Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)

Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)Chỉ số khả tổng trong phạm trù môđun artin trên vành giao hoán (NCKH)

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❇⑩❖ ❈⑩❖ ❚✃◆● ❑➌❚ ✣➋ ❚⑨■ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❈❻P ✣❸■ ❍➴❈ ❈❍➓ ❙➮ ❑❍❷ ❚✃◆● ❚❘❖◆● P❍❸▼ ❚❘Ò ▼➷✣❯◆ ❆❘❚■◆ ❚❘➊◆ ❱⑨◆❍ ●■❆❖ ❍❖⑩◆ ▼➣ sè✿ ✣❍✷✵✶✺✲❚◆✵✻✲✵✷ ❈❤õ ♥❤✐➺♠ ✤➲ t➔✐✿ ❚❤❙✳ ◆❈❙✳ ❚r➛♥ ✣ù❝ ❉ô♥❣ ❚❤→✐ ◆❣✉②➯♥✱ ✸✴✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❇⑩❖ ❈⑩❖ ❚✃◆● ❑➌❚ ✣➋ ❚⑨■ ❑❍❖❆ ❍➴❈ ❱⑨ ❈➷◆● ◆●❍➏ ❈❻P ✣❸■ ❍➴❈ ❈❍➓ ❙➮ ❑❍❷ ❚✃◆● ❚❘❖◆● P❍❸▼ ❚❘Ò ▼➷✣❯◆ ❆❘❚■◆ ❚❘➊◆ ❱⑨◆❍ ●■❆❖ ❍❖⑩◆ ▼➣ sè✿ ✣❍✷✵✶✺✲❚◆✵✻✲✵✷ ❳→❝ ♥❤➟♥ ❝õ❛ tê ❝❤ù❝ ❝❤õ tr➻ ❈❤õ ♥❤✐➺♠ ✤➲ t ỵ t õ ỵ t ◆❈❙✳ ❚r➛♥ ✣ù❝ ❉ô♥❣ ❚❤→✐ ◆❣✉②➯♥✱ ✸✴✷✵✶✽ ✐ ❉❆◆❍ ❙⑩❈❍ ◆❍Ú◆● ❚❍⑨◆❍ ❱■➊◆ ❚❍❆▼ ●■❆ ◆●❍■➊◆ ❈Ù❯ ✣➋ ❚⑨■ ❱⑨ ✣❒◆ ❱➚ P❍➮■ ❍ÑP ❈❍➑◆❍ ■✳ ❚❤➔♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❚❚ ❍å ✈➔ t➯♥ ✶ ✷ ✸ ✣ì♥ ✈à ❝ỉ♥❣ t→❝ ❱❛✐ trá ❚❤❙✳ ❚r➛♥ ✣ù❝ ❉ơ♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥✱ ❚r÷í♥❣ ✣❍❑❍ ❈❤õ ♥❤✐➺♠ ●❙✳❚❙✳ ▲➯ ❚❤❛♥❤ rữớ ữợ ỗ t P ỗ rữớ ữ ỵ ỡ ố ủ tỹ ❤✐➺♥ ❚➯♥ ✤ì♥ ✈à ◆ë✐ ❞✉♥❣ ♣❤è✐ ❤đ♣ ✣↕✐ ❞✐➺♥ ữ ú ù ữợ ♥❣❤✐➯♥ ❝ù✉ ●❙✳ ❚❙❑❍✳ ◆❣✉②➵♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ✈➔ ❚ü ❈÷í♥❣ ❈ỉ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❍ñ♣ t→❝ ♥❣❤✐➯♥ ❝ù✉✱ ✈✐➳t ❝❤✉♥❣ ❜➔✐ ❜→♦ ❚❙✳ ❚r➛♥ ✣é ▼✐♥❤ ❈❤➙✉ ❚❤→✐ ◆❣✉②➯♥ ▼ư❝ ❧ư❝ ❚❤ỉ♥❣ t✐♥ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✐✐✐ ■♥❢♦r♠❛t✐♦♥ ♦♥ r❡s❡❛r❝❤ r❡s✉❧ts ✈✐✐ ▼ð ✤➛✉ ✶ ❈❤❛♣t❡r ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✻ ✶✳✶ ổ ố ỗ ữỡ t tr ✈➔♥❤ ✻ ✶✳✷ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❑✐➸✉ ✤❛ t❤ù❝ ✈➔ ❧å❝ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❈❤❛♣t❡r ✷✳ ✣♦ t➼♥❤ ❦❤æ♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✷✶ ✷✳✶ ❑✐➸✉ ✤❛ t❤ù❝ ❞➣②✿ ✣à❛ ♣❤÷ì♥❣ ❤â❛ ✈➔ ✤➛② ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ❚➼♥❤ ❝❤➜t ❧➯♥✲①✉è♥❣ ✈➔ ✤➦❝ trữ ỗ ✳ ✳ ✸✶ ❈❤❛♣t❡r ✸✳ ❱➲ ❝❤➾ sè ❦❤↔ q✉② ✈➔ ❝❤➾ sè ❦❤↔ tê♥❣ ❝õ❛ ♠æ✤✉♥ ✹✶ ✸✳✶ ❈❤➾ sè ❦❤↔ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ❈❤➾ sè ❦❤↔ tê♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ✺✸ ✐✐✐ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚❍➷◆● ❚■◆ ❑➌❚ ◗❯❷ ◆●❍■➊◆ ❈Ù❯ ✶✳ ❚❤æ♥❣ t✐♥ ❝❤✉♥❣✿ ✲ ❚➯♥ ✤➲ t➔✐✿ ❈❤➾ sè ❦❤↔ tê♥❣ tr♦♥❣ ♣❤↕♠ trò ♠ỉ✤✉♥ ❆rt✐♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✲ ▼➣ sè✿ ✣❍✷✵✶✺✕❚◆✵✻✕✵✷ ✲ ❈❤õ ♥❤✐➺♠✿ ❚❤❙✳ ◆❈❙✳ ❚r➛♥ ✣ù❝ ❉ô♥❣ ✲ ❚ê ❝❤ù❝ ❝❤õ tr➻✿ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✲ ❚❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥✿ ✵✾✴✷✵✶✺ ✲ ✵✾✴✷✵✶✼✳ ✷✳ ▼ư❝ t✐➯✉✿ ✲ ●✐ỵ✐ t❤✐➺✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤→✐ ♥✐➺♠ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ❝õ❛ ✤♦ t➼♥❤ ❦❤ỉ♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ♥➔♦ ✤â ❝õ❛ M ✈➔ ✈ỵ✐ ♠ư❝ ✤➼❝❤ M ✳ ❈❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② t❤ỉ♥❣ q✉❛ ✤à❛ ♣❤÷ì♥❣ ❤â❛ ✈➔ ✤➛② ✤õ ①✉è♥❣✑ ❝õ❛ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ❣✐ú❛ M M/xM m✲❛❞✐❝✱ ✈ỵ✐ x t➼♥❤ ❝❤➜t ✏❧➯♥ ✕ ❧➔ ♣❤➛♥ tû t❤❛♠ sè M ✳ ◆❣♦➔✐ r❛✱ ✈ỵ✐ ❣✐↔ t❤✐➳t R ❧➔ t❤÷ì♥❣ ❝õ❛ ✈➔♥❤ ●♦r❡♥st❡✐♥ ✤à❛ ♣❤÷ì♥❣✱ ❝❤ó♥❣ tỉ✐ ♠✐➯✉ t↔ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ❝õ❛ ❦❤✉②➳t t❤✐➳✉ ❝õ❛ M t❤æ♥❣ q✉❛ ❝→❝ ♠æ✤✉♥ M✳ ✲ ❈❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ✈➲ sü ❜➜t ❜✐➳♥ ❝õ❛ ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ ✐✤➯❛♥ t❤❛♠ sè ❝õ❛ ♠æ✤✉♥ ❈♦❤❡♥✕▼❛❝❛✉❧❛②✳ ✲ ❳➙② ❞ü♥❣ ✤à♥❤ ♥❣❤➽❛ ❝❤➾ sè ❦❤↔ tê♥❣ ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ✈➲ ❦❤→✐ ♥✐➺♠ ♥➔②✳ ✲ ◆➙♥❣ ❝❛♦ ♥➠♥❣ ❧ü❝ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❝→♥ ❜ë ❣✐↔♥❣ ❞↕② ✣↕✐ số ỵ tt số t P❤ö❝ ✈ö ❤✐➺✉ q✉↔ ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ →♥ ❚✐➳♥ s➽ ❝õ❛ ❝❤õ ♥❤✐➺♠ ✤➲ t➔✐❀ ▼ð rë♥❣ ❤ñ♣ t→❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❣✐ú❛ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ợ ỡ s tr ữợ ✸✳ ❚➼♥❤ ♠ỵ✐✱ t➼♥❤ s→♥❣ t↕♦✿ ✲ ❑✐➸✉ ✤❛ t❤ù❝ ữủ ỹ ữớ ợ t ♥➠♠ ✶✾✾✷ ✤➸ ✤♦ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝õ❛ ♠æ✤✉♥✳ ❚r♦♥❣ ✤â✱ ❦✐➸✉ ✤❛ t❤ù❝ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ t❤ỉ♥❣ q✉❛ ✤à❛ ♣❤÷ì♥❣ ❤â❛✱ ✤➛② ✤õ m✲❛❞✐❝✳ ❚r♦♥❣ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❦❤→✐ ♥✐➺♠ ♠ỵ✐ ❧➔ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ✤➸ ✤♦ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ♠æ✤✉♥✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② t❤ỉ♥❣ q✉❛ ✤à❛ ♣❤÷ì♥❣ ❤â❛ ✈➔ ✤➛② ✤õ ①✉è♥❣✑ ❝õ❛ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ❣✐ú❛ M✳ ♥➔♦ ✤â ❝õ❛ M ✈➔ ▼➦t ❦❤→❝✱ ✈ỵ✐ ❣✐↔ t❤✐➳t M/xM R m✲❛❞✐❝✱ ✈ỵ✐ x ❧➔ ♣❤➛♥ tû t❤❛♠ sè ❧➔ t❤÷ì♥❣ ❝õ❛ ✈➔♥❤ ●♦r❡♥st❡✐♥ ✤à❛ ♣❤÷ì♥❣✱ ❝❤ó♥❣ tỉ✐ ♠✐➯✉ t↔ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ❝õ❛ ♠æ✤✉♥ ❦❤✉②➳t t❤✐➳✉ ❝õ❛ t➼♥❤ ❝❤➜t ✏❧➯♥ ✕ M t❤æ♥❣ q✉❛ ❝→❝ M✳ ✲ ❑➳t q✉↔ ✈➲ sü ❜➜t ❜✐➳♥ ❝õ❛ ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ ✐✤➯❛♥ t❤❛♠ sè ❝õ❛ tr➯♥ ✈➔♥❤ ❈♦❤❡♥✕▼❛❝❛✉❧❛② ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ◆✳●✳ ◆♦rt❤❝♦tt✳ ❚r♦♥❣ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♠ð rë♥❣ ❦➳t q✉↔ ♥➔② ❝❤♦ ♠æ✤✉♥✳ ✲ ✣➣ ❝â ♥❤✐➲✉ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐ ✈➲ ❝❤➾ sè ❦❤↔ q✉②✳ ❚r♦♥❣ ✤➲ t➔✐ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❦❤→✐ ♥✐➺♠ ♠ỵ✐ ❧➔ ❝❤➾ sè ❦❤↔ tê♥❣ ✤è✐ ✈ỵ✐ ♠ỉ✤✉♥ ❆rt✐♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✈➔ ♠ët sè ❝→❝ t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥✳ ✹✳ ❑➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✿ ❈❤♦ (R, m) ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r✱ M ❧➔ ♠ët R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ t q t ữủ t ❳➙② ❞ü♥❣ ✤à♥❤ ♥❣❤➽❛ ❦✐➸✉ ✤❛ t❤ù❝ ❞➣② ❝õ❛ ♠æ✤✉♥ • ◆➳✉ R