( PUNCT cana-2413 76 23 οΏ½ οΏ½ PROPN cana-2413 76 24 Μƒ Μƒ NOUN cana-2413 76 25 οΏ½ οΏ½ PROPN cana-2413 76 26 ) ) PUNCT cana-2413 76 27 ; ; PUNCT cana-2413 77 1 𝑆𝑓 𝑆𝑓 PROPN cana-2413 77 2 = = PUNCT cana-2413 77 3 { { PUNCT cana-2413 77 4 𝑓 𝑓 PROPN cana-2413 77 5 , , PUNCT cana-2413 77 6 𝑓2 𝑓2 NOUN cana-2413 77 7 , , PUNCT cana-2413 77 8 β‹― β‹― VERB cana-2413 77 9 } } PUNCT cana-2413 77 10 in in ADP cana-2413 77 11 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 77 12 ) ) PUNCT cana-2413 77 13 . . PUNCT cana-2413 78 1 that that PRON cana-2413 78 2 is be AUX cana-2413 78 3 οΏ½ οΏ½ PROPN cana-2413 78 4 Μƒ Μƒ PROPN cana-2413 78 5 οΏ½ οΏ½ PROPN cana-2413 78 6 οΏ½ οΏ½ PROPN cana-2413 78 7 Μƒ Μƒ PROPN cana-2413 78 8 οΏ½ οΏ½ PROPN cana-2413 78 9 πœ‡ πœ‡ X cana-2413 78 10 ( ( PUNCT cana-2413 78 11 οΏ½ οΏ½ PROPN cana-2413 78 12 Μƒ Μƒ NOUN cana-2413 78 13 οΏ½ οΏ½ PROPN cana-2413 78 14 ) ) PUNCT cana-2413 79 1 = = PROPN cana-2413 79 2 lim lim PROPN cana-2413 79 3 𝑔 𝑔 PROPN cana-2413 79 4 οΏ½ οΏ½ PROPN cana-2413 79 5 βƒ– βƒ– PROPN cana-2413 79 6 οΏ½ οΏ½ PROPN cana-2413 79 7 𝑄 𝑄 PROPN cana-2413 79 8 οΏ½ οΏ½ PROPN cana-2413 79 9 Μ…Μƒ Μ…Μƒ NOUN cana-2413 79 10 οΏ½ οΏ½ PROPN cana-2413 79 11 (𝑛 (𝑛 NOUN cana-2413 79 12 ) ) PUNCT cana-2413 79 13 𝑔 𝑔 PROPN cana-2413 79 14 ( ( PUNCT cana-2413 79 15 οΏ½ οΏ½ PROPN cana-2413 79 16 Μ„Μƒ Μ„Μƒ NOUN cana-2413 79 17 οΏ½ οΏ½ NOUN cana-2413 79 18 (𝑛 (𝑛 NOUN cana-2413 79 19 ) ) PUNCT cana-2413 79 20 ) ) PUNCT cana-2413 80 1 = = SYM cana-2413 80 2 lim lim PROPN cana-2413 80 3 𝑔 𝑔 PROPN cana-2413 80 4 οΏ½ οΏ½ PROPN cana-2413 80 5 βƒ– βƒ– PROPN cana-2413 80 6 οΏ½ οΏ½ PROPN cana-2413 80 7 ( ( PUNCT cana-2413 80 8 𝑓(̅𝑛 𝑓(̅𝑛 NOUN cana-2413 80 9 ) ) PUNCT cana-2413 80 10 ) ) PUNCT cana-2413 81 1 βˆ’1( βˆ’1( PROPN cana-2413 81 2 οΏ½ οΏ½ PROPN cana-2413 81 3 Μ„Μƒ Μ„Μƒ NOUN cana-2413 81 4 οΏ½ οΏ½ NOUN cana-2413 81 5 (𝑛 (𝑛 NOUN cana-2413 81 6 ) ) PUNCT cana-2413 81 7 ) ) PUNCT cana-2413 81 8 ; ; PUNCT cana-2413 81 9 οΏ½ οΏ½ PROPN cana-2413 81 10 Μ„Μƒ Μ„Μƒ NOUN cana-2413 81 11 οΏ½ οΏ½ NOUN cana-2413 81 12 (𝑛 (𝑛 NOUN cana-2413 81 13 ) ) PUNCT cana-2413 81 14 = = SYM cana-2413 81 15 οΏ½ οΏ½ PROPN cana-2413 81 16 Μƒ Μƒ PROPN cana-2413 81 17 οΏ½ οΏ½ PROPN cana-2413 81 18 𝑋𝑛 𝑋𝑛 PROPN cana-2413 81 19 οΏ½ οΏ½ PROPN cana-2413 81 20 Μƒ Μƒ PROPN cana-2413 81 21 οΏ½ οΏ½ PROPN cana-2413 81 22 ⁄ ⁄ PROPN cana-2413 81 23 , , PUNCT cana-2413 81 24 see see VERB cana-2413 81 25 [ [ X cana-2413 81 26 1 1 X cana-2413 81 27 ] ] PUNCT cana-2413 81 28 and and CCONJ cana-2413 81 29 [ [ X cana-2413 81 30 9 9 NUM cana-2413 81 31 ] ] PUNCT cana-2413 81 32 . . PUNCT cana-2413 82 1 now now ADV cana-2413 82 2 , , PUNCT cana-2413 82 3 if if SCONJ cana-2413 82 4 𝐼 𝐼 PROPN cana-2413 82 5 οΏ½ οΏ½ PROPN cana-2413 82 6 Μƒ Μƒ PROPN cana-2413 82 7 οΏ½ οΏ½ PROPN cana-2413 82 8 πœ‡ πœ‡ ADP cana-2413 82 9 ⊲ ⊲ NOUN cana-2413 82 10     SPACE cana-2413 82 11 οΏ½ οΏ½ PROPN cana-2413 82 12 Μƒ Μƒ PROPN cana-2413 82 13 οΏ½ οΏ½ PROPN cana-2413 82 14 οΏ½ οΏ½ PROPN cana-2413 82 15 Μƒ Μƒ PROPN cana-2413 82 16 οΏ½ οΏ½ PROPN cana-2413 82 17 πœ‡ πœ‡ X cana-2413 82 18 ( ( PUNCT cana-2413 82 19 οΏ½ οΏ½ PROPN cana-2413 82 20 Μƒ Μƒ NOUN cana-2413 82 21 οΏ½ οΏ½ NOUN cana-2413 82 22 ) ) PUNCT cana-2413 82 23 such such ADJ cana-2413 82 24 that that DET cana-2413 82 25 π‘Žπ‘›π‘›π‘” π‘Žπ‘›π‘›π‘” NOUN cana-2413 82 26 ( ( PUNCT cana-2413 82 27 𝐼 𝐼 PROPN cana-2413 82 28 οΏ½ οΏ½ PROPN cana-2413 82 29 Μƒ Μƒ PROPN cana-2413 82 30 οΏ½ οΏ½ PROPN cana-2413 82 31 πœ‡ πœ‡ ADP cana-2413 82 32 ) ) PUNCT cana-2413 82 33 = = SYM cana-2413 82 34 0 0 NUM cana-2413 82 35 , , PUNCT cana-2413 82 36 𝐼 𝐼 PROPN cana-2413 82 37 οΏ½ οΏ½ PROPN cana-2413 82 38 Μƒ Μƒ PROPN cana-2413 82 39 οΏ½ οΏ½ PROPN