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Trong chương này chúng tôi trình bày lại một số kiến thức cơ bản để làm cơ sở cho Chương 2 như tập nguyên tố liên kết, tập nguyên tố gắn kết, dãy chính quy, dãy lọc chính quy và dãy đối chính quy chiều > E. Trong toàn bộ luận văn này chúng tôi luôn giả thiết rằng là một vành giao hoán và có đơn vị 10.
❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❱➹ ❚❍➚ ❚❍➑ ❙■◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ❚❾P ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ▲■➊◆ ◗❯❆◆ ✣➌◆ ❉❶❨ ✣➮■ ❈❍➑◆❍ ◗❯❨ ❈❍■➋❯ ▲❰◆ ❍❒◆ ❦ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ♥➠♠ ✷✵✶✾ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❱➹ ❚❍➚ ❚❍➑ ❙■◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ❚❾P ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ▲■➊◆ ◗❯❆◆ ✣➌◆ ❉❶❨ ✣➮■ ❈❍➑◆❍ ◗❯❨ ❈❍■➋❯ ▲❰◆ ❍❒◆ ❦ ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ✣↕✐ sè ỵ tt số số ữớ ữợ P ❧ö❝ ▼Ð ✣❺❯ ✷ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✹ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ❚➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉➣② ❝❤➼♥❤ q✉② ✈➔ ❞➣② ❧å❝ ❝❤➼♥❤ q✉② ❉➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ ❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✵ ✶✹ ✶✾ ✷ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ❚❾P ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ▲■➊◆ ◗❯❆◆ ✣➌◆ ❉❶❨ ✣➮■ ❈❍➑◆❍ ◗❯❨ ❈❍■➋❯ ▲❰◆ ❍❒◆ ❦ ✷✼ ✷✳✶ ❑➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t t AttR (0:A (x1 n1 , , xi ni ) R) i=0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷ ❑➳t q✉↔ ê♥ ✤à♥❤ ❝❤♦ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t t n AttR (TorR i (R/I, (0:A J ))) i=0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✹✶ ✶ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ✧ ê♥ ✤à♥❤ ❝õ❛ ♠ët sè t➟♣ ✐❞✤❡❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❧✐➯♥ q✉❛♥ ✤➳♥ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ ❦ ✧✱❧í✐ ✤➛✉ t✐➯♥ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ❣✐→♦ ❚❙✳ P❤↕♠ ỳ trỹ t ữợ ❞➝♥ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ tæ✐ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ◆❣♦➔✐ r❛ tỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ ❚❤➛②✱ ❈ỉ ❑❤♦❛ ❚♦→♥ ❚r÷í♥❣ ✣↕✐ ❍å❝ ◗✉② ◆❤ì♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ✤ó♥❣ t❤í✐ ❤↕♥ q✉② ✤à♥❤✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝↔♠ ì♥ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥✱ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ ❜➯♥ tæ✐✱ ✤ë♥❣ ✈✐➯♥ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳ ◆❣➔② ✻ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✾ ❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ❱➹ ❚❍➚ ❚❍➑ ❙■◆❍ ✷ ▼Ð ✣❺❯ ❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❝❤♦ (R, m) tr ữỡ ợ ✐✤➯❛♥ ❝ü❝ ✤↕✐ m✳ I ✱ J ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ R, M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ A Rổ rt ỵ tt tự ♠ỉ✤✉♥ ❆rt✐♥ ✤÷đ❝ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✤÷❛ ✈➔♦ ♥➠♠ ✶✾✼✸ ữủ ữ ởt ố ợ ỵ tt t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r✳ ▼ët R−♠ỉ✤✉♥ N ✤÷đ❝ ❣å✐ ❧➔ t❤ù ❝➜♣ ♥➳✉ N = ✈➔ ✈ỵ✐ ♠å✐ r ∈ R✱ ♣❤➨♣ ♥❤➙♥ ❜ð✐ r tr➯♥ N ❧➔ t♦➔♥ ❝➜✉ ❤♦➦❝ ❧ô② ❧✐♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② AnnR(N ) = p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❑❤✐ ✤â✱ t❛ ♥â✐ N ❧➔ p−t❤ù ❝➜♣✳ ▼ët ❜✐➵✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ K ❧➔ ♠ët ♣❤➙♥ t➼❝❤ t❤➔♥❤ tê♥❣ ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ K = K1 + + Kn, tr♦♥❣ ✤â Ki ❧➔ pi−t❤ù ❝➜♣✳ ▼å✐ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ K ✤➲✉ ❝â t❤➸ ✤÷❛ ✤÷đ❝ ✈➲ ❞↕♥❣ tè✐ t❤✐➸✉ tù❝ ❧➔ ❝→❝ pi ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈ỵ✐ ♠å✐ i = 1, , n ✈➔ ❦❤æ♥❣ ❝â Ki ♥➔♦ t❤ø❛✳ ❑❤✐ ✤â t➟♣ {p1, , pn} ✤÷đ❝ ❣å✐ ❧➔ t➟♣ tố t ổ K ỵ ❧➔ AttR(K)✳ ❚❤❡♦ ▼❛❝❞♦♥❛❧❞✱ ♠å✐ ♠æ✤✉♥ ❆rt✐♥ ✤➲✉ ❝â ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣✳ ◆➠♠ ✶✾✼✾✱ ▼✳ ❇r♦❞♠❛♥♥ ❬✺❪ ❝❤ù♥❣ ♠✐♥❤ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t AssR (M/J n M ) n ợ ữ ởt t q✉↔ ✤è✐ ♥❣➝✉✱ ❘✳ ❨✳ ❙❤❛r♣ ❬✶✺❪ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t AttR (0:AJ n) ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ❚ø ✤➙②✱ ♠ët ❝➙✉ ọ ữủ t r ợ ộ (x1, , xr ) ❝→❝ ♣❤➛♥ tû ❝õ❛ ✈➔♥❤ R✱ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t AttR(0 :A (xn1 , , xnr )) ❝â ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥ ❦❤ỉ♥❣❄ ❚✉② ♥❤✐➯♥✱ ❑❛t③♠❛♥ ❬✺❪ ①➙② ❞ü♥❣ ♠ët ✈➼ ❞ö ✈➲ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ (R, m) ❝â ❝❤✐➲✉ ❜➡♥❣ ✺ ✈➔ ❤❛✐ ♣❤➛♥ tû x, y ∈ m s❛♦ ❝❤♦ AssR(H(x,y)R (R)) ❧➔ t➟♣ ✈æ ❤↕♥✳ ❉♦ ✤â✱ AssR (R/(xn , y n )R) ❧➔ t➟♣ ✈æ n∈N ❤↕♥✳ ❙✉② r❛ AttR(0 :A (xn, yn)R) ❧➔ t➟♣ ✈æ ❤↕♥✱ tr♦♥❣ ✤â A = E(R/m) ❧➔ n∈N ❜❛♦ ♥ë✐ ①↕ ❝õ❛ R/m ❧➔ ♠ët R−♠æ✤✉♥ ❆rt✐♥✳ ❉♦ ✤â✱ t➟♣ AttR(0 :A (xn, yn)R) ❦❤ỉ♥❣ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ❱➻ ✈➟②✱ ❝❤ó♥❣ t❛ ❝➛♥ t➻♠ ✤✐➲✉ ❦✐➺♥ ❝õ❛ A ✈➔ r ✤➸ AttR (0 :A (xn1 , , xnr )R) ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ n , ,n ∈N ◆❤➔♥ ✈➔ ❍♦➔♥❣ ❬✶✸❪ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ A−✤è✐ ❞➣② ❝❤✐➲✉ > k ♥❤÷ s❛✉✿ ▼ët ❞➣② (x1, , xr ) ❝→❝ ♣❤➛♥ tû ❝õ❛ m ✤÷đ❝ ❣å✐ ❧➔ A−✤è✐ ❞➣② ❝❤✐➲✉ > k ♥➳✉ xi ∈ p ✈ỵ✐ ♠å✐ p ∈ (AttR (0 :A (x1 , , xi−1 )R))>k ✈ỵ✐ ♠å✐ i = 1, , r ❚✐➳♣ t❤❡♦✱ ◆❤➔♥ ✈➔ ❍♦➔♥❣ ❬✶✸❪ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ( AttR (0 :A (xn1 , , xnr )R))≥k n , ,n ∈N ❧➔ t➟♣ ❤ú✉ ❤↕♥✱ tr♦♥❣ ✤â (x1, , xr ) ❧➔ ♠ët A−✤è✐ ❞➣② ❝❤✐➲✉ > k✳ ❉✉♥❣ ✈❛ ◆❤➔♥ ❬✶✷❪ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ dimR(0 :A I) > k t❤➻ ♠å✐ A−✤è✐ ❞➣② ❝❤✐➲✉ > k tr♦♥❣ I ❝â t❤➸ ❦➨♦ ❞➔✐ ✤➳♥ ❝ü❝ ✤↕✐ ✈➔ ♠å✐ A−✤è✐ ❞➣② ❝❤✐➲✉ > k ❝ü❝ ✤↕✐ tr♦♥❣ I ✤➲✉ ❝â ❝ò♥❣ ✤ë ❞➔✐✳ ✣ë ❞➔✐ ❝❤✉♥❣ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ✤ë rë♥❣ ❝❤✐➲✉ > k A tr I ỵ Width>k (I, A)✳ ◆➳✉ dimR(0 :A I) ≤ k t❤➻ ợ số ữỡ r õ t t ✤÷đ❝ ♠ët A−✤è✐ ❞➣② ❝❤✐➲✉ > k tr♦♥❣ I ❝â r r trữớ ủ t q ữợ Width>k (I, A) = +∞✳ ❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ AttR(0 :A J n) t❛ ❝â t❤➸ ❝❤➾ r❛ r➡♥❣ Width>k (I, (0 :A J n )) n ợ ỗ ❝â ✷ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✱ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t✱ ❞➣② ❝❤➼♥❤ q✉② ✈➔ ❞➣② ❧å❝ ❝❤➼♥❤ q✉②✱ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ✈➔ ✤ë rë♥❣ ❝❤✐➲✉ > k✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ♥❣✉②➯♥ tè t ❣➢♥ ❦➳t AttR(0 :A (xn1 , , xni )R) ✈➔ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝❤♦ t➟♣ ♥❣✉②➯♥ tè (x1 , , xr ) 1 r r 1 ❣➢♥ ❦➳t i=0 t i=0 i ✳ n AttR (TorR i (R/I, (0 :A J ))) r r ✹ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤➸ ❧➔♠ ❝ì sð ❝❤♦ ❈❤÷ì♥❣ ✷ ♥❤÷ t➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✱ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t✱ ❞➣② ❝❤➼♥❤ q✉②✱ ❞➣② ❧å❝ ❝❤➼♥❤ q✉② ✈➔ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k✳ ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ❣✐↔ t❤✐➳t r➡♥❣ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✈➔ ❝â ✤ì♥ ✈à = 0✳ ✶✳✶ ❚➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✈➲ t➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥✳ ▼ët ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛ ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ tố t M tỗ t ởt ♣❤➛♥ tû x ∈ M ✱ x = s❛♦ ❝❤♦ AnnR (x) = p✳ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè t M ỵ AssR(M ) Ass(M )✳ ◆❤÷ ✈➟②✱ R AssR (M ) = {p ∈ SpecR | ∃x ∈ M, x = 0, p = AnnR (x)} ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ P❤➛♥ tû a ❝õ❛ R ữủ ởt ữợ ổ Rổ M tỗ t x M, x=0 s ax = ữợ ổ M ữủ ỵ ZDR(M ) ữ ✈➟②✱ ZDR (M ) = {a ∈ R | ∃x, = x ∈ M, ax = 0} ❈❤ó þ ✶✳✶✳✸✳ ❚❛ ❧✉ỉ♥ ❝â AssR(R/p) = {p} , ✈ỵ✐ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❜➜t ❦➻ ❝õ❛ R✳ ỵ ỵ R ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ R−♠æ✤✉♥ ❦❤→❝ ❦❤æ♥❣✳ ❑❤✐ ✤â ✭✐✮ P❤➛♥ tû ❝ü❝ ✤↕✐ tr♦♥❣ ❤å ❝→❝ ✐✤➯❛♥ F = {AnnR(x) | = x ∈ M } ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳ ✣➦❝ ❜✐➺t✱ Ass (M ) = ∅ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M = ữợ M ❧➔ ❤ñ♣ ❝õ❛ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✱ tù❝ ❧➔ p ZDR (M ) = p∈AssR (M ) ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ p = Ann(x) ❧➔ ♠ët ♣❤➛♥ tû ❝ü❝ ✤↕✐ ❝õ❛ F t❤➻ p ❧➔ ♠æt ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ a, b ∈ R ♠➔ ab ∈ p, b ∈/ p t❛ ❝â abx = 0, bx = 0✳ ❉♦ = bx ∈ M ♥➯♥ Ann(bx) ∈ F ✳ ❱➻ Ann(x) ❧➔ ♣❤➛♥ tû ❝ü❝ ✤↕✐ ❝õ❛ F ✈➔ Ann(x) ⊂ Ann(bx) ❞♦ ✤â Ann(bx) = Ann(x)✳ ❍ì♥ ♥ú❛✱ t❛ ❝â abx = ♥➯♥ a ∈ Ann(bx)✳ ❉♦ ✤â a ∈ Ann(x)✳ ❙✉② r❛ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❱➟② p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳ ❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ AssR (M ) = ∅ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M = 0✳ ●✐↔ sû M = õ tỗ t = x M ✈➔ ❞♦ ✤â F = ∅✳ ❱➻ R ❧➔ tr tỗ t ởt tỷ ỹ p ∈ F ✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ p ∈ AssR(M ) ♥➯♥ AssR(M ) = ∅✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ AssR(M ) = t tỗ t = x ∈ M s❛♦ ❝❤♦ AssR(x) = p✱ ✈ỵ✐ p ♥❣✉②➯♥ tè✳ ❉♦ ✤â M = 0✳ ✭✐✐✮ ●✐↔ sû α ∈ ZDR(M )✱ t❛ ❝❤ù♥❣ ♠✐♥❤ α ∈ p✳ ❚❤➟t ✈➟②✱ ✈➻ p∈Ass (M ) α ∈ ZDR (M ) tỗ t x = s x = 0✳ ❙✉② r❛ α ∈ Ann(x)✳ ❱➻ t❤➳ R ✻ tỗ t p Ass(M ) s Ann(x) p✳ ❙✉② r❛ α ∈ ZDR (M ) ⊆ p p∈AssR (M ) ❉♦ ✤â✱ p p∈AssR (M ) ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû α ∈ p t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ởt ữợ pAss (M ) M ✳ ❚❤➟t ✈➟②✱ ✈➻ α ∈ p ♥➯♥ t❤➻ tỗ t p Ass(M ) s p pAss (M ) õ tỗ t x = 0, x ∈ M ✤➸ αx = 0✳ ❙✉② r❛ α ∈ ZDR(M ) ❉♦ ✤â✱ R R p ⊆ ZDR (M ) p∈AssR (M ) ❱➟② ZDR(M ) = p p∈AssR (M ) ❇ê ✤➲ ✶✳✶✳✺✳ ❈❤♦ R ❧➔ ✈➔♥❤ ✈➔ M, N ❧➔ ❝→❝ R−♠æ✤✉♥✳ ❑❤✐ ✤â ✭✐✮ ◆➳✉ N ⊂ M t❤➻ AssR(N ) ⊂ AssR(M )✳ ✭✐✐✮ ❈❤♦ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè tr➯♥ ✈➔♥❤ R✳ ❑❤✐ ✤â p ∈ AssR(M ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tỗ t ởt Rổ N M s N ∼= R/p ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû p ∈ AssR(N ) õ tỗ t = x N ✤➸ p = AnnR (x)✳ ❱➻ x ∈ N ⊂ M ♥➯♥ x ∈ M ✳ ❉♦ ✤â p ∈ AssR (M )✳ ❱➟② AssR (N ) ⊂ AssR (M )✳ ✭✐✐✮ ●✐↔ sû p ∈ AssR(M )✳ ❑❤✐ õ tỗ t = x M s p = AnnR(x)✳ ✣➦t N = Rx ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❚❛ ①➨t →♥❤ ①↕ f ♥❤÷ s❛✉ f : R −→ Rx a −→ ax ❘ã r f Rt ỵ ❝➜✉ ♠æ✤✉♥ t❛ ❝â N ∼= R/Kerf = R/Ann(x) = R/p✳ ◆❣÷đ❝ ❧↕✐✱ ❞♦ AssR(R/p) = {p} = AssR(N ) tỗ t tỷ x = x N ⊆ M, ✤➸ p = AnnR (x)✳ ❱➟② p ∈ AssR (M )✳ ❇ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ỵ r M ♠ët ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r R✱ S ❧➔ t➟♣ ❝♦♥ ♥❤➙♥ ✤â♥❣ ❝õ❛ R✳ ❑❤✐ ✤â AssS −1 R S −1 M = pS −1 R | p ∈ AssR (M ) , p ∩ S = ∅ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ♥➯♥ S −1R ❝ô♥❣ ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✳ ❉♦ ✤â✱ t➟♣ AssS R S −1M ❧➔ ①→❝ ✤à♥❤✳ ▲➜② p ∈ AssR(M ) s❛♦ ❝❤♦ p ∩ S = õ tỗ t m = 0, m ∈ M s❛♦ ❝❤♦ p = (0 :R m)✳ ❚❛ ❝â pS −1R = (0 :S R m/1) ∈ Spec(S −1R)✳ ❉♦ ✤â✱ pS −1R ∈ AssS R S −1M ✳ ❙✉② r❛ pS −1R | p ∈ AssR (M ) , p ∩ S = ∅ ⊆ −1 −1 −1 AssS −1 R S −1 M ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû q ∈ AssS R S −1M ✳ ❑❤✐ ✤â✱ q ∈ Spec(S −1)R ✈➔ ❝â ❞✉② ♥❤➜t p ∈ Spec(R) s❛♦ ❝❤♦ p ∩ S = ∅ ✈ỵ✐ q = pS 1R t tỗ t m M, s ∈ S s❛♦ ❝❤♦ q = (0 :S R m/s) ❱➻ s/1 ❧➔ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ ❝õ❛ S −1 R ♥➯♥ q = (0 :S R m/1) ❱➻ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ♥➯♥ p ❧➔ ♠ët ✐✤➯❛♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ✤÷đ❝ s✐♥❤ ❜ð✐ ❝→❝ ♣❤➛♥ tû p1, p2, , pn ❑❤✐ ✤â✱ pim/1 = 0S M , ✈ỵ✐ ♠å✐ i = 1, 2, n ▼➦t ❦❤→❝✱ ✈ỵ✐ ♠é✐ i = 1, 2, n t tỗ t ♠ët si ∈ S s❛♦ ❝❤♦ sipim = ✣➦t t := s1 sn ∈ S ✳ ❑❤✐ ✤â✱ tpim = 0, ✈ỵ✐ ♠å✐ i = 1, 2, n ❙✉② r❛ ptm = ❞♦ ✤â p ⊆ (0 :R tm) ◆❣÷đ❝ ❧↕✐✱ ❧➜② r ∈ (0 :R tm) ❑❤✐ ✤â rtm = ♥➯♥ (rt/1)(m/1) = 0S M ❙✉② r❛ rt/1 ∈ (0 :S R m/1) = pS −1R ❱➻ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➯♥ rt ∈ p ❍ì♥ ♥ú❛✱ t ∈ S ⊆ R/p ♥➯♥ r ∈ p s✉② r❛ p ⊇ (0 :R tm) ❉♦ ✤â p = (0 :R tm), s✉② r❛ p ∈ AssR(M ) ❱➟② −1 −1 −1 −1 −1 −1 pS −1 R | p ∈ AssR (M ) , p ∩ S = ∅ ⊇ AssS −1 R S −1 M ❍➺ q✉↔ ✶✳✶✳✼✳ ❈❤♦ M ❧➔ ♠ët R ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✳ ❑❤✐ ✤â✱ AssRp (Mp ) = {qRp | q ∈ AssR (M ) , q p} ỵ ỵ t q t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t t AttR (0:A (x1n1 , , xini ) R) i=0 ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tæ✐ ♥➯✉ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè t ❣➢♥ ❦➳t AttR (0:A (x1n , , xin ) R) ✈➔ ♠ët sè ❜ê ✤➲ ❞ò♥❣ ✤➸ ❝❤ù♥❣ i=0 t q tr rữợ t ú t ú þ r➡♥❣ ♥➳✉ A ❧➔ ♠ët R−♠æ✤✉♥ ❆rt✐♥ t❤➻ A ❝â ♠ët ❝➜✉ tró❝ tü ♥❤✐➯♥ ❧➔ R−♠ỉ✤✉♥✳ ❱ỵ✐ ❝➜✉ tró❝ tü ♥❤✐➯♥ ♥➔②✱ ♠ët t➟♣ ❝♦♥ ❝õ❛ A ❧➔ ♠ët R−♠æ✤✉♥ ❝♦♥ ❝õ❛ A ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❧➔ R−♠æ✤✉♥ ❝♦♥ ❝õ❛ A ❉♦ ✤â✱ A ❧➔ ♠ët R−♠ỉ✤✉♥ ❆rt✐♥✳ ❍ì♥ ♥ú❛✱ i AttR A = p ∩ R : p ∈ AttR A ❇ê ✤➲ f : R R ởt ỗ ❝➜✉ ✈➔♥❤ ✈➔ M ❧➔ R −♠ỉ✤✉♥✳ ❳➨t M ♥❤÷ ♠ët R−♠æ✤✉♥ ❝↔♠ s✐♥❤ ❜ð✐ f ❑❤✐ ✤â✱ AssR (M ) = {p ∩ R | p ∈ AssR (M )} ❇ê ✤➲ ✷✳✶✳✷✳ ✭❬✶✼❪✱ ✸✳✹✳✶✹✮ ❈❤♦ M ❧➔ R−♠æ✤✉♥✳ ❑❤✐ ✤â✱ D Exti R (R/I, M ) ∼ = Tori R (R/I, D (M )) Exti R (R/I, D (M )) ∼ = D Tori R (R/I, M ) ú ỵ ỵ ố ts t õ AttR Tor i R R/IR, A = AssR ExtR i R/IR, D (A) ✭✐✐✮ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✷✳✶✳✷ ❝❤♦ tr÷í♥❣ ❤đ♣ i = 0✱ t❛ ✤÷đ❝ D(0 :A I) ∼ = D(HomR (R/I, A)) ∼ = (R/I) ⊗ D(A) ∼ = D(A)/ID(A) ✷✾ ❉♦ ✤â✱ AttR (0 :A I) = AssR (D(A)/ID(A)) ❇ê ✤➲ ✷✳✶✳✹✳ ✭❬✻❪✱ ❇ê ✤➲ ✷✳✺✮ ❈❤♦ k ≥ ❧➔ ♠ët sè ♥❣✉②➯♥✳ ●✐↔ sû (x1, , xr ) ❧➔ ♠ët A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k✳ ❑❤✐ ✤â (x1, , xr ) ❧➔ ♠ët D(A)p−❞➣② ❧å❝ ❝❤➼♥❤ q✉② ②➳✉ ✈ỵ✐ ♠å✐ p ∈ Var(AnnRA) t❤ä❛ ♠➣♥ dim(R/(p ∩ R)) ≥ k✳ Ð ✤➙② xi ❧➔ ↔♥❤ ❝õ❛ xi tr♦♥❣ Rp ✈ỵ✐ i = 1, , r ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû r➡♥❣ (x1, , xr ) ❧➔ A−❞➣② ✤è✐ q ợ > k tỗ t p ∈ Var(AnnRA) t❤ä❛ ♠➣♥ dim(R/(p ∩ R)) ≥ k s❛♦ ❝❤♦ (x1, , xr ) ❦❤æ♥❣ ❧➔ D(A)p q õ tỗ t i {1, , r} s❛♦ ❝❤♦ xi ∈ qRp ✈ỵ✐ qRp ∈ AssR D (A)p / (x1 , , xi−1 ) D (A)p , qRp pRp ✳ ❱➻ p D (A)p / (x1 , , xi−1 ) D (A)p ∼ = (D(A)/(x1 , , xi−1 )D(A))p ∼ = (D(0 :A (x1 , , xi−1 )R))p ❧➔ Rp−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✱ ♥➯♥ q ∈ AssR(D(0 :A r❛ q ∈ AttR(0 :A (x1, , xi−1)R) ❑❤✐ ✤â✱ t❛ ❝â xi ✳ ❙✉② (x1 , , xi−1 )R)) ∈ q ∩ R ∈ AttR (0 :A (x1 , , xi−1 )R) ❱➔ ✈➻ q p ♥➯♥ dim(R/q ∩ R) > dim(R/p ∩ R) ≥ k✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t (x1, , xr ) ❧➔ A−❞➣② ✤è✐ ❝❤➼♥❤ q✉②✳ ❱➟② ❜ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❑➳t q✉↔ ❝❤➼♥❤ ✤➛✉ t✐➯♥ tr ỵ s ỵ ỵ (R, m) ✤à❛ ♣❤÷ì♥❣ ✈➔ A ❧➔ ♠ët R−♠ỉ✤✉♥ ❆rt✐♥✳ ❈❤♦ k ≥ ❧➔ ♠ët sè ♥❣✉②➯♥ ✈➔ (x1, , xr ) ❧➔ ♠ët A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ t ≤ r t❛ ❝â t t n1 ni AttR (0:A (x1 , , xi ) R) i=0 ✈ỵ✐ ♠å✐ n1, , nt ∈ N = ≥k AttR (0:A (x1 , , xi ) R) i=0 ≥k ✸✵ AttR (0:A (x1 n , , xi n )R) ✳ ❚❛ ❧➜② ♠ët sè ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ p ∈ i=0 ≥k ♥❣✉②➯♥ ≤ i0 ≤ t s❛♦ ❝❤♦ p ∈ (AttR (0:A (x1n , , xi n )R))≥k ✈ỵ✐ ♠å✐ i < i0 ❑❤✐ ✤â✱ t i i0 AttR (0 :A (x1 n1 , , xi0 ni0 )R) = p ∩ R | p ∈ AttR (0 :A (x1 n1 , , xi0 ni0 )R) õ tỗ t p AttR(0 :A (x1n , , xi n þ ✷✳✶✳✸ t❛ ❝â i0 ✱ s❛♦ ❝❤♦ p ∩ R = p ❚ø ❈❤ó )R) AttR (0 :A (x1 n1 , , xi0 ni0 )R) = AssR (D(A)/(x1 n1 , , xi0 ni0 )D(A)) ❙✉② r❛ p ∈ AssR(D(A)/(x1n , , xi n i0 ✳ ❉♦ ✤â✱ )D(A)) pRp ∈ AssR (D(A)p /(x1 n1 , , xi0 ni0 )D(A)p ) p ❱➻ (x1, , xr ) ❧➔ ♠ët A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ♥➯♥ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✹ t❛ ❝â (x1, , xi ) ❧➔ D(A)p−❞➣② ❧å❝ ❝❤➼♥❤ q✉②✱ ✈ỵ✐ xi ❧➔ ↔♥❤ ❝õ❛ xi tr♦♥❣ Rp ◆➳✉ p ∈ AttR(0 :A (x1, , xi)R) ✈ỵ✐ ♠å✐ i < i0✱ t❤➻ rã r➔♥❣ t❛ ❝â t p∈ AttR (0:A (x1 , , xi ) R) i=0 