cana-2413 82 40 πœ‡ πœ‡ ADP cana-2413 82 41 = = X cana-2413 82 42 𝑄 𝑄 PROPN cana-2413 82 43 οΏ½ οΏ½ PROPN cana-2413 82 44 Μƒ Μƒ PROPN cana-2413 82 45 οΏ½ οΏ½ NOUN cana-2413 82 46 (𝑛 (𝑛 NOUN cana-2413 82 47 ) ) PUNCT cana-2413 82 48 𝑔 𝑔 PROPN cana-2413 82 49 ( ( PUNCT cana-2413 82 50 𝐼 𝐼 PROPN cana-2413 82 51 οΏ½ οΏ½ PROPN cana-2413 82 52 Μƒ Μƒ PROPN cana-2413 82 53 οΏ½ οΏ½ PROPN cana-2413 82 54 πœ‡ πœ‡ ADP cana-2413 82 55 ) ) PUNCT cana-2413 82 56 and and CCONJ cana-2413 82 57 𝐼 𝐼 PROPN cana-2413 82 58 οΏ½ οΏ½ PROPN cana-2413 82 59 Μƒ Μƒ PROPN cana-2413 82 60 οΏ½ οΏ½ PROPN cana-2413 82 61 πœ‡ πœ‡ ADP cana-2413 82 62 ⊲ ⊲ PROPN cana-2413 82 63     SPACE cana-2413 82 64 οΏ½ οΏ½ PROPN cana-2413 82 65 Μƒ Μƒ PROPN cana-2413 82 66 οΏ½ οΏ½ NOUN cana-2413 82 67 (𝑛 (𝑛 NOUN cana-2413 82 68 ) ) PUNCT cana-2413 82 69 = = SYM cana-2413 82 70     SPACE cana-2413 82 71 𝑄 𝑄 PROPN cana-2413 82 72 οΏ½ οΏ½ PROPN cana-2413 82 73 Μƒ Μƒ PROPN cana-2413 82 74 οΏ½ οΏ½ NOUN cana-2413 82 75 (𝑛 (𝑛 NOUN cana-2413 82 76 ) ) PUNCT cana-2413 82 77 𝑔 𝑔 PROPN cana-2413 82 78 ( ( PUNCT cana-2413 82 79 οΏ½ οΏ½ PROPN cana-2413 82 80 Μƒ Μƒ PROPN cana-2413 82 81 οΏ½ οΏ½ PROPN cana-2413 82 82 ) ) PUNCT cana-2413 82 83 , , PUNCT cana-2413 82 84 representing represent VERB cana-2413 82 85 𝐼 𝐼 PROPN cana-2413 82 86 οΏ½ οΏ½ PROPN cana-2413 82 87 Μƒ Μƒ PROPN cana-2413 82 88 οΏ½ οΏ½ PROPN cana-2413 82 89 πœ‡ πœ‡ ADP cana-2413 82 90 at at ADP cana-2413 82 91 a a DET cana-2413 82 92 level level NOUN cana-2413 82 93 𝑛 𝑛 NOUN cana-2413 82 94 in in ADP cana-2413 82 95 the the DET cana-2413 82 96 inverse inverse NOUN cana-2413 82 97 limit limit NOUN cana-2413 82 98 . . PUNCT cana-2413 83 1 hence hence ADV cana-2413 83 2 π‘Žπ‘›π‘›π‘” π‘Žπ‘›π‘›π‘” PROPN cana-2413 83 3 ( ( PUNCT cana-2413 83 4 𝑄 𝑄 PROPN cana-2413 83 5 οΏ½ οΏ½ PROPN cana-2413 83 6 Μƒ Μƒ PROPN cana-2413 83 7 οΏ½ οΏ½ NOUN cana-2413 83 8 (𝑛 (𝑛 NOUN cana-2413 83 9 ) ) PUNCT cana-2413 83 10 𝑔 𝑔 PROPN cana-2413 83 11 ( ( PUNCT cana-2413 83 12 𝐼 𝐼 PROPN cana-2413 83 13 οΏ½ οΏ½ PROPN cana-2413 83 14 Μƒ Μƒ PROPN cana-2413 83 15 οΏ½ οΏ½ PROPN cana-2413 83 16 πœ‡ πœ‡ ADP cana-2413 83 17 ) ) PUNCT cana-2413 83 18 ) ) PUNCT cana-2413 84 1 = = SYM cana-2413 84 2 0 0 PUNCT cana-2413 85 1 and and CCONJ cana-2413 85 2 there there PRON cana-2413 85 3 exists exist VERB cana-2413 85 4 𝐼0 𝐼0 ADJ cana-2413 85 5 πœ‡ πœ‡ X cana-2413 85 6 ( ( PUNCT cana-2413 85 7 𝑛 𝑛 NOUN cana-2413 85 8 ) ) PUNCT cana-2413 85 9     SPACE cana-2413 85 10 ⊲ ⊲ NOUN cana-2413 85 11     SPACE cana-2413 85 12 οΏ½ οΏ½ PROPN cana-2413 85 13 Μƒ Μƒ PROPN cana-2413 85 14 οΏ½ οΏ½ NOUN cana-2413 85 15 (𝑛 (𝑛 NOUN cana-2413 85 16 ) ) PUNCT cana-2413 85 17 ( ( PUNCT cana-2413 85 18 as as ADP cana-2413 85 19 above above ADJ cana-2413 85 20 ) ) PUNCT cana-2413 85 21 finite finite VERB cana-2413 85 22 such such ADJ cana-2413 85 23 that that DET cana-2413 85 24 π‘Žπ‘›π‘›π‘” π‘Žπ‘›π‘›π‘” NOUN cana-2413 85 25 ( ( PUNCT cana-2413 85 26 𝑄 𝑄 PROPN cana-2413 85 27 οΏ½ οΏ½ PROPN cana-2413 85 28 Μƒ Μƒ PROPN cana-2413 85 29 οΏ½ οΏ½ NOUN cana-2413 85 30 (𝑛 (𝑛 NOUN cana-2413 85 31 ) ) PUNCT cana-2413 85 32 𝑔 𝑔 PROPN cana-2413 85 33 ( ( PUNCT cana-2413 85 34 𝐼0(𝑛 𝐼0(𝑛 PROPN cana-2413 85 35 ) ) PUNCT cana-2413 85 36 ) ) PUNCT cana-2413 85 37 ) ) PUNCT cana-2413 86 1 = = PUNCT cana-2413 86 2 0 0 X cana-2413 86 3 . . PUNCT cana-2413 87 1 then then ADV cana-2413 87 2 there there PRON cana-2413 87 3 exists exist VERB cana-2413 87 4 𝐼 𝐼 PROPN cana-2413 87 5 0 0 NUM cana-2413 87 6 οΏ½ οΏ½ PROPN cana-2413 87 7 Μƒ Μƒ PROPN cana-2413 87 8 οΏ½ οΏ½ PROPN cana-2413 87 9 πœ‡ πœ‡ ADP cana-2413 87 10     SPACE cana-2413 87 11 ⊲ ⊲ NOUN cana-2413 87 12     SPACE cana-2413 87 13 𝐼 𝐼 PROPN cana-2413 87 14 οΏ½ οΏ½ PROPN cana-2413 87 15 Μƒ Μƒ PROPN cana-2413 87 16 οΏ½ οΏ½ PROPN cana-2413 87 17 πœ‡ πœ‡ X cana-2413 87 18 ( ( PUNCT cana-2413 87 19 finite finite PROPN cana-2413 87 20 ) ) PUNCT cana-2413 87 21 such such ADJ cana-2413 87 22 that that DET cana-2413 87 23 π‘Žπ‘›π‘›π‘” π‘Žπ‘›π‘›π‘” NOUN cana-2413 87 24 ( ( PUNCT cana-2413 87 25 𝐼 𝐼 PROPN cana-2413 87 26 0 0 NUM cana-2413 87 27 οΏ½ οΏ½ PROPN cana-2413 87 28 Μƒ Μƒ PROPN cana-2413 87 29 οΏ½ οΏ½ PROPN cana-2413 87 30 πœ‡ πœ‡ ADP cana-2413 87 31 ) ) PUNCT cana-2413 87 32     SPACE cana-2413 87 33 = = SYM cana-2413 87 34 0 0 NUM cana-2413 87 35 and and CCONJ cana-2413 87 36 οΏ½ οΏ½ PROPN cana-2413 87 37 Μƒ Μƒ PROPN cana-2413 87 38 οΏ½ οΏ½ PROPN cana-2413 87 39 οΏ½ οΏ½ PROPN cana-2413 87 40 Μƒ Μƒ PROPN cana-2413 87 41 οΏ½ οΏ½ PROPN cana-2413 87 42 πœ‡ πœ‡ X cana-2413 87 43 ( ( PUNCT cana-2413 87 44 οΏ½ οΏ½ PROPN cana-2413 87 45 Μƒ Μƒ NOUN cana-2413 87 46 οΏ½ οΏ½ PROPN cana-2413 87 47 ) ) PUNCT cana-2413 87 48 is be AUX cana-2413 87 49 zip zip NOUN cana-2413 87 50 ring ring NOUN cana-2413 87 51 . . PUNCT cana-2413 88 1 it it PRON cana-2413 88 2 is be AUX cana-2413 88 3 noted note VERB cana-2413 88 4 that that SCONJ cana-2413 88 5 one one PRON cana-2413 88 6 may may AUX cana-2413 88 7 start start VERB cana-2413 88 8 the the DET cana-2413 88 9 argument argument NOUN cana-2413 88 10 at at ADP cana-2413 88 11 any any DET cana-2413 88 12 π‘š π‘š NOUN cana-2413 88 13 larger large ADJ cana-2413 88 14 than than ADP cana-2413 88 15 𝑛. 𝑛. NOUN cana-2413 88 16 communications communication NOUN cana-2413 88 17 on on ADP cana-2413 88 18 applied apply VERB cana-2413 88 19 nonlinear nonlinear ADJ cana-2413 88 20 analysis analysis NOUN cana-2413 88 21 issn issn NOUN cana-2413 88 22 : : PUNCT cana-2413 88 23 1074 1074 NUM cana-2413 88 24 - - PUNCT cana-2413 88 25 133x 133x NUM cana-2413 88 26 vol vol NOUN cana-2413 88 27 32 32 NUM cana-2413 88 28 no no NOUN cana-2413 88 29 . . PUNCT cana-2413 89 1 2s 2s NUM cana-2413 89 2 ( ( PUNCT cana-2413 89 3 2025 2025 NUM cana-2413 89 4 ) ) PUNCT cana-2413 89 5 407 407 NUM cana-2413 89 6 https://internationalpubls.com https://internationalpubls.com X cana-2413 89 7 now now ADV cana-2413 89 8 , , PUNCT cana-2413 89 9 associating associate VERB cana-2413 89 10 to to ADP cana-2413 89 11 an an DET cana-2413 89 12 open open ADJ cana-2413 89 13 set set NOUN cana-2413 89 14 𝑋(𝑓 𝑋(𝑓 NUM cana-2413 89 15 ) ) PUNCT cana-2413 89 16 the the DET cana-2413 89 17 micro micro NOUN cana-2413 89 18 - - NOUN cana-2413 89 19 localizations localization NOUN cana-2413 89 20 οΏ½ οΏ½ PROPN cana-2413 89 21 Μƒ Μƒ PROPN cana-2413 89 22 οΏ½ οΏ½ PROPN cana-2413 89 23 οΏ½ οΏ½ PROPN cana-2413 89 24 Μƒ Μƒ PROPN cana-2413 89 25 οΏ½ οΏ½ PROPN cana-2413 89 26 πœ‡ πœ‡ X cana-2413 89 27 ( ( PUNCT cana-2413 89 28 οΏ½ οΏ½ PROPN cana-2413 89 29 Μƒ Μƒ NOUN cana-2413 89 30 οΏ½ οΏ½ PROPN cana-2413 89 31 ) ) PUNCT cana-2413 89 32 , , PUNCT cana-2413 89 33 res re NOUN cana-2413 89 34 . . PUNCT cana-2413 90 1 𝑄𝑓 𝑄𝑓 NOUN cana-2413 90 2 πœ‡ πœ‡ ADP cana-2413 90 3 ( ( PUNCT cana-2413 90 4 𝑆 𝑆 PROPN cana-2413 90 5 ) ) PUNCT cana-2413 90 6 we we PRON cana-2413 90 7 obtain obtain VERB cana-2413 90 8 sheaf sheaf NOUN cana-2413 90 9 οΏ½ οΏ½ PROPN cana-2413 90 10 Μƒ Μƒ PROPN cana-2413 90 11 οΏ½ οΏ½ NOUN cana-2413 90 12 𝑋 𝑋 NOUN cana-2413 90 13 πœ‡ πœ‡ X cana-2413 90 14 , , PUNCT cana-2413 90 15 res re NOUN cana-2413 90 16 . . PUNCT cana-2413 90 17 𝑂𝑋 𝑂𝑋 PROPN cana-2413 90 18 πœ‡ πœ‡ ADP cana-2413 90 19 having have VERB cana-2413 90 20 as as ADP cana-2413 90 21 the the DET cana-2413 90 22 completed complete VERB cana-2413 90 23 stalks stalk NOUN cana-2413 90 24 at at ADP cana-2413 90 25 𝑃 𝑃 NOUN cana-2413 90 26 ∈ ∈ NOUN cana-2413 90 27 𝑆𝑝𝑒𝑐𝑔(𝐺(𝑆 𝑆𝑝𝑒𝑐𝑔(𝐺(𝑆 NOUN cana-2413 90 28 ) ) PUNCT cana-2413 90 29 ) ) PUNCT cana-2413 91 1 ( ( PUNCT cana-2413 91 2 or or CCONJ cana-2413 91 3 𝑇 𝑇 PROPN cana-2413 91 4 = = SYM cana-2413 91 5 π‘ƒπ‘Ÿπ‘œπ‘—π‘” π‘ƒπ‘Ÿπ‘œπ‘—π‘” PROPN cana-2413 91 6     SPACE cana-2413 91 7 𝐺(𝑆 𝐺(𝑆 NOUN cana-2413 91 8 ) ) PUNCT cana-2413 91 9 the the DET cana-2413 91 10 ring ring PROPN cana-2413 91 11 οΏ½ οΏ½ PROPN cana-2413 91 12 Μƒ Μƒ PROPN cana-2413 91 13 οΏ½ οΏ½ PROPN cana-2413 91 14 𝑝 𝑝 ADJ cana-2413 91 15 πœ‡ πœ‡ X cana-2413 91 16 ( ( PUNCT cana-2413 91 17 οΏ½ οΏ½ PROPN cana-2413 91 18 Μƒ Μƒ NOUN cana-2413 91 19 οΏ½ οΏ½ NOUN cana-2413 91 20 ),res ),res PROPN cana-2413 91 21 . . PUNCT cana-2413 92 1 𝑄𝑃 𝑄𝑃 PROPN cana-2413 92 2 πœ‡ πœ‡ PROPN cana-2413 92 3 ( ( PUNCT cana-2413 92 4 𝑆 𝑆 PROPN cana-2413 92 5 ) ) PUNCT cana-2413 92 6 , , PUNCT cana-2413 92 7 see see VERB cana-2413 92 8 [ [ X cana-2413 92 9 9 9 NUM cana-2413 92 10 ] ] PUNCT cana-2413 92 11 . . PUNCT cana-2413 93 1 note note VERB cana-2413 93 2 that that DET cana-2413 93 3 𝑂𝑋 𝑂𝑋 PROPN cana-2413 93 4 πœ‡ πœ‡ ADP cana-2413 93 5 is be AUX cana-2413 93 6 a a DET cana-2413 93 7 sheaf sheaf NOUN cana-2413 93 8 of of ADP cana-2413 93 9 zariski zariski NOUN cana-2413 93 10 rings ring NOUN cana-2413 93 11 and and CCONJ cana-2413 93 12 the the DET cana-2413 93 13 𝑋 𝑋 PROPN cana-2413 93 14 βˆ’adic βˆ’adic PROPN cana-2413 93 15 completion completion PROPN cana-2413 93 16 οΏ½ οΏ½ PROPN cana-2413 93 17 Μƒ Μƒ PROPN cana-2413 93 18 οΏ½ οΏ½ PROPN cana-2413 93 19 𝑇,𝑝 𝑇,𝑝 NUM cana-2413 93 20 πœ‡ πœ‡ ADP cana-2413 93 21 βˆ§π‘‹ βˆ§π‘‹ PROPN cana-2413 93 22 = = SYM cana-2413 93 23 lim lim PROPN cana-2413 93 24 𝑔 𝑔 PROPN cana-2413 93 25 οΏ½ οΏ½ PROPN cana-2413 93 26 βƒ– βƒ– PROPN cana-2413 93 27 οΏ½ οΏ½ PROPN cana-2413 93 28 lim lim PROPN cana-2413 93 29 𝑔 𝑔 PROPN cana-2413 93 30 𝑓 𝑓 PRON cana-2413 93 31 οΏ½ οΏ½ PROPN cana-2413 93 32 Μƒ Μƒ PROPN cana-2413 93 33 οΏ½ οΏ½ PROPN cana-2413 93 34 οΏ½ οΏ½ PROPN cana-2413 93 35 Μƒ Μƒ PROPN cana-2413 93 36 οΏ½ οΏ½ PROPN cana-2413 93 37 πœ‡ πœ‡ X cana-2413 93 38 ( ( PUNCT cana-2413 93 39 οΏ½ οΏ½ PROPN cana-2413 93 40 Μƒ Μƒ NOUN cana-2413 93 41 οΏ½ οΏ½ PROPN cana-2413 93 42 ) ) PUNCT cana-2413 93 43 𝑋𝑛 𝑋𝑛 PROPN cana-2413 93 44 lim𝑔 lim𝑔 VERB cana-2413 93 45 𝑓 𝑓 DET cana-2413 93 46 οΏ½ οΏ½ PROPN cana-2413 93 47 Μƒ Μƒ PROPN cana-2413 93 48 οΏ½ οΏ½ PROPN cana-2413 93 49 οΏ½ οΏ½ PROPN cana-2413 93 50 Μƒ Μƒ PROPN cana-2413 93 51 οΏ½ οΏ½ PROPN cana-2413 93 52 πœ‡ πœ‡ X cana-2413 93 53 ( ( PUNCT cana-2413 93 54 οΏ½ οΏ½ PROPN cana-2413 93 55 Μƒ Μƒ NOUN cana-2413 93 56 οΏ½ οΏ½ PROPN cana-2413 93 57 ) ) PUNCT cana-2413 94 1 = = PROPN cana-2413 94 2 lim lim PROPN cana-2413 94 3 𝑔 𝑔 PROPN cana-2413 94 4 οΏ½ οΏ½ PROPN cana-2413 94 5 βƒ– βƒ– PROPN cana-2413 94 6 οΏ½ οΏ½ PROPN cana-2413 94 7 lim𝑔 lim𝑔 VERB cana-2413 94 8 𝑓 𝑓 PRON cana-2413 94 9 οΏ½ οΏ½ PROPN cana-2413 94 10 Μƒ Μƒ PROPN cana-2413 94 11 οΏ½ οΏ½ PROPN cana-2413 94 12 οΏ½ οΏ½ PROPN cana-2413 94 13 Μƒ Μƒ PROPN cana-2413 94 14 οΏ½ οΏ½ PROPN cana-2413 94 15 πœ‡ πœ‡ X cana-2413 94 16 ( ( PUNCT cana-2413 94 17 οΏ½ οΏ½ PROPN cana-2413 94 18 Μƒ Μƒ NOUN cana-2413 94 19 οΏ½ οΏ½ NOUN cana-2413 94 20 ) ) PUNCT cana-2413 94 21 𝑋𝑛 𝑋𝑛 PROPN cana-2413 94 22 οΏ½ οΏ½ PROPN cana-2413 94 23 Μƒ Μƒ PROPN cana-2413 94 24 οΏ½ οΏ½ PROPN cana-2413 94 25 οΏ½ οΏ½ PROPN cana-2413 94 26 Μƒ Μƒ PROPN cana-2413 94 27 οΏ½ οΏ½ PROPN cana-2413 94 28 πœ‡ πœ‡ X cana-2413 94 29 ( ( PUNCT cana-2413 94 30 οΏ½ οΏ½ PROPN cana-2413 94 31 Μƒ Μƒ NOUN cana-2413 94 32 οΏ½ οΏ½ PROPN cana-2413 94 33 ) ) PUNCT cana-2413 94 34 = = PROPN cana-2413 94 35 lim lim PROPN cana-2413 94 36 𝑔 𝑔 PROPN cana-2413 94 37 οΏ½ οΏ½ PROPN cana-2413 94 38 βƒ– βƒ– PROPN cana-2413 94 