k ữủ tỗ t i < i0 s❛♦ ❝❤♦ p ∈/ AttR(0 :A (x1, , xi)R) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② pRp ∈ / AssR (D(A)p /(x1 , , xi )D(A)p ) p ❉♦ ✤â✱ (x1, , xi ) ❧➔ D(A)p−❞➣② ❝❤➼♥❤ q✉②✳ ❱➻ t❤➳✱ AssR (D(A)p /(x1 n1 , , xi0 ni0 )D(A)p ) = AssR (D(A)p /(x1 , , xi0 )D(A)p ) p p ✈➔ pRp ∈ AssR (D(A)p /(x1 , , xi0 )D(A)p ) p ❚ø ✤â✱ s✉② r❛ p ∈ AssR (D(A)/(x1 , , xi0 )D(A)) = AttR (0 :A (x1 , x2 , , xi0 )R) ✸✶ ❚❛ ❝â p = p ∩ R ∈ (AttR (0:A (x1, , xi )R))≥k ❉♦ ✤â✱ t❛ ❝â ✤÷đ❝ ❜❛♦ ❤➔♠ t❤ù❝ s❛✉ t t n1 ni ⊆ AttR (0:A (x1 , , xi ) R) i=0 AttR (0:A (x1 , , xi ) R) i=0 ≥k ≥k ❚÷ì♥❣ tü ♥❤÷ tr➯♥ ❜❛♦ ❤➔♠ t❤ù❝ ♥❣÷đ❝ ❧↕✐ ❝ơ♥❣ ✤ó♥❣ t t n1 ni ⊇ AttR (0:A (x1 , , xi ) R) i=0 AttR (0:A (x1 , , xi ) R) i=0 ≥k ≥k ❉♦ ✤â✱ t t n1 ni AttR (0:A (x1 , , xi ) R) i=0 = ≥k AttR (0:A (x1 , , xi ) R) i=0 ≥k ỵ ữủ ự ứ t q ỵ tr t õ ởt số q s q ỵ k ≥ ❧➔ ♠ët sè ♥❣✉②➯♥ ✈➔ (x1 , , xr ) ❧➔ ♠ët A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k✳ ❑❤✐ ✤â✱ AttR (0:A (x1 n1 , , xr nr ) R) n1 , ,nr k ởt t ỳ ú ỵ ỵ õ t ữủ t ữ s ❈❤♦ k ≥ ❧➔ ♠ët sè ♥❣✉②➯♥ ✈➔ (x1, , xr ) ❧➔ ♠ët A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ t ≤ r t❤➻ t➟♣ t AttR (0:A (x1 n1 , , xi ni )) i=0 ≥k ❧➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ n1, , nt r ú ỵ tr k = 0✱ t❤➻ t❛ ❝â ✤÷ì❝ ❤➺ q✉↔ s❛✉ ✸✷ ❍➺ q✉↔ ✷✳✶✳✽✳ ❈❤♦ (x1, , xr ) ❧➔ ♠ët A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ t ≤ r t❤➻ t➟♣ t AttR (0:A (x1 n1 , , xi ni )) i=0 ❧➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ n1, , nt ✷✳✷ ❑➳t q✉↔ ê♥ ✤à♥❤ ❝❤♦ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t t n AttR (TorR i (R/I, (0:A J ))) i=0 ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ♣❤➛♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❦➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ✐✤➯❛♥ t ♥❣✉②➯♥ tè ❣➢♥ ❦➳t AttR(TorRi (R/I, (0:AJ n)))✳ rữợ t ú t ởt i=0 số trủ s ỵ ✸✳✶✮ ❚➟♣ AttR (0:AJ n) ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ ♥ ❧ỵ♥✳ ❇ê ✤➲ ✷✳✷✳✷✳ ✭❬✼❪✱ ❇ê ✤➲ ✹✳✷✮ ❈❤♦ k ≥ ❧➔ ♠ët sè ♥❣✉②➯♥✳ ❑❤✐ ✤â Width>k (I, (0 :A J n )) ♥❤➟♥ ❣✐→ trà ❧➔ ♠ët ❤➡♥❣ sè ❦❤✐ ♥ ✤õ ❧ỵ♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê tỗ t ởt số n0 s AttR :(0: J ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ ♠å✐ n ≥ n0 ❱➻ t❤➳✱ dim :(0: J ) I ♥❤➟♥ ❣✐→ trà ❧➔ ♠ët ❤➡♥❣ sè d✱ ✈ỵ✐ ♠å✐ n ≥ n0 ❈❤ó♥❣ t❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ❝õ❛ d ✰✮ ❚r÷í♥❣ ❤đ♣ ✶✿ ◆➳✉ d ≤ k t❤➻ Width>k (I, (0 :A J n)) = +∞, ✈ỵ✐ ♠å✐ n ≥ n0 ❑❤✐ ✤â ❦➳t ❧✉➟♥ tr➯♥ ❧➔ ✤ó♥❣✳ ✰✮ ❚r÷í♥❣ ❤đ♣ ✷✿ ●✐↔ sû d > k ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ①➨t r = lim infWidth>k (I, (0:A J n )) ú ỵ r r ❧➔ ♠ët sè ♥❣✉②➯♥✳ ❚❛ ✤✐ ❝❤ù♥❣ n→∞ ♠✐♥❤ r = Width>k (I, (0 :A J n)) ❜➡♥❣ q✉② ♥↕♣ t r tỗ t ởt số ♥❣✉②➯♥ a ≥ n0 s❛♦ ❝❤♦ AttR (0 :A J n) ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ ♠å✐ n ≥ a A A n n) I ✸✸ ◆➳✉ r = t❤➻ t❛ ❝â I ⊆ n≥a I⊆ >k ✈ỵ✐ ♠ët n ≥ a ♥➔♦ ✤â t❤➻ I ⊆ p p∈AttR (0:A J n ) ✈ỵ✐ ✈ỉ ❤↕♥ n✳ ●✐↔ sû I ⊆ p p∈AttR (0:A J n ) >k ✈ỵ✐ ♠å✐ p p∈AttR (0:A J n ) >k ✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ❝â tè✐ ✤❛ ❤ú✉ ❤↕♥ sè n ∈ {1, 2, , a − 1} t❤ä❛ p ✳ ✣✐➲✉ ♥➔② ❧➔ ❦❤æ♥❣ t❤➸✳ ❉♦ ✤â✱ I ⊆ p p∈AttR (0:A J n ) p∈AttR (0:A J n ) >k >k ✈ỵ✐ ♠å✐ n ≥ a ✈➔ ✈➻ t❤➳ Width>k (I, (0 :A = 0, ✈ỵ✐ ♠å✐ n ≥ a ❱➟② ❦➳t q✉↔ tr➯♥ ✤ó♥❣ ✈ỵ✐ r = ◆➳✉ r > 0, ✈➻ Width>k (I, (0 :A J n)) = r ≥ ✈ỵ✐ ✈ỉ ❤↕♥ n ✈➔ AttR (0 :A J n) ❧➔ ê♥ ✤à♥❤ ợ n a tỗ t x1 I ✈➔ x1 ∈/ p ✈ỵ✐ J n )) ♠å✐ n ≥ a✳ ❇➙② ❣✐í t❛ ①➨t r = n→∞ lim infWidth>k r➡♥❣ p∈AttR (0:A J n ) I, (0 :(0:A J n ) x1 R) >k ú ỵ Width>k I, (0 :(0:A J n ) x1 R) = Width>k (I, (0 :A J n )) − ✈➔ r = r − ❇ð✐ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â Width>k I, (0 :(0: J ) x1R) = r ✈ỵ✐ n ✤õ ❧ỵ♥✳ ❉♦ ✤â✱ Width>k (I, (0 :A J n)) = r ✈ỵ✐ n ❧ỵ♥✳ ❱➟② t❛ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ A n ❇ê ✤➲ ✷✳✷✳✸✳ ✭❬✼❪✱ ❇ê ✤➲ ✹✳✸✮ ❈❤♦ k ≥ ❧➔ ♠ët sè ♥❣✉②➯♥ ✈➔ r ❧➔ ❣✐→ trà ê♥ ✤à♥❤ ❝õ❛ Width>k (I, (0 :A J n))✳ ●✐↔ sû ≤ r < õ tỗ t ởt x1, , xr tr♦♥❣ I ❧➔ (0 :A J n)−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k✱ ✈ỵ✐ ♠å✐ n ✤õ ❧ỵ♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ s➩ ❝❤ù♥❣ q t r r tỗ t ởt ❞➣② r ♣❤➛♥ tû tr♦♥❣ I t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜ê ✤➲✳ ❇ð✐ ❇ê ✤➲ ✷✳✷✳✶ ✈➔ ❇ê ✤➲ ✷✳✷✳✷ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ AttR(0 :A J n) ❧➔ ê♥ ✤à♥❤ ✈➔ r = Width>k (I, (0 :A J n ))✱ ✈ỵ✐ ♠å✐ n ≥ a ✈ỵ✐ a ❧➔ ♠ët sè ♥❣✉②➯♥✳ ◆➳✉ r = t tỗ t x1 I x1 / p ✈ỵ✐ ♠å✐ n ≥ a✱ p∈AttR (0:A J n ) ❞♦ ✤â x1 ❧➔ ♠ët ♣❤➛♥ tû t❤✉ë❝ I t❤ä❛ ♠➣♥ ❜ê ✤➲✳ >k ✸✹ ●✐↔ sû r > 1✱ ❜ð✐ ❝❤ù♥❣ ♠✐♥❤ ð tr➯♥✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝❤å♥ ♠ët ♣❤➛♥ tû x1 ∈ I s❛♦ ❝❤♦ x1 ❧➔ (0 :A J n )−♣❤➛♥ tû ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ợ ỡ k ợ n a ữ þ r➡♥❣ Width>k I, (0 :(0:A J n ) x1 R) = Width>k (I, (0 :A J n )) − = r − 1, ✈ỵ✐ ♠å✐ n ≥ a tt q tỗ t x2, , xr tr♦♥❣ I ❧➔ ♠ët I, (0 :(0: J ) x1 R) −❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k ❦❤✐ n ✤õ ❧ỵ♥✳ ❱➻ t❤➳✱ x1 , , xr ❧➔ (0 :A J n )−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ✈ỵ✐ ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k n ợ ữủ ự A n ▼➺♥❤ ✤➲ ✷✳✷✳✹✳ ✭❬✻❪✮ ❈❤♦ r = depth(I, M ) ✈➔ x1, , xr ❧➔ ♠ët M −❞➣② ❝❤➼♥❤ q✉②✳ ❑❤✐ ✤â✱ AssR (ExtrR (R/I, M )) = AssR (HIr (M )) = AssR (M/(x1 , , xr )M ) ∩ Var(I) ❇ê ✤➲ ✷✳✷✳✺✳ ❬✺❪ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â✱ t➟♣ AssR (M/J n M ) ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ❧ỵ♥✳ ❇ê ✤➲ ✷✳✷✳✻✳ ✭❬✶✺❪✱ ❇ê ✤➲ ✹✳✺✮ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ R✳ ✣➦t Mn = M/J nM ◆➳✉ dim (Mn/IMn) ≤ k, t❤➻ ✈ỵ✐ ♠é✐ t ≥ t➟♣ I, J t AssR Exti R (R/I, Mn ) i=0 ≥k ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ❧ỵ♥✳ ❇ê ✤➲ ✷✳✷✳✼✳ ✭❬✼❪✱ ❇ê ✤➲ ✹✳✻✮ ❈❤♦ (x1, , xr ) ❧➔ A−❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k tr♦♥❣ I ❑❤✐ ✤â✱ t ( t AttR (TorR i (R/I, A)))≥k i=0 ✈ỵ✐ ♠å✐ t ≤ r =( i=0 AssR (D(A)/(x1 , , xi )D(A)))≥k ∩ Var(I), ✸✺ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ i0 ≤ t s❛♦ ❝❤♦ t AttR (TorR i (R/I, A)))≥k p ∈ ( ❚❛ ❧➜② ♠ët sè ♥❣✉②➯♥ i=0 p ∈ AttR (TorR i0 (R/I, A)) ✈➔ p ∈/ AttR(TorRi(R/I, A)), ✈ỵ✐ ♠å✐ i < i0 ❉♦ p ∈ AttR(TorRi (R/I, A)) tỗ t p AttR (TorR i0 (R/I, A)) s❛♦ ❝❤♦ p ∩ R = p✳ ❙✉② r❛ AnnR(A) ⊆ p ✈➔ I R ⊆ p✳ ❉♦ dimR/(p∩R) = dimR/p ≥ k, ♥➯♥ →♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶✳✹ t❛ ❝â (x1, , xi ) ❧➔ D(A)p−❞➣② ❧å❝ ❝❤➼♥❤ q✉②✳ ◆➳✉ p ∈/ AssR(D(A)/(x1, , xi)D(A)) ✈ỵ✐ ♠å✐ i < i0 t❤➻ pRp ∈ / AssR (D(A)p /(x1 , , xi )D(A)p ), p ✈ỵ✐ ♠å✐ i < i0✳ ❚ø ✤â s✉② r❛ (x1, , xi ) ❧➔ ♠ët D(A)p−❞➣② ❝❤➼♥❤ q✉② ✈➔ ❝❤ó♥❣ t❛ ❝â ✤ë s➙✉ depth(I Rp, D(A)p) ≥ i0 ▼➦t ❦❤→❝✱ ✈➻ p ∈ AttR(TorRi (R/I, A)) ♥➯♥ t❛ ✤÷đ❝ 0 p ∈ Var(AnnR TorR i0 (R/I, A)) = Var(AnnR (Exti0 (R/I R, D(A)))) R = SuppR (Exti0 (R/I R, D(A))) R ❱➻ t❤➳✱ ExtiR (Rp/I Rp, D(A)p) = 0✳ ❙✉② r❛ ✤ë s➙✉ depth(I Rp, D(A)p) ≤ i0 ❉♦ ✤â i0 = depth(I Rp, D(A)p) ❱➻ p ∈ AttR(TorRi (R/I, A)) ♥➯♥ p ∈ AssR(ExtiR (R/I R, D(A))) ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✹ t❛ ❝â p 0 pRp ∈ AssR (Exti0 (Rp /I Rp , D(A)p )) p Rp = AssR (D(A)p /(x1 , , xi0 )D(A)p ) ∩ Var(I Rp ) p ✸✻ ❱➻ t❤➳✱ p ∈ AssR(D(A)/(x1, , xi )D(A)) ∩ Var(I R)✳ ❚❤❡♦ ợ ỗ tỹ R R✱ t❛ ✤÷đ❝ p ∈ AssR (D(A)/(x1 , , xi0 )D(A)) ∩ Var(I) ❱➟② t❛ ❝â ❜❛♦ ❤➔♠ t❤ù❝ t t AttR (TorR i (R/I, A)))≥k ( AssR (D(A)/(x1 , , xi )D(A)))≥k Var(I) ( i=0 ữủ ợ t ý p ( i=0 t AssR (D(A)/(x1 , , xi )D(A)))k Var(I) i=0 ổ tỗ t ởt sè ♥❣✉②➯♥ i0 ≤ t s❛♦ ❝❤♦ p ∈ AssR (D(A)/(x1 , , xi0 )D(A))≥k ∩ Var(I) ✈➔ p∈ / AssR (D(A)/(x1 , , xi )D(A)), ✈ỵ✐ ♠å✐ i < i0✳ ❱➻ t❤➳✱ pRp ∈/ AssR (D(A)p/(x1, , xi)D(A)p) ✈ỵ✐ ♠å✐ i < i0 ❉♦ ✤â p AssRp (D(A)p /(x1 , , xi )D(A)p ) = (AssRp (D(A)p /(x1 , , xi )D(A)p ))≥1 , ✈ỵ✐ ♠å✐ i < i0✳ ❑❤✐ ✤â t❛ ✤÷đ❝ AssR (D(A)p /(x1 , , xi )D(A)p ) = (AssR (D(A)p /(x1 , , xi )D(A)p ))≥1 p p ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✹✱ x1, , xi ❧➔ ♠ët D(A)p−❞➣② ❧å❝ ❝❤➼♥❤ q✉② ❞♦ ✤â x1, , xi ❧➔ ♠ët D(A)p−❞➣② ❝❤➼♥❤ q✉②✳ ❱➻ p ∈ AssR(D(A)/(x1, , xi )D(A)) ∩ Var(I)✱ ♥➯♥ 0 pRp ∈ AssR (D(A)p /(x1 , , xi0 )D(A)p ) ∩ Var(I Rp ) p ❉♦ ✤â = depth(I Rp , D(A)p /(x1 , , xi0 )D(A)p ) = depth(I Rp , D(A)p ) − i0 ✸✼ ❙✉② r❛ depth(I Rp, D(A)p) = i0✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✳✹✱ t❛ ❝â AssR (D(A)p /(x1 , , xi0 )D(A)p ) ∩ Var(I Rp ) p =AssR (Exti0 (Rp /I Rp , D(A)p )) Rp p ❉♦ ✤â pRp ∈ AssR (ExtiR (Rp/I Rp, D(A)p))✳ ❱➻ t❤➳ p p p ∈ AssR (Exti0 (R/I R, D(A))) R ❚ø ✤â t❛ ❝â p ∈ AttR(TorRi (R/I, A))✳ ❙✉② r❛ p ∈ AttR(TorRi (R/I, A)) ✈➔ t❛ ♥❤➟♥ ✤÷đ❝ ❜❛♦ ❤➔♠ 0 t AssR (D(A)/(x1 , , xi )D(A)))≥k ∩ Var(I) ⊆ (∪ti=0 AttR (TorR i (R/I, A)))≥k ( i=0 ❱➟②✱ t AssR (D(A)/(x1 , , xi )D(A)))≥k ∩ Var(I) = (∪ti=0 AttR (TorR i (R/I, A)))≥k ( i=0 ❇ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❙❛✉ ✤➙② ❧➔ ỵ tự ỵ ỵ k ❧➔ ♠ët sè ♥❣✉②➯♥ ✈➔ r ❧➔ ❣✐→ trà ê♥ ✤à♥❤ ❝õ❛ Width>k (I, (0 :A J n)) ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ t ≤ r✱ t➟♣ t n AttR (TorR i (R/I, (0 :A J ))))≥k ( i=0 ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ú t õ trữớ ủ ữợ rữớ ❤ñ♣ ✶✳ ◆➳✉ r = ∞✱ t❤➻ dimR :(0: J ) I A n ≤k ✈ỵ✐ n ✤õ ❧ỵ♥✳ ❱➻ dimR :(0:A J n ) I = dimR D :(0:A J n ) I = dimR (D(0 :A J n )/ID(0 :A J n )) = dimR ((D(A)/J n D(A)) /I(D(A)/J n D(A))) , ✸✽ ❝❤ó♥❣ t❛ t ữủ dimR ((D(A)/J nD(A)) /I(D(A)/J nD(A))) k ợ n ợ ú ỵ t õ t AttR TorR R/IR, (0:A J n ) i i=0 ≥k t AssR ExtiR R/IR, D (0:A J n ) = i=0 ≥k t AssR ExtiR R/IR, D(A)/J n D (A) = i=0 ≥k ❉♦ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✷✳✻✱ AttR TorR R/IR, (0:A J n ) i i=0 ❦❤✐ n ✤õ ❧ỵ♥✳ ❍ì♥ ♥ú❛✱ ❜ð✐ ❇ê ✤➲ ✷✳✷✳✶ t❛ ❝â t ≥k ❧➔ ê♥ ✤à♥❤ t n AttR TorR i (R/I, (0:A J )) i=0 = ≥k t AttR TorR R/IR, (0:A J n ) i p∩R | p∈ i=0 ≥k , ❧➔ n ợ rữớ ủ ◆➳✉ r = 0, t❤➻ n AttR TorR R/IR, (0:A J ) = AssR Ext0R R/IR, D (0:A J n ) = AssR Ext0R R/IR, D(A)/J n D (A) = AssR (D(A)/J n D(A)) ∩ V (I R) ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✺ t❤➻ t➟♣ AssR (D(A)/J nD(A)) ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥ ♥➯♥ s✉② r❛ AttR TorR0 R/IR, (0:AJ n) ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ❉♦ ✤â✱ t❛ t❤✉ ✤÷đ❝ TorR0 (R/I, (0:AJ n)) ❧➔ n ợ rữớ ủ ◆➳✉ ≤ r < ∞✳ ❇ð✐ ❇ê ✤➲ ✷✳✷✳✷ tỗ t ởt số ữỡ a s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ≥ a t❛ ❝â ❝→❝ ✤✐➲✉ s❛✉ ✤➙② ❧➔ ✤ó♥❣ ✸✾ ✭✐✮ r = Width>k (I, (0 :A J n )) ỗ t↕✐ ♠ët ❞➣② (x1, , xr ) ∈ I ❧➔ ♠ët ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k ❝õ❛ (0 :A J n) ❑❤✐ ✤â✱ D (0 :A J n ) /(x1 , , xi )D (0 :A J n ) =(D(A)/J n D(A))/(x1 , , xi )(D(A)/J n D(A)) =(D(A)/(x1 , , xi )D(A))/J n (D(A)/(x1 , , xi )D(A)) ✈ỵ✐ ♠å✐ i = 1, , r ✈➔ ✈ỵ✐ ♠å✐ n ≥ a ❚❤❡♦ ✤â ✈➔ ❇ê ✤➲ ✷✳✷✳✼ t❛ ✤÷đ❝ ✤➥♥❣ t❤ù❝ s❛✉ ❧➔ ✤ó♥❣ t n AttR TorR i (R/I, (0:A J )) i=0 ≥k t AssR (D (0:A J n ) / (x1 , , xi ) D (0:A J n )) = i=0 ∩ Var (I) ≥k t AssR ((D (A) / (x1 , , xi ) D (A)) /J n (D (A) / (x1 , , xi ) D (A))) = i=0 ∩ Var (I) ≥k ✈ỵ✐ ♠å✐ t ≤ r ✈➔ ✈ỵ✐ ♠å✐ n ≥ a ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✷✳✷✳✺ t❤➻ t➟♣ t n AttR TorR i (R/I, (0:A J )) i=0 ≥k ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ❱➟② ✤à♥❤ ỵ ữủ ú q r ❧➔ ❣✐→ trà ê♥ ✤à♥❤ ❝õ❛ Width>0 (I, (0 :A J n))✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ t ≤ r, t➟♣ t i=0 n AttR TorR i (R/I, (0:A J )) ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ✹✵ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ ♠ët sè t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❧✐➯♥ q✉❛♥ ✤➳♥ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ợ ỡ k t ữủ ởt số t q✉↔ ❝❤➼♥❤ ♥❤÷ s❛✉✿ ✶✳ ❈❤÷ì♥❣ ✶✱ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✱ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t✱ ❞➣② ❝❤➼♥❤ q✉② ✈➔ ❞➣② ❧å❝ ❝❤➼♥❤ q✉②✱ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ✈➔ ✤ë rë♥❣ ❝❤✐➲✉ > k✳ ✷✳ ❚r♦♥❣ ❝❤÷ì♥❣ ✷✱ tr➻♥❤ ❜➔② ✈➲ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ t ❦➳t AttR(0 :A (xn1 , , xni )R) ✈➔ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝❤♦ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t i=0 t i=0 i ✳ n AttR (TorR i (R/I, (0 :A J ))) ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼✳ ❇r♦❞♠❛♥♥✱ ❆s②♠♣t♦t✐❝ st❛❜✐❧✐t② ♦❢ ▼❛t❤✳ ❙♦❝✳✱ ✭✶✮ ✼✹ ✭✶✾✼✾✮✱ ✶✻✲✶✽✳ ✱ Pr♦❝✳ ❆♠❡r✳ AssR (M/I n M ) ❬✷❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥✱ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❝❡rt❛✐♥ ❊①t✲♠♦❞✉♥❧❡s✱❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✭✹✮ ✸✻ ✭✷✵✵✽✮✱ ✶✺✷✼✲✶✺✸✻✳ ❬✸❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣✱ ✧▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥✲ tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✽✽✳ ❬✹❪ ◆✳ ❚✳ ❈✉♦♥❣✱ P✳ ❙❝❤❡♥③❡❧ ❛♥❞ ◆✳ ❱✳ ❚r✉♥❣✱ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ▼♦❞✲✉❧♥✱ ▼❛t❤✳ ◆❛❝❤r✳✱ ✽✺ ✭✶✾✼✽✮✱ ✺✼✲✼✸✳ ❬✺❪ ▼✳ ❑❛t③♠❛♥✱ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ✐♥❢✐♥✐t❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐❡s ♦❢ ❛ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✺✷ ✭✷✵✵✷✮✱ ✶✻✶✲✶✻✻✳ ❬✻❪ P✳ ❍✳ ❑❤❛♥❤✱ ❆♥ ✐♥❞❡♣❡♥❞❡♥t r❡s✉❧t ❢♦r ❛tt❛❝❤❡❞ ♣r✐♠❡s ♦❢ ❝❡rt❛✐♥ ❚♦r✲♠♦❞✉❧❡s✱ ❇✉❧❧✳ ❑♦r❡❛♥ ▼❛t❤✳ ❙♦❝✳ ✺✷ ✭✷✵✶✺✮✱ ✺✸✶✲✺✹✵✳ ❬✼❪ P✳ ❍✳ ❑❤→♥❤✱ t❤❡ st❛❜✐❧✐t② ♦❢ ❝❡rt❛✐♥ s❡ts ♦❢ ❛tt❛❝❤❡❞ ♣r✐♠❡ ✐❞❡❛❧s r❡❧❛t❡❞ t♦ ❝♦s❡q✉❡♥❝❡ ✐♥ ❞✐♠❡♥s✐♦♥ ❃❦✱ ❇✉❧❧✳ ❑♦r❡❛♥ ▼❛t❤✳ ❙♦❝✳ ✺✸ ✭✷✵✶✻✮✱ ✶✸✽✺✲✶✸✾✹✳ ❬✽❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞✱ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠♠✉✲ t❛t✐✈❡ r✐♥❣✱ ❙②♠♣✳ ▼❛t❤✳ ✶✶ ✭✶✾✼✸✮✱ ✷✸✲✹✸✳ ✹✷ ❬✾❪ ❍✳ ▼❛ts✉♠✉r❛ ✭✶✾✽✾✮✱❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✳ ❬✶✵❪ ▲✳ ▼❡❧❦❡rss♦♥ ❛♥❞ P✳ ❙❝❤❡♥③❡❧✱ ❆s②♠♣t♦t✐❝ ♣r✐♠❡ ✐❞❡❛❧s r❡❧❛t❡❞ t♦ ❞❡✲ r✐✈❡❞ ❢✉♥❝t✐♦♥s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✹✮ ✶✶✼ ✭✶✾✾✸✮✱ ✾✸✺✲✾✸✽✳ ❬✶✶❪ ❲✳ ❇r✉♥s ❛♥❞ ❍✳❏✳ ❍❡r③♦❣ ✭✶✾✾✸✮✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❘✐♥❣s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✲✈❡rs✐t② Pr❡ss✳ ❬✶✷❪ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ◆✳ ❚✳ ❉✉♥❣✱ ❆ ❋✐♥✐t❡♥❡ss ❘❡s✉❧t ❢♦r ❆tt❛❝❤❡❞ Pr✐♠❡s ♦❢ ❈❡rt❛✐♥ ❚♦r✲▼♦❞✉❧❡s✱ ❆❧❣❡❜r❛ ❈♦❧❧♦q✉✐✉♠✱ ✶✾ ✭✷✵✶✷✮✱ ✼✽✼✲✼✾✻✳ ❬✶✸❪ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ◆✳ ❱✳ ❍♦❛♥❣✱ ❆ ❋✐♥✐t❡♥❡ss ❘❡s✉❧t ❢♦r ❆tt❛❝❤ Pr✐♠❡s ♦❢ ❆rt✐♥✐❛♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✱ ❏✳ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s ✭✶✮ ✶✸ ✭✷✵✶✹✮✱ ✶✹ ♣❛❣❡s✳ ❬✶✹❪ ▲✳ ❏✳ ❘❛t❧✐❢❢✱ ❖♥ Pr✐♠❡ ❞✐✈✐s♦rs ♦❢ I n✱ n ❧❛r❣❡✱ ▼✐❝❤✐❣❛♥ ▼❛t❤✳ ❏✳✱ ✷✸ ✭✶✾✼✻✮✱ ✸✸✼✲✸✺✷✳ ❬✶✺❪ ❘✳ ❨✳ ❙❤❛r♣✱ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ ❝❡rt❛✐♥ s❡ts ♦❢ ❛tt❝❤❡❞ ♣r✐♠❡ ✐❞❡❛❧s✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✭✷✮ ✸✹ ✭✶✾✽✻✮✱ ✷✶✷✲✷✶✽✳ ❬✶✻❪ ❘✳ ❨✳ ❙❤❛r♣✱ ❙t❡♣s ✐♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ❜② ❘♦❞♥❡②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✵✶✳ ❬✶✼❪ ❏✳ ❙tr♦♦❦❡r✱ ❍♦♠♦❧♦❣✐❝❛❧ q✉❡st✐♦♥ ✐♥ ❧♦❝❛❧ ❛❧❣❡❜r❛✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡r✲ s✐t② Pr❡ss✱ ✶✾✾✵✳ ❬✶✽❪ ❆✳ ❖♦✐s❤✐✱ ▼❛t❧✐s ❞✉❛❧❧✐t② ❛♥❞ t❤❡ ✇✐❞t❤ ♦❢ ❛ ♠♦❞✉❧❡✱ ❍✐r♦s❤✐♠❛ ▼❛t❤✳ ❏ ✻ ✭✶✾✼✻✮✱ ✺✼✸✲✺✽✼✳ ... sü ❝õ❛ M ✳ ●✐↔ sû u ∈ M ❜➜t ❦ý✱ ✈➻ xk u ∈ xk M = x 2k M tỗ t v M s❛♦ ❝❤♦ xk u = x 2k v ✱ s✉② r❛ xk (u − xk v) = 0✳ ❉♦ ✤â u − xk v ∈ M1 ❚❛ ❝â u = xk v + (u − xk v) ∈ M2 + M1 ❙✉② r❛✱ M = M1 + M2✳... ❝õ❛ M ❧➔ ứ õ tỗ t số tỹ k s❛♦ ❝❤♦ xk M = xk+1M = · · · = x 2k M = ✳ ✣➦t M1 = Ker(ϕx ,M ) ✈➔ M2 = xk M ❑❤✐ ✤â✱ M1 ✈➔ M2 ❧➔ ❤❛✐ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❱➻ xk M1 = ✈➔ xk M = ♥➯♥ M1 = M ✱ ✈➔ ✈➻ M... ▼ët ❜✐➵✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ K ❧➔ ♠ët ♣❤➙♥ t➼❝❤ t❤➔♥❤ tê♥❣ ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ K = K1 + + Kn, tr♦♥❣ ✤â Ki ❧➔ pi−t❤ù ❝➜♣✳ ▼å✐ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ K ✤➲✉ ❝â t❤➸ ✤÷❛ ✤÷đ❝ ✈➲ ❞↕♥❣ tè✐