39 οΏ½ οΏ½ PROPN cana-2413 94 40 lim𝑔 lim𝑔 VERB cana-2413 94 41 𝑓 𝑓 PRON cana-2413 94 42 οΏ½ οΏ½ PROPN cana-2413 94 43 Μ…Μƒ Μ…Μƒ PROPN cana-2413 94 44 οΏ½ οΏ½ PROPN cana-2413 94 45 οΏ½ οΏ½ PROPN cana-2413 94 46 Μƒ Μƒ PROPN cana-2413 94 47 οΏ½ οΏ½ NOUN cana-2413 94 48 (𝑛 (𝑛 NOUN cana-2413 94 49 ) ) PUNCT cana-2413 94 50 πœ‡ πœ‡ X cana-2413 94 51 ( ( PUNCT cana-2413 94 52 οΏ½ οΏ½ PROPN cana-2413 94 53 Μƒ Μƒ NOUN cana-2413 94 54 οΏ½ οΏ½ NOUN cana-2413 94 55 Μ… Μ… NOUN cana-2413 94 56 ) ) PUNCT cana-2413 94 57 = = SYM cana-2413 94 58 οΏ½ οΏ½ PROPN cana-2413 94 59 Μƒ Μƒ PROPN cana-2413 94 60 οΏ½ οΏ½ PROPN cana-2413 94 61 𝑝 𝑝 ADJ cana-2413 94 62 πœ‡ πœ‡ X cana-2413 94 63 ( ( PUNCT cana-2413 94 64 οΏ½ οΏ½ PROPN cana-2413 94 65 Μƒ Μƒ NOUN cana-2413 94 66 οΏ½ οΏ½ PROPN cana-2413 94 67 ) ) PUNCT cana-2413 94 68 , , PUNCT cana-2413 94 69 the the DET cana-2413 94 70 micro micro NOUN cana-2413 94 71 - - NOUN cana-2413 94 72 localization localization NOUN cana-2413 94 73 at at ADP cana-2413 94 74 β„Ž(𝐺(𝑆 β„Ž(𝐺(𝑆 NOUN cana-2413 94 75 ) ) PUNCT cana-2413 94 76 βˆ’ βˆ’ PROPN cana-2413 95 1 𝑃 𝑃 NOUN cana-2413 95 2 ) ) PUNCT cana-2413 95 3 . . PUNCT cana-2413 96 1 as as ADP cana-2413 96 2 above above ADV cana-2413 96 3 , , PUNCT cana-2413 96 4 at at ADP cana-2413 96 5 a a DET cana-2413 96 6 level level NOUN cana-2413 96 7 𝑛 𝑛 NOUN cana-2413 96 8 , , PUNCT cana-2413 96 9 obtaining obtain VERB cana-2413 96 10 𝑄𝑃 𝑄𝑃 ADJ cana-2413 96 11 πœ‡ πœ‡ X cana-2413 96 12 ( ( PUNCT cana-2413 96 13 𝑆 𝑆 PROPN cana-2413 96 14 ) ) PUNCT cana-2413 96 15 from from ADP cana-2413 96 16 οΏ½ οΏ½ PROPN cana-2413 96 17 Μƒ Μƒ PROPN cana-2413 96 18 οΏ½ οΏ½ PROPN cana-2413 96 19 𝑝 𝑝 ADJ cana-2413 96 20 πœ‡ πœ‡ X cana-2413 96 21 ( ( PUNCT cana-2413 96 22 οΏ½ οΏ½ PROPN cana-2413 96 23 Μƒ Μƒ NOUN cana-2413 96 24 οΏ½ οΏ½ PROPN cana-2413 96 25 ) ) PUNCT cana-2413 96 26 and and CCONJ cana-2413 96 27 one one PRON cana-2413 96 28 may may AUX cana-2413 96 29 easily easily ADV cana-2413 96 30 prove prove VERB cana-2413 96 31 that that SCONJ cana-2413 96 32 οΏ½ οΏ½ PROPN cana-2413 96 33 Μƒ Μƒ PROPN cana-2413 96 34 οΏ½ οΏ½ PROPN cana-2413 96 35 𝑝 𝑝 ADJ cana-2413 96 36 πœ‡ πœ‡ X cana-2413 96 37 ( ( PUNCT cana-2413 96 38 οΏ½ οΏ½ PROPN cana-2413 96 39 Μƒ Μƒ NOUN cana-2413 96 40 οΏ½ οΏ½ PROPN cana-2413 96 41 ) ) PUNCT cana-2413 96 42 , , PUNCT cana-2413 96 43 res re NOUN cana-2413 96 44 . . PUNCT cana-2413 97 1 𝑄𝑃 𝑄𝑃 PROPN cana-2413 97 2 πœ‡ πœ‡ PROPN cana-2413 97 3 ( ( PUNCT cana-2413 97 4 𝑆 𝑆 PROPN cana-2413 97 5 ) ) PUNCT cana-2413 97 6 , , PUNCT cana-2413 97 7 are be AUX cana-2413 97 8 zip zip NOUN cana-2413 97 9 rings ring NOUN cana-2413 97 10 . . PUNCT cana-2413 98 1 hence hence ADV cana-2413 98 2 the the DET cana-2413 98 3 following following ADJ cana-2413 98 4 result result NOUN cana-2413 98 5 holds hold VERB cana-2413 98 6 : : PUNCT cana-2413 98 7 proposition proposition NOUN cana-2413 98 8 7 7 NUM cana-2413 98 9 . . PUNCT cana-2413 99 1 i.(𝑇 i.(𝑇 PROPN cana-2413 99 2 , , PUNCT cana-2413 99 3     SPACE cana-2413 99 4 οΏ½ οΏ½ PROPN cana-2413 99 5 Μƒ Μƒ PROPN cana-2413 99 6 οΏ½ οΏ½ NOUN cana-2413 99 7 𝑇 𝑇 PROPN cana-2413 99 8 πœ‡ πœ‡ X cana-2413 99 9 ) ) PUNCT cana-2413 99 10 is be AUX cana-2413 99 11 zip zip NOUN cana-2413 99 12 graded grade VERB cana-2413 99 13 affine affine NOUN cana-2413 99 14 scheme scheme NOUN cana-2413 99 15 . . PUNCT cana-2413 100 1 ii.(𝑇 ii.(𝑇 PROPN cana-2413 100 2 , , PUNCT cana-2413 100 3     SPACE cana-2413 100 4 𝑂𝑇 𝑂𝑇 PROPN cana-2413 100 5 πœ‡ πœ‡ X cana-2413 100 6 ) ) PUNCT cana-2413 100 7 is be AUX cana-2413 100 8 zip zip NOUN cana-2413 100 9 filtered filter VERB cana-2413 100 10 affine affine NOUN cana-2413 100 11 scheme scheme NOUN cana-2413 100 12 . . PUNCT cana-2413 101 1 references reference NOUN cana-2413 101 2 [ [ X cana-2413 101 3 1 1 NUM cana-2413 101 4 ] ] PUNCT cana-2413 101 5 asensio asensio PROPN cana-2413 101 6 m. m. PROPN cana-2413 101 7 j. j. PROPN cana-2413 101 8 , , PUNCT cana-2413 101 9 van van PROPN cana-2413 101 10 den den PROPN cana-2413 101 11 bergh bergh PROPN cana-2413 101 12 m. m. PROPN cana-2413 101 13 and and CCONJ cana-2413 101 14 van van PROPN cana-2413 101 15 oystaeyen oystaeyen PROPN cana-2413 101 16 f. f. PROPN cana-2413 101 17 , , PUNCT cana-2413 101 18 a a DET cana-2413 101 19 new new ADJ cana-2413 101 20 algebraic algebraic ADJ cana-2413 101 21 approach approach NOUN cana-2413 101 22 to to ADP cana-2413 101 23 micro micro NOUN cana-2413 101 24 - - NOUN cana-2413 101 25 localization localization NOUN cana-2413 101 26 of of ADP cana-2413 101 27 filtered filter VERB cana-2413 101 28 rings ring NOUN cana-2413 101 29 , , PUNCT cana-2413 101 30 trans trans PROPN cana-2413 101 31 . . PROPN cana-2413 102 1 amer amer PROPN cana-2413 102 2 . . PUNCT cana-2413 102 3 math math PROPN cana-2413 102 4 . . PUNCT cana-2413 103 1 soc soc PROPN cana-2413 103 2 . . PUNCT cana-2413 104 1 316 316 NUM cana-2413 104 2 ( ( PUNCT cana-2413 104 3 1989 1989 NUM cana-2413 104 4 ) ) PUNCT cana-2413 104 5 , , PUNCT cana-2413 104 6 15 15 NUM cana-2413 104 7 - - SYM cana-2413 104 8 25 25 NUM cana-2413 104 9 . . PUNCT cana-2413 105 1 [ [ X cana-2413 105 2 2 2 NUM cana-2413 105 3 ] ] PUNCT cana-2413 105 4 hortshorne hortshorne PROPN cana-2413 105 5 r. r. PROPN cana-2413 105 6 , , PUNCT cana-2413 105 7 algebraic algebraic ADJ cana-2413 105 8 geometry geometry NOUN cana-2413 105 9 , , PUNCT cana-2413 105 10 g.t.m g.t.m NOUN cana-2413 105 11 . . PROPN cana-2413 105 12 52 52 NUM cana-2413 105 13 , , PUNCT cana-2413 105 14 springer springer NOUN cana-2413 105 15 verlag verlag PROPN cana-2413 105 16 , , PUNCT cana-2413 105 17 new new PROPN cana-2413 105 18 york york PROPN cana-2413 105 19 , , PUNCT cana-2413 105 20 1977 1977 NUM cana-2413 105 21 . . PUNCT cana-2413 106 1 [ [ X cana-2413 106 2 3 3 X cana-2413 106 3 ] ] X cana-2413 106 4 huishi huishi PROPN cana-2413 106 5 l. l. PROPN cana-2413 106 6 and and CCONJ cana-2413 106 7 van van PROPN cana-2413 106 8 oystaeyen oystaeyen PROPN cana-2413 106 9 f. f. PROPN cana-2413 106 10 , , PUNCT cana-2413 106 11 zariskian zariskian ADJ cana-2413 106 12 filterations filteration NOUN cana-2413 106 13 , , PUNCT cana-2413 106 14 comm comm NOUN cana-2413 106 15 . . PUNCT cana-2413 107 1 in in ADP cana-2413 107 2 algebra algebra NOUN cana-2413 107 3 , , PUNCT cana-2413 107 4 17(12 17(12 NUM cana-2413 107 5 ) ) PUNCT cana-2413 107 6 ( ( PUNCT cana-2413 107 7 1989 1989 NUM cana-2413 107 8 ) ) PUNCT cana-2413 107 9 . . PUNCT cana-2413 108 1 [ [ X cana-2413 108 2 4 4 X cana-2413 108 3 ] ] X cana-2413 108 4 leroy leroy PROPN cana-2413 108 5 a. a. PROPN cana-2413 108 6 and and CCONJ cana-2413 108 7 matczuk matczuk PROPN cana-2413 108 8 j. j. PROPN cana-2413 108 9 , , PUNCT cana-2413 108 10 zip zip NOUN cana-2413 108 11 property property NOUN cana-2413 108 12 of of ADP cana-2413 108 13 certain certain ADJ cana-2413 108 14 extensions extension NOUN cana-2413 108 15 , , PUNCT cana-2413 108 16 journal journal NOUN cana-2413 108 17 of of ADP cana-2413 108 18 pure pure ADJ cana-2413 108 19 and and CCONJ cana-2413 108 20 applied applied ADJ cana-2413 108 21 algebra algebra NOUN cana-2413 108 22 , , PUNCT cana-2413 108 23 vol vol NOUN cana-2413 108 24 . . NOUN cana-2413 108 25 220 220 NUM cana-2413 108 26 , , PUNCT cana-2413 108 27 no no INTJ cana-2413 108 28 . . PUNCT cana-2413 109 1 1(2016 1(2016 NUM cana-2413 109 2 ) ) PUNCT cana-2413 110 1 , , PUNCT cana-2413 110 2 p. p. NOUN cana-2413 110 3 335 335 NUM cana-2413 110 4 - - SYM cana-2413 110 5 345 345 NUM cana-2413 110 6 . . PUNCT cana-2413 111 1 [ [ X cana-2413 111 2 5 5 NUM cana-2413 111 3 ] ] X cana-2413 111 4 lunqun lunqun NOUN cana-2413 111 5 o. o. PROPN cana-2413 111 6 , , PUNCT cana-2413 111 7 jinwang jinwang PROPN cana-2413 111 8 l. l. PROPN cana-2413 111 9 and and CCONJ cana-2413 111 10 yuemin yuemin PROPN cana-2413 111 11 x. x. PROPN cana-2413 111 12 , , PUNCT cana-2413 111 13 extension extension NOUN cana-2413 111 14 of of ADP cana-2413 111 15 zip zip NOUN cana-2413 111 16 modules module NOUN cana-2413 111 17 , , PUNCT cana-2413 111 18 j. j. PROPN cana-2413 111 19 of of ADP cana-2413 111 20 advances advance NOUN cana-2413 111 21 in in ADP cana-2413 111 22 mathematics mathematics PROPN cana-2413 111 23 ( ( PUNCT cana-2413 111 24 china china PROPN cana-2413 111 25 ) ) PUNCT cana-2413 111 26 , , PUNCT cana-2413 111 27 43(5 43(5 X cana-2413 111 28 ) ) PUNCT cana-2413 111 29 ( ( PUNCT cana-2413 111 30 2014 2014 NUM cana-2413 111 31 ) ) PUNCT cana-2413 111 32 , , PUNCT cana-2413 111 33 683 683 NUM cana-2413 111 34 - - SYM cana-2413 111 35 694 694 NUM cana-2413 111 36 . . PUNCT cana-2413 112 1 [ [ X cana-2413 112 2 6 6 NUM cana-2413 112 3 ] ] X cana-2413 112 4 nastasescu nastasescu PROPN cana-2413 112 5 c. c. PROPN cana-2413 112 6 and and CCONJ cana-2413 112 7 van van PROPN cana-2413 112 8 oystaen oystaen PROPN cana-2413 112 9 f. f. PROPN cana-2413 112 10 , , PUNCT cana-2413 112 11 graded grade VERB cana-2413 112 12 and and CCONJ cana-2413 112 13 filtered filter VERB cana-2413 112 14 rings ring NOUN cana-2413 112 15 and and CCONJ cana-2413 112 16 modulus modulus NOUN cana-2413 112 17 , , PUNCT cana-2413 112 18 l.n l.n PROPN cana-2413 112 19 . . PROPN cana-2413 112 20 in in ADP cana-2413 112 21 mathematics mathematics PROPN cana-2413 112 22 , , PUNCT cana-2413 112 23 springer springer NOUN cana-2413 112 24 - - PUNCT cana-2413 112 25 verlag verlag PROPN cana-2413 112 26 , , PUNCT cana-2413 112 27 berlin berlin PROPN cana-2413 112 28 , , PUNCT cana-2413 112 29 heidelberg heidelberg PROPN cana-2413 112 30 , , PUNCT cana-2413 112 31 new new PROPN cana-2413 112 32 york york PROPN cana-2413 112 33 , , PUNCT cana-2413 112 34 1977 1977 NUM cana-2413 112 35 . . PUNCT cana-2413 113 1 [ [ X cana-2413 113 2 7 7 X cana-2413 113 3 ] ] X cana-2413 113 4 nastasescu nastasescu NOUN cana-2413 113 5 c. c. PROPN cana-2413 113 6 and and CCONJ cana-2413 113 7 van van PROPN cana-2413 113 8 oystaen oystaen PROPN cana-2413 113 9 f. f. PROPN cana-2413 113 10 , , PUNCT cana-2413 113 11 graded grade VERB cana-2413 113 12 ring ring NOUN cana-2413 113 13 theory theory NOUN cana-2413 113 14 , , PUNCT cana-2413 113 15 m. m. NOUN cana-2413 113 16 library library PROPN cana-2413 113 17 28 28 NUM cana-2413 113 18 , , PUNCT cana-2413 113 19 north north NOUN cana-2413 113 20 holland holland PROPN cana-2413 113 21 , , PUNCT cana-2413 113 22 amesterdam amesterdam NOUN cana-2413 113 23 , , PUNCT cana-2413 113 24 1981 1981 NUM cana-2413 113 25 . . PUNCT cana-2413 114 1 [ [ X cana-2413 114 2 8 8 NUM cana-2413 114 3 ] ] SYM cana-2413 114 4 nawal nawal PROPN cana-2413 114 5 m. m. PROPN cana-2413 114 6 noureldeen noureldeen PROPN cana-2413 114 7 , , PUNCT cana-2413 114 8 radwan radwan PROPN cana-2413 114 9 a. a. PROPN cana-2413 114 10 e. e. PROPN cana-2413 114 11 , , PUNCT cana-2413 114 12 and and CCONJ cana-2413 114 13 ahmed ahmed PROPN cana-2413 114 14 aboubakr aboubakr PROPN cana-2413 114 15 , , PUNCT cana-2413 114 16 on on ADP cana-2413 114 17 micro micro NOUN cana-2413 114 18 - - NOUN cana-2413 114 19 localization localization NOUN cana-2413 114 20 of of ADP cana-2413 114 21 graded grade VERB cana-2413 114 22 and and CCONJ cana-2413 114 23 filtered filter VERB cana-2413 114 24 formal formal ADJ cana-2413 114 25 modules module NOUN cana-2413 114 26 , , PUNCT cana-2413 114 27 accepted accept VERB cana-2413 114 28 and and CCONJ cana-2413 114 29 to to PART cana-2413 114 30 appear appear VERB cana-2413 114 31 in in ADP cana-2413 114 32 applied applied ADJ cana-2413 114 33 mathematics mathematic NOUN cana-2413 114 34 and and CCONJ cana-2413 114 35 information information NOUN cana-2413 114 36 sciences science NOUN cana-2413 114 37 ( ( PUNCT cana-2413 114 38 2024 2024 NUM cana-2413 114 39 ) ) PUNCT cana-2413 114 40 . . PUNCT cana-2413 115 1 [ [ X cana-2413 115 2 9 9 X cana-2413 115 3 ] ] PUNCT cana-2413 115 4 radwan radwan PROPN cana-2413 115 5 a. a. PROPN cana-2413 115 6 e. e. PROPN cana-2413 115 7 and and CCONJ cana-2413 115 8 van van PROPN cana-2413 115 9 oystaeyen oystaeyen PROPN cana-2413 115 10 f. f. PROPN cana-2413 115 11 , , PUNCT cana-2413 115 12 micro micro ADJ cana-2413 115 13 - - NOUN cana-2413 115 14 structure structure ADJ cana-2413 115 15 sheaves sheaf NOUN cana-2413 115 16 , , PUNCT cana-2413 115 17 formal formal ADJ cana-2413 115 18 schemes scheme NOUN cana-2413 115 19 and and CCONJ cana-2413 115 20 quantum quantum NOUN cana-2413 115 21 sections section NOUN cana-2413 115 22 over over ADP cana-2413 115 23 projective projective ADJ cana-2413 115 24 schemes scheme NOUN cana-2413 115 25 , , PUNCT cana-2413 115 26 in in ADP cana-2413 115 27 p p NOUN cana-2413 115 28 of of ADP cana-2413 115 29 contact contact NOUN cana-2413 115 30 , , PUNCT cana-2413 115 31 france france PROPN cana-2413 115 32 - - PUNCT cana-2413 115 33 belgium belgium NOUN cana-2413 115 34 , , PUNCT cana-2413 115 35 1992 1992 NUM cana-2413 115 36 . . PUNCT cana-2413 116 1 [ [ X cana-2413 116 2 10 10 NUM cana-2413 116 3 ] ] PUNCT cana-2413 116 4 radwan radwan PROPN cana-2413 116 5 a. a. PROPN cana-2413 116 6 e. e. PROPN cana-2413 116 7 , , PUNCT cana-2413 116 8 filtered filter VERB cana-2413 116 9 and and CCONJ cana-2413 116 10 graded grade VERB cana-2413 116 11 micro micro ADJ cana-2413 116 12 - - ADJ cana-2413 116 13 affine affine ADJ cana-2413 116 14 schemes scheme NOUN cana-2413 116 15 , , PUNCT cana-2413 116 16 j. j. PROPN cana-2413 116 17 inst inst PROPN cana-2413 116 18 . . PUNCT cana-2413 116 19 math math NOUN cana-2413 116 20 . . PUNCT cana-2413 117 1 and and CCONJ cana-2413 117 2 comp comp PROPN cana-2413 117 3 . . PUNCT cana-2413 118 1 sci sci PROPN cana-2413 118 2 . . PROPN cana-2413 118 3 , , PUNCT cana-2413 118 4 5(1994 5(1994 NUM cana-2413 118 5 ) ) PUNCT cana-2413 118 6 , , PUNCT cana-2413 118 7 73 73 NUM cana-2413 118 8 - - SYM cana-2413 118 9 81 81 NUM cana-2413 118 10 . . PUNCT cana-2413 119 1 [ [ X cana-2413 119 2 11 11 NUM cana-2413 119 3 ] ] X cana-2413 119 4 sharp sharp PROPN cana-2413 119 5 r. r. PROPN cana-2413 119 6 y. y. PROPN cana-2413 119 7 , , PUNCT cana-2413 119 8 graded grade VERB cana-2413 119 9 annihilators annihilator NOUN cana-2413 119 10 and and CCONJ cana-2413 119 11 uniformly uniformly ADV cana-2413 119 12 f f X cana-2413 119 13 - - PUNCT cana-2413 119 14 compatible compatible ADJ cana-2413 119 15 ideals ideal NOUN cana-2413 119 16 , , PUNCT cana-2413 119 17 acta acta PROPN cana-2413 119 18 math math PROPN cana-2413 119 19 . . PUNCT cana-2413 120 1 ( ( PUNCT cana-2413 120 2 vietnamica vietnamica PROPN cana-2413 120 3 ) ) PUNCT cana-2413 120 4 , , PUNCT cana-2413 120 5 40 40 NUM cana-2413 120 6 ( ( PUNCT cana-2413 120 7 2015 2015 NUM cana-2413 120 8 ) ) PUNCT cana-2413 120 9 , , PUNCT cana-2413 120 10 179 179 NUM cana-2413 120 11 - - SYM cana-2413 120 12 195 195 NUM cana-2413 120 13 . . PUNCT