r❛ ❦❤✐ ❧➔ ❝❛t❡♥❛r② t❤➻ R M ✱ ❦➼ ❤✐➺✉ ❧➔ sp(M )✳ sp(M ) ≥ dim(nSCM(M ))✳ ❉➜✉ ❜➡♥❣ ①↔② ❧➔ ❝❛t❡♥❛r② ♣❤ê ❞ö♥❣ ✈➔ ♠å✐ tợ tự R p ∈ Supp M ✭✐✮ ◆➳✉ ✈➔ R ❧➔ ❝❛t❡♥❛r②✳ dim(R/p) ≥ sp(M ) t❤➻ Mp ❧➔ Rp ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈ ❞➣②✳ ✭✐✐✮ ◆➳✉ dim(R/p) ≤ sp(M ) • sp(M ) ≤ sp(M )✳ ●✐↔ sû ❑❤✐ ✤â sp(Mp ) ≤ sp(M ) − dim(R/p)✳ R/p ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t • t❤➻ p ❝õ❛ ❧➔ ✉♥♠✐①❡❞ ✈ỵ✐ ♠å✐ ✐✤➯❛♥ M✳ sp(M ) > ✈➔ x ∈ m ❧➔ f ✕❞➣② ❝❤➦t ❝õ❛ Di−1 /Di sp(M/xM ) ≤ sp(M ) − 1✳ ❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ♣❤ê ❞ư♥❣ ✈➔ ♠å✐ t❤ỵ ❤➻♥❤ t❤ù❝ ❝õ❛ R ✈ỵ✐ ♠å✐ R i ≤ t✳ ❧➔ ❝❛t❡♥❛r② ❧➔ ❈♦❤❡♥✕▼❛❝❛✉❧❛②✳ • ❚➼♥❤ t♦→♥ • ❈❤ù♥❣ ♠✐♥❤ t➼♥❤ ❜➜t ❜✐➳♥ ❝õ❛ ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ ✐✤➯❛♥ t❤❛♠ sè ❝õ❛ sp(M ) t❤æ♥❣ q✉❛ ♠æ✤✉♥ ❦❤✉②➳t t❤✐➳✉ ❝õ❛ M✳ ♠æ✤✉♥ ữ r số tê♥❣ ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ ✤➳♥ ❦❤→✐ ♥✐➺♠ ♥➔②✳ ✺✳ ❙↔♥ ♣❤➞♠✿ ✺✳✶✳ ❙↔♥ ♣❤➞♠ ❦❤♦❛ ❤å❝✿ ✲ ❳✉➜t ❜↔♥ ✶ ❜➔✐ ❜→♦ t❤✉ë❝ ❞❛♥❤ ♠ư❝ ❙❈■✳ • ▲❡ ❚❤❛♥❤ ◆❤❛♥✱ ❚r❛♥ ❉✉❝ ❉✉♥❣✱ ❚r❛♥ ❉♦ ▼✐♥❤ ❈❤❛✉ ✭✷✵✶✻✮✱ ✧❆ ♠❡❛s✉r❡ ♦❢ ♥♦♥✲s❡q✉❡♥t✐❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛②♥❡ss ♦❢ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠♦❞✉❧❡s✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✹✻✽✱ ♣♣✳ t tr ữợ ❚r➛♥ ✣ù❝ ❉ô♥❣ ✭✷✵✶✺✮✱ ✧❖♥ t❤❡ ✐♥✈❛r✐❛♥t ♦❢ t❤❡ ✐♥❞❡① ♦❢ r❡❞✉❝✐❜✐❧✐t② ❢♦r ♣❛r❛♠❡t❡r ✐❞❡❛❧s ♦❢ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✑✱ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✶✹✼✭✵✷✮✱ ♣♣✳ ✶✾✾✕✷✵✷✳ ✺✳✷✳ ❙↔♥ ♣❤➞♠ ✤➔♦ t↕♦✿ ✲ ❈â ✵✶ ❑▲❚◆ ✣↕✐ ❤å❝ ✤➣ ♥❣❤✐➺♠ t❤✉✳ ✈✐ • ◆❣✉②➵♥ ❇→ ▲♦♥❣ ✭✷✵✶✼✮✱ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥û❛ ♥❤â♠ sè✱ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✻✳ P❤÷ì♥❣ t❤ù❝ ❝❤✉②➸♥ ❣✐❛♦✱ ✤à❛ ❝❤➾ ù♥❣ ❞ư♥❣✱ t→❝ ✤ë♥❣ ✈➔ ❧đ✐ ➼❝❤ ♠❛♥❣ ❧↕✐ ❝õ❛ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉✿ ✲ ❈→❝ ❜➔✐ ❜→♦ ❦❤♦❛ ❤å❝ ❧➔ s↔♥ ♣❤➞♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ✤÷đ❝ t tr t tr ữợ ữủ tợ ❝→❝ ✤ë❝ ❣✐↔ t❤ỉ♥❣ q✉❛ t❤÷ ✈✐➺♥ tr✉②➲♥ t❤è♥❣ ✈➔ t❤÷ ✈✐➺♥ ✤✐➺♥ tû✳ ❈→❝ ❜➔✐ ❜→♦ ✤â ❧➔ t✐➲♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ t✐➳♣ t❤❡♦ ❝❤♦ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ự số ỵ tt số ❈→❝ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐ ❝ô♥❣ ❧➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤ ✈✐➯♥✱ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝✱ ❣✐↔♥❣ ✈✐➯♥✱ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤ ✈➔ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ♥❣❤✐➯♥ ự số ỵ tt số t ự tr ỵ ✈➔ t➯♥✱ ✤â♥❣ ❞➜✉ ✮ ✭ ❈❤õ ♥❤✐➺♠ ✤➲ t➔✐ ỵ t r ự ❉ô♥❣ ✈✐✐ ■◆❋❖❘▼❆❚■❖◆ ❖◆ ❘❊❙❊❆❘❈❍ ❘❊❙❯▲❚❙ ✶✳ ●❡♥❡r❛❧ ✐♥❢♦r♠❛t✐♦♥✿ ✲ Pr♦❥❡❝t t✐t❧❡✿ ❚❤❡ s✉♠ ✕ ✐rr❡❞✉❝✐❜✐❧✐t② ✐♥❞❡① ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦♥ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣ ✲ ❈♦❞❡ ♥✉♠❜❡r✿ ✲ ❈♦♦r❞✐♥❛t♦r✿ ✣❍✷✵✶✺✕❚◆✵✻✕✵✷ ▼❙❝✳ P❤❉✳ ❙t✉❞❡♥t ❚r❛♥ ❉✉❝ ❉✉♥❣ ✲ ■♠♣❧❡♠❡♥t✐♥❣ ✐♥st✐t✉t✐♦♥✿ ✲ ❉✉r❛t✐♦♥✿ ❚◆❯ ✲ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡s ✵✾✴✷✵✶✺ ✲ ✾✴✷✵✶✼✳ ✷✳ ❖❜❥❡❝t✐✈❡✭s✮✿ ✲ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧ t②♣❡ ♦❢ ❞❡♥♦t❡❞ ❜② sp(M )✱ ✐♥ ♦r❞❡r t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r M M✱ ✇❤✐❝❤ ✐s ✐s ❞✐❢❢❡r❡♥t ❢r♦♠ t❤❡ s❡q✉❡♥t✐❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛②♥❡ss✳ ❲❡ st✉❞② t❤❡ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧ t②♣❡ ✉♥❞❡r ❧♦❝❛❧✐③❛t✐♦♥ ❛♥❞ m✲❛❞✐❝ ❝♦♠♣❧❡t✐♦♥✳ ❲❡ ✐♥✈❡st✐❣❛t❡ ❛♥ ❛s❝❡♥t ✕ ❞❡✲ s❝❡♥t ♣r♦♣❡rt② ♦❢ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧ t②♣❡ ❜❡t✇❡❡♥ ❝❡rt❛✐♥ ♣❛r❛♠❡t❡r M ❛♥❞ M/xM ❢♦r x ♦❢ M ✳ ▼♦r❡♦✈❡r✱ ✇❤❡♥ R ✐s ❛ q✉♦t✐❡♥t ♦❢ ❛ ●♦r❡♥st❡✐♥ ❧♦❝❛❧ r✐♥❣✱ ✇❡ ❞❡s❝r✐❜❡ sp(M ) ✐♥ t❡r♠ ♦❢ t❤❡ ❞❡❢✐❝✐❡♥❝② ♠♦❞✉❧❡ ♦❢ M✳ ✲ Pr♦✈❡ r❡s✉❧t ❛❜♦✉t t❤❡ ✐♥✈❛r✐❛♥t ♦❢ t❤❡ ✐♥❞❡① ♦❢ r❡❞✉❝✐❜✐❧✐t② ♦❢ ♣❛r❛♠❡t❡r ✐❞❡❛❧s ♦❢ ❈♦❤❡♥ ✕ ▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✳ ✲ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♥♦t✐♦♥ ♦❢ t❤❡ s✉♠ ✕ ✐rr❡❞✉❝✐❜❧✐t② ✐♥❞❡① ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ✇❡ st✉❞② s♦♠❡ ♣r♦♣♦s✐t✐♦♥ ♦❢ t❤✐s ♥♦t✐♦♥✳ ✲ ❉❡✈❡❧♦♣ ❛❜✐❧✐t② r❡s❡❛r❝❤ ♦❢ ❛❧❣❡❜r❛✐❝ ❛♥❞ ❛r✐t❤♠❡t✐❝ t❡❛❝❤❡rs ♦❢ ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡s❀ ❛tt❡♥❞ ❡❢❢✐❝✐❡♥t❧② ✉♣ t♦ ❝♦♦r❞✐♥❛t♦r✬s t❤❡s✐s❀ ❡①t❡♥❞ s❝✐❡♥t✐❢✐❝ ❝♦♦♣❡r❛t✐♦♥ ❜❡t✇❡❡♥ ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✈❡rs✐t② ❛♥❞ ♦t❤❡rs❀ s❡r✈❡ t♦ ❣r❛❞✉❛t❡ ♣r♦❣r❛♠ ✐♥ tr❛✐♥✐♥❣ ❛♥❞ r❡s❡❛r❝❤ ♦❢ ❚❤❛✐ ◆❣✉②❡♥ ❯♥✐✲ ✈❡rs✐t②✳ ✸✳ ❈r❡❛t✐✈❡♥❡ss ❛♥❞ ✐♥♥♦✈❛t✐✈❡♥❡ss✿ ✈✐✐✐ ✲ P♦❧②♥♦♠✐❛❧ t②♣❡ ✐s ❛ ❝♦♥❝❡♣t ✐♥tr♦❞✉❝❡❞ ❜② Pr♦❢❡ss♦r ◆❣✉②❡♥ ❚✉ ❈✉♦♥❣ ✐♥ ✶✾✾✷ t♦ ♠❡❛s✉r❡ t❤❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②♥❡ss ♦❢ ♠♦❞✉❧❡s✳ ■♥ ✐t✱ t❤❡ ♣♦❧②✲ ♥♦♠✐❛❧ t②♣❡ ✐s st✉❞✐❡❞ t❤r♦✉❣❤ ❧♦❝❛❧✐③❛t✐♦♥✱ m ✲ ❛❞✐❝ ❝♦♠♣❧❡t✐♦♥✳ ■♥ t❤✐s ❛rt✐❝❧❡✱ ✇❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❛ ♥❡✇ ❝♦♥❝❡♣t t❤❛t ✐s s❡q✉❡♥t✐❛❧ ♣♦❧♦②♥♦♠✐❛❧ t②♣❡ t♦ ♠❡❛s✉r❡ t❤❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②♥❡ss ♦❢ ♠♦❞✉❧❡s✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❤❛✈❡ st✉❞✐❡❞ t❤❡ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧ t②♣❡ t❤r♦✉❣❤ ❧♦❝❛❧✐③❛t✐♦♥ ❛♥❞ m ✲ ❛❞✐❝ ❝♦♠♣❧❡t✐♦♥✱ t❤❡ ❛s❝❡♥t✲❞❡s❝❡♥t t❤❡♦r❡♠ ♦❢ t❤❡ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧ t②♣❡ ❜❡t✇❡❡♥ M ❤❛♥❞✱ ✐❢ R ❛♥❞ M/xM ❢♦r x ✐s ❛ ♣❛r❛♠❡t❡r ❡❧❡♠❡♥t ♦❢ M✳ ❖♥ t❤❡ ♦t❤❡r ✐s t❤❡ ❧♦❝❛❧ ●♦r❡♥st❡✐♥ r✐♥❣✱ ✇❡ ❞❡s❝r✐❜❡ t❤❡ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦✲ ♠✐❛❧ t②♣❡ ♦❢ M ✐♥ t❡r♠ ♦❢ t❤❡ ❞❡❢✐❝✐❡♥❝② ♠♦❞✉❧❡ ♦❢ M✳ ✲ ❚❤❡ r❡s✉❧t ♦❢ ✐♥✈❛r✐❛♥t ♦❢ t❤❡ ✐♥❞❡① ♦❢ r❡❞✉❝✐❜✐❧✐t② ♦❢ ♣❛r❛♠❡t❡r ✐❞❡❛❧s ♦❢ ❈♦❤❡♥✕▼❛❝❛✉❧❛② r✐♥❣ ❤❛s ❜❡❡♥ ♣r♦✈❡❞ ❜② ◆✳●✳ ◆♦rt❤❝♦tt✳ ■♥ t❤✐s t♦♣✐❝✱ ✇❡ ❡①t❡♥❞ t❤✐s r❡s✉❧t t♦ t❤❡ ♠♦❞✉❧❡✳ ✲ ❚❤❡r❡ ❤❛✈❡ ❜❡❡♥ ♠❛♥② st✉❞✐❡s ♦❢ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛r♦✉♥❞ t❤❡ ✇♦r❧❞ ♦♥ t❤❡ ✐♥❞❡① ♦❢ r❡❞✉❝✐❜✐❧✐t②✳ ■♥ t❤✐s ❛rt✐❝❧❡✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ❝♦♥❝❡♣t t❤❛t ✐s t❤❡ s✉♠ ✕ ✐rr❡❞✉❝✐❜❧✐t② ✐♥❞❡① ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦♥ t❤❡ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣ ❛♥❞ s♦♠❡ r❡❧❛t❡❞ ♣r♦♣❡rt✐❡s✳ ✹✳ ❘❡s❡❛r❝❤ r❡s✉❧ts✿ ▲❡t (R, m) ❜❡ ❛ ◆♦❡t❤❡r✐❛♥ ❧♦❝❛❧ r✐♥❣✱ M ❜❡ ❛ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ R✲♠♦❞✉❧❡✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ ♣r♦❥❡❝t ❛r❡✿ • ■♥tr♦❞✉❝❡❞ t❤❡ ♥♦t✐♦♥ t❤❡ s❡q✉❡♥t✐❛❧ ♣♦❧②♥♦♠✐❛❧ t②♣❡ ♦❢ ❜② • ■❢ M✱ ❞❡♥♦t❡❞ sp(M )✳ R ✐s ❝❛t❡♥❛r② t❤❡♥ tr✉❡ ✇❤❡♥ R sp(M ) ≥ dim(nSCM(M ))✳ ❚❤❡ ❡q✉❛❧✐t② ❤♦❧❞s ✐s ✉♥✐✈❡rs❛❧❧② ❝❛t❡♥❛r② ❛♥❞ ❛❧❧ ❢♦r♠❛❧ ❢✐❜❡rs ♦❢ R ❛r❡ ❈♦❤❡♥✕▼❛❝❛✉❧❛②✳ • ▲❡t p ∈ Supp M ✭✐✮ ■❢ ❛♥❞ R ✐s ❝❛t❡♥❛r②✳ dim(R/p) ≥ sp(M ) t❤❡♥ Mp ✐s s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❈❤÷ì♥❣ ✸ ❱➲ ❝❤➾ sè ❦❤↔ q✉② ✈➔ ❝❤➾ sè ❦❤↔ tê♥❣ ❝õ❛ ♠æ✤✉♥ r sốt ữỡ t ổ tt ợ ❝ü❝ ✤↕✐ ❞✉② ♥❤➜t ❑r✉❧❧ dim M = d✱ A ❧➔ m ✈➔ M R✲♠æ✤✉♥ ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✤õ m✲❛❞✐❝ ❝õ❛ R ✈➔ R (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ❧➔ ♠ët R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ❝❤✐➲✉ ❆rt✐♥✳ ❱ỵ✐ ♠é✐ ✐✤➯❛♥ ❝❤ù❛ I✳ ❑➼ ❤✐➺✉ R ✈➔ I✱ M ❦➼ ❤✐➺✉ Var(I) ❧➛♥ ❧÷đt ❧➔ ✤➛② M✳ ✸✳✶ ❈❤➾ sè ❦❤↔ q✉② ▼ët tr♦♥❣ ♥❤ú♥❣ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝õ❛ ✤↕✐ sè ❣✐❛♦ ❤♦→♥ ỵ t t q ữủ ự ♠✐♥❤ ❜ð✐ ❊✳ ◆♦❡t❤❡r ♥➠♠ ✶✾✷✶✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❜➔ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❜➜t ❦➻ ✐✤➯❛♥ I ❝õ❛ ✈➔♥❤ ◆♦❡t❤❡r R ✤➲✉ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ t❤➔♥❤ ❣✐❛♦ t❤✉ ❣å♥ ❝õ❛ ❝→❝ ✐✤➯❛♥ ❜➜t ❦❤↔ q✉② ✈➔ sè ✐✤➯❛♥ ❜➜t ❦❤↔ q✉② ①✉➜t ❤✐➺♥ tr♦♥❣ ♣❤➙♥ t➼❝❤ ❧➔ ❦❤æ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❜✐➸✉ ❞✐➵♥✳ ❇➜t ❜✐➳♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ ✐✤➯❛♥ I ✳ ◆➠♠ ✶✾✺✼✱ ❉✳ ●✳ ◆♦rt❤❝♦tt ❝❤➾ r❛ r➡♥❣ ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ ✐✤➯❛♥ t❤❛♠ sè tr♦♥❣ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✤à❛ ♣❤÷ì♥❣ ♣❤ư t❤✉ë❝ ❞✉② ♥❤➜t ✈➔♦ ✈➔♥❤ ✈➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❝→❝❤ ❝❤å♥ ✐✤➯❛♥ t❤❛♠ sè✳ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ s➩ ♠ð rë♥❣ ❦➳t q✉↔ tr➯♥ ❝õ❛ ❉✳ ●✳ ◆♦rt❤❝♦tt ❝❤♦ ♠æ✤✉♥✳ ✹✶ ✹✷ ◆❤➢❝ ❧↕✐ ♠æ✤✉♥ ❝♦♥ ♥➳✉ N ❝❤ù❛ N ❝õ❛ M ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ q✉② ❝õ❛ M ❣å✐ ❧➔ ❦❤æ♥❣ t❤➸ ❜✐➸✉ ❞✐➵♥ t❤➔♥❤ ❣✐❛♦ ❝õ❛ ❤❛✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ N✳ M ❙è ♠æ✤✉♥ ❜➜t ❦❤↔ q✉② ①✉➜t ❤✐➺♥ tr♦♥❣ ♠ët ♣❤➙♥ t➼❝❤ t❤✉ ❣å♥ ❝õ❛ ♠ỉ✤✉♥ ❝♦♥ N ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ N ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ irM (N )✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✶✳ ❈❤♦ q ❧➔ ✐✤➯❛♥ t❤❛♠ sè ❝õ❛ M✳ ❈❤➾ sè ❦❤↔ q✉② ❝õ❛ q ❧➔ sè t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ q✉② ①✉➜t ❤✐➺♥ tr♦♥❣ ♣❤➙♥ t➼❝❤ ❜➜t ❦❤↔ q✉② t❤✉ ❣å♥ ❝õ❛ qM ✱ ú ỵ M irM (qM )✳ ❚❛ ❦➼ ❤✐➺✉ Soc(M ) ❧➔ tê♥❣ t➜t ❝↔ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ ✤ì♥ ❝õ❛ ✤➳ ❝õ❛ M ✳ ❱ỵ✐ ✐✤➯❛♥ t❤❛♠ sè q ❝õ❛ ♠ỉ✤✉♥ M t❛ ❧✉ỉ♥ ❝â irM (qM ) = R ([qM :M m]/qM ) = dimR/m Soc(M/qM ) ❈❤♦ M, N ❧➔ ❝→❝ R✲♠æ✤✉♥ ✈➔ ❣✐↔ sû (x1, x2, , xn) ❧➔ M ✲ ❞➣②✳ ◆➳✉ xnN = t❤➻ HomR(N, M/(x1, , xn−1)M ) = 0✳ ❇ê ✤➲ ✸✳✶✳✶✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû h ∈ HomR (N, M/(x1 , , xn−1 )M ) ✈➔ a ❧➔ ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ N ✳ ❚❛ t❤➜② xn h(a) = h(xn a) = h(0) = ✈➻ xn ∈ Ann(N)✳ ▼➦t ❦❤→❝✱ t❤❡♦ ❣✐↔ t❤✐➳t✱ ❉♦ ✈➟②✱ h(a) = (x1 , , xn ) ❧➔ M ✲❞➣② t❤➻ xn ∈ / ZdvR (M/(x1 , , xn−1 )M )✳ ✈➔ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✸✳✶✳✷✳ ❈❤♦ M, N ❧➔ ❝→❝ R✲♠æ✤✉♥ ✈➔ ❣✐↔ sû (x1, x2, , xn) ❧➔ M ✲❞➣② s❛♦ ❝❤♦ xiN = 0✱ i = 1, 2, , n✳ ❑❤✐ ✤â ExtnR (N, M ) ∼ = HomR (N, M/(x1 , , xn )M ) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ n✳ ❚r÷í♥❣ ❤đ♣ n = ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❱ỵ✐ n ≥ 1✱ ①➨t ❞➣② ❦❤ỵ♣ ♥❣➢♥ x → M −→ M → M/x1 M → 0, tr♦♥❣ ✤â →♥❤ ①↕ tø M ✤➳♥ M ❧➔ →♥❤ ①↕ ♥❤➙♥ ❝❤♦ ❜ð✐ x1 t t ữủ ợ s s γ x n−1 n−1 ExtR (N, M ) → ExtR (N, M/x1 M ) → ExtnR (N, M ) → ExtnR (N, M ) ✹✸ ❉➵ t❤➜②✱ →♥❤ ①↕ ✤â γ x ExtnR (N, M ) → ExtnR (N, M ) ❧➔ →♥❤ ①↕ ✈➻ x1 N = 0✱ ❞♦ γ ❧➔ ❧➔ t♦➔♥ ❝➜✉✳ ❚❤❡♦ ❣✐↔ t❤✐➳t✱ ∼ Extn−1 R (N, M ) = HomR (N, M/(x1 , , xn−1 )M ) HomR (N, M/(x1 , , xn−1 )M ) = ♥❤÷♥❣ ✤ì♥ ❝➜✉✱ ❞♦ ✤â γ ❧➔ ✤➥♥❣ ❝➜✉✳ ✣➦t t❤❡♦ ❇ê ✤➲ ✸✳✶✳✶✳ ❙✉② r❛ M = M/x1 M ✱ t❛ ❝â n ∼ Extn−1 R (N, M ) = ExtR (N, M ) ▲↕✐ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â ∼ ∼ Extn−1 R (N, M ) = HomR (N, M /(x2 , , xn )M ) = HomR (N, M/(x1 , , xn )M ) ❉♦ ✈➟②✱ ExtnR (N, M ) ∼ = HomR (N, M/(x1 , , xn )M ) ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✷✳ ❈❤♦ M =0 ❧➔ R✲♠ỉ✤✉♥ ✈ỵ✐ depth M ❂ t✳ ❑❤✐ ✤â r (M ) = dimk ExttR (k , M) ❣å✐ ❧➔ ❦✐➸✉ ❝õ❛ M ✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿ ✭✐✮ R ❧➔ ✈➔♥❤ ●♦r❡♥st❡✐♥✳ ✭✐✐✮ R ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝â ỵ ❈❤♦ x = (x1, x2, , xt) ❧➔ M ✲❞➣② ❝ü❝ ✤↕✐✳ ❑❤✐ ✤â r(M ) = dimk Soc(M/xM ) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❜✐➳t Soc(N ) = (0 :N m) ∼ = Hom(R/m, N ) = Hom(k, N ) ✈ỵ✐ Rổ N ú ỵ r t = depth M ❝ü❝ ✤↕✐✳ ❉♦ xi ∈ m t❤➻ ❧➔ ✤ë ❞➔✐ ❝❤✉♥❣ ❝õ❛ xi ∈ Ann(R/m) = Ann(k) ✈ỵ✐ ♠å✐ M ✲❞➣② i = 1, 2, , t✳ ❉♦ ✤â✱ r(M ) = dimk ExttR (k, M ) = dimk HomR (k, M/(x1 , , xt )M ) ✹✹ t❤❡♦ ❇ê ✤➲ ✸✳✶✳✷✳ ❑❤✐ ✤â r(M ) = dimk HomR (k, M/(x)M ) = dimk Soc(M/(x)M ) ❈❤♦ M ❧➔ ♠æ✤✉♥ ❈♦❤❡♥ ▼❛❝❛✉❧❛②✳ ❑❤✐ ✤â ❝❤➾ sè ❦❤↔ q✉② ❝õ❛ ✐✤➯❛♥ t❤❛♠ sè q ❝õ❛ M ✤ë❝ ❧➟♣ ✈ỵ✐ ❝→❝❤ ❝❤å♥ q ✈➔ ①→❝ ✤à♥❤ ❜ð✐ irM (qM ) = r(M ) ỵ ự (R, m) ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✤à❛ ♣❤÷ì♥❣ t❤➻ ♠å✐ ❤➺ t❤❛♠ sè ❝õ❛ t❤➻ dim M M ❂ ✤➲✉ ❧➔ M ✲❞➣②✳ ▼➦t ❦❤→❝✱ ❞♦ M depth M ❧➔ ♠æ✤✉♥ ❈♦❤❡♥ ✲▼❛❝❛✉❧❛② q = (x1 , x2 , , xd )✳ ❂ ❞✳ ❚❛ ❣✐↔ sû ❉♦ ✈➟②✱ irM (qM ) = dimk Soc(M/(x1 , , xd )M ) = dimk ExtdR (k, M ) = r(M ) ❚❛ s✉② r❛ ✤✐➲✉ ự ỵ ỵ t t ữủ t q s M ❧➔ ♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ (R, m)✳ ❚❤❡♥ irM (qM ) = ❢♦r ❛❧❧ ♣❛r❛♠❡t❡r ✐❞❡❛❧ q ♦❢ M ✐❢ ❛♥❞ ♦♥❧② ✐❢ R ✐s ●♦r❡♥st❡✐♥✳ ❍➺ q số tờ ỵ tt tự ổ ữủ ợ t ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❬✶✼❪ ❝â t❤➸ ①❡♠ ❧➔ ✤è✐ ♥❣➝✉ ỵ tt t sỡ r ♠å✐ t✐➸✉ R✲♠æ✤✉♥ A = A1 + + An , A ✤➲✉ ❝â ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ ❧➔ pi ✲t❤ù ❆rt✐♥ tr♦♥❣ ✤â Ai ✤ë❝ ❧➟♣ ✈ỵ✐ ✈✐➺❝ ❝❤å♥ ❜✐➸✉ ❞✐➵♥ tè✐ t✐➸✉ ❝õ❛ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ A ✈➔ ❦➼ ❤✐➺✉ ❜ð✐ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✶✳ ❬✶✼❪ ▼ỉ✤✉♥ ❝♦♥ B B ❝➜♣✳ ❚➟♣ ❤đ♣ {p1 , , pn } A✳ ❚➟♣ ✤â ❣å✐ ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ AttR A✳ ❝õ❛ A ✤÷đ❝ ❣å✐ ❧➔ ❜➜t ❦❤↔ tê♥❣ ♥➳✉ ❦❤ỉ♥❣ t❤➸ ❜✐➸✉ ữợ tờ ổ tỹ sỹ tù❝ ✹✺ ❧➔ ♥➳✉ B = C + D, ✈ỵ✐ C, D ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ B t❤➻ B =C ❤♦➦❝ B = D✳ ❇ê ✤➲ ✸✳✷✳✶✳ t✐➸✉ ❬✶✼❪ ▼å✐ ♠æ✤✉♥ ❆rt✐♥ A = A1 + + An ✱ A tr♦♥❣ ✤â ♠é✐ ✤➲✉ ❝â ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ tê♥❣ tè✐ Ai ❧➔ ❜➜t ❦❤↔ tê♥❣ ✈➔ ❦❤æ♥❣ t❤ø❛✳ ◆❣♦➔✐ r❛✱ sè t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ tê♥❣ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ ✤ë❝ ❧➟♣ ✈ỵ✐ ❝→❝❤ ❝❤å♥ ❜✐➸✉ ❞✐➵♥ tè✐ t✐➸✉ ❝õ❛ A✳ ❚ø ❜ê ✤➲ tr➯♥✱ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ ✈➲ ❝❤➾ sè ❦❤↔ tê♥❣ ❝❤♦ ♠ët ♠æ✤✉♥ ❆t✐♥✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✷✳ ❙è t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ tê♥❣ tr♦♥❣ ♠ët ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ tê♥❣ tè✐ t✐➸✉ ❝õ❛ A ✤÷đ❝ ❣å✐ ❧➔ ❝❤➾ sè ❦❤↔ tê♥❣ ❝õ❛ A ✈➔ ❦➼ ❦✐➺✉ ❧➔ irR (A)✳ A ❈❤ó þ r➡♥❣ ♠å✐ ♠æ✤✉♥ ❝♦♥ ❜➜t ❦❤↔ tê♥❣ ❝õ❛ ✤â✱ ✈ỵ✐ ♠é✐ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ tê♥❣ tè✐ t✐➸✉ ❝õ❛ A✱ ✤➲✉ ❧➔ t❤ù ❝➜♣✳ ❉♦ ❜➡♥❣ ❝→❝❤ ❜ä ✤✐ ♥❤ú♥❣ t❤➔♥❤ ♣❤➛♥ ❜➜t ❦❤↔ tê♥❣ ❝â ❝ò♥❣ ❝➠♥ ❧✐♥❤ ❤â❛ tû✱ t❛ t❤✉ ✤÷đ❝ ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛ A r➡♥❣ ❉♦ ✤â✱ t❛ ❝â ❝â ❝➜✉ tró❝ tü ♥❤✐➯♥ ♥❤÷ ❝♦♥ ❝õ❛ ♥❤÷ A✳ A ①➨t ♥❤÷ R✲♠ỉ✤✉♥✳ R✲♠ỉ✤✉♥ ❉♦ ✤â A ❧➔ ❦❤↔ tê♥❣ ❝õ❛ ♠ỉ✤✉♥ ❆rt✐♥ R✲♠ỉ✤✉♥✳ ❚❛ ❜✐➳t ❱ỵ✐ ❝➜✉ tró❝ ♥➔②✱ ♠ët ♠æ✤✉♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♥â ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ R✲♠æ✤✉♥ A irR (A) ≥ Card(AttR A) A ①➨t ❆rt✐♥✳ ❇ê ✤➲ s❛✉ ❝❤➾ r❛ r➡♥❣ ❝❤➾ sè ❜↔♦ t♦➔♥ q✉❛ ✤➛② ✤õ m✲❛❞✐❝✳ ❇ê ✤➲ ✸✳✷✳✷✳ irR (A) = irR (A)✳ ❈❤ù♥❣ ♠✐♥❤✳ ❍✐➸♥ ♥❤✐➯♥ ❞♦ ❞➔♥ ♠æ✤✉♥ ❝♦♥ ❝õ❛ A ✈ỵ✐ ♠é✐ t❤➔♥❤ ♣❤➛♥ ❧➔ R✲♠ỉ✤✉♥ ❜➜t ❦❤↔ tê♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❧➔ R✲ ♠æ✤✉♥ ❜➜t tờ A ữ ỵ r số ❦❤↔ q✉② ❝õ❛ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❧➔ ❦❤æ♥❣ ❜↔♦ t♦➔♥ q✉❛ ✤➛② ✤õ ❝❤✐➲✉ m✲❛❞✐❝✳ ❈❤➥♥❣ ❤↕♥✱ ❝❤♦ (R, m) ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ◆♦❡t❤❡r ①➙② ❞ü♥❣ ❜ð✐ ❋❡rr❛♥❞ s tỗ t tè ❧✐➯♥ ❦➳t ❝õ❛ t↕✐ R✱ P ∈ Ass(R) ✈ỵ✐ dim R/P = ❉♦ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➯♥ ♥â ❧➔ ❜➜t ❦❤↔ q✉②✳ ❉♦ ✤â Q ∈ Ass(R) s❛♦ ❝❤♦ irR (0) = dim(R/Q) = dim R = tỗ irR (0) ≥ Card(Ass(R)) ≥ > = irR (0) ❚❛ ❣✐↔ sû E = E(R/m) HomR (−, E) ❞÷ ❝õ❛ R✳ R✲♠æ✤✉♥ ❧➔ ❜❛♦ ♥ë✐ ①↕ ❝õ❛ ❧➔ ❤➔♠ tỷ ố ts ú ỵ r M ❧➔ R✲♠æ✤✉♥ R/m✳ ❑➼ ❤✐➺✉ k = R/m DR (−) = ❧➔ tr÷í♥❣ t❤➦♥❣ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ DR (M ) ❧➔ ❆rt✐♥✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ A ❧➔ R✲♠æ✤✉♥ ❆rt✐♥✳ ✣➦t k = R/m✳ ❑❤✐ ✤â ✭✐✮ dimk Soc(M ) = dimk DR(M )/mDR(M ) ✭✐✐✮ ◆➳✉ B ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ A s❛♦ ❝❤♦ R(A/B) < ∞ ✈➔ B + mA = A t❤➻ B = A ✭✐✐✐✮ ◆➳✉ A ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ t❤➻ irR(A) = dimk (A/mA) ❇ê ✤➲ ✸✳✷✳✸✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ ✭✐✮✳ ❘ã r➔♥❣ DR (Soc(M )) ∼ = HomR (HomR (R/m, M ), E) R/m ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ E ❧➔ R✲♠ỉ✤✉♥ ♥ë✐ ①↕✱ →♣ ❞ư♥❣ ❬✷✽✱ ❇ê ✤➲ ✷✳✷❪ t❛ ❝â HomR (HomR (R/m, M ), E) ∼ = R/m⊗R HomR (M, E) ∼ = DR (M )/mDR (M ) ❉♦ ✤â DR (Soc(M )) ∼ = DR (M )/mDR (M ) ✈❡❝tì✱ t❛ ❝â ✤➥♥❣ ❝➜✉ k ✲❦❤æ♥❣ ❚ø Soc(M ) ❧➔ k ✲❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì DR (Soc(M )) = HomR (Soc(M ), E(k)) = Homk (Soc(M ), k) ▼➔ dimk Soc(M ) < ∞✱ t❛ t❤✉ ✤÷đ❝ dimk Soc(M ) = dimk (Homk (Soc(M ), k)) = dimk DR (Soc(M )) = dimk (DR (M )/mDR (M )) ❣✐❛♥ ✹✼ ✭✐✐✮✳ ●✐↔ sû A ⊆ B✱ A = B ❈❤♦ a ∈ A \ B k tỗ t số A = B + mA✱ ▼➔ s❛♦ ❝❤♦ A/B ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ mk−1 A ⊆ B ✈➔ mk A ⊆ B ❉♦ ♥➯♥ t❛ ❝â mk−1 A = mk−1 (B + mA) ⊆ mk−1 B + mk A ⊆ B, ♠➙✉ t❤✉➝♥✳ ✭✐✐✐✮ ●✐↔ sû ❣✐❛♥ ✈❡❝tì dim(A/mA) = n A/mA✱ tr♦♥❣ ✤â ●✐↔ sû {e1 , , en } ei = ei + mA ✈ỵ✐ ❧➔ ❝ì sð ❝õ❛ i = 1, , n ✣➦t k ✲❦❤æ♥❣ Bi = Rei ❑❤✐ ✤â (B1 + + Bn + mA)/mA = A/mA ❉♦ ✤â A = (B1 + + Bn ) + mA ▲↕✐ ❞♦ R (A) < ∞✱ ♥➯♥ t❤❡♦ ✭✐✐✮ t❛ ❝â A = B1 + + Bn ❇✐➸✉ ❞✐➵♥ ♥➔② ❧➔ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ tê♥❣ tè✐ t✐➸✉ ❝õ❛ A✳ ▼➦t ❦❤→❝✱ ❞♦ R (A) < ∞✱ ❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ ◆➳✉ Bi A✳ ❉♦ ✤â ♠é✐ Bi ✈➔ {x, y} ❧➔ {x, y} ❧➔ ❝→❝ ❤➺ s✐♥❤ ❝õ❛ A ❤♦➦❝ ❧➔ Bi = D✱ R✲♠æ✤✉♥ ❝➜♣ tè✐ t✐➸✉ ❝õ❛ A✳ Bi ✱ Q i ∈ {1, , n} tr♦♥❣ ✤â C, D ❧➔ ❝→❝ x ∈ C, y ∈ D tr♦♥❣ ✤â ❚ø ❧↕✐ ỵ t õ Bi ❆rt✐♥✱ ❉♦ ✈➟② p ∈ AttR (A)✳ p✲t❤ù ❝➜♣ ❝õ❛ Bi = Rx ♦r Bi = Ry ✱ tù❝ A ❚❛ ❦➼ ❤✐➺✉ ∗ p (A) ❧➔ ❦➼ ❤✐➺✉ ①✉➜t ❤✐➺♥ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ t❤ù ♣❤➛♥ ∗ p (A)✱ ✈➔ Q ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ ♥❤ó♥❣ tè✐ t✐➸✉ ❝õ❛ ❚❛ ❣å✐ ♠æ✤✉♥ ❝♦♥ t❤ù ❝➜♣ ❝õ❛ A ♥➳✉ Q ∈ ♥➳✉ ei = x + y, {e1 , , en } ♠➙✉ t❤✉➝♥✳ ❝õ❛ t➟♣ t➜t ❝↔ ♠æ✤✉♥ ❝♦♥ A ❑❤✐ ✤â ❦❤æ♥❣ ❧➔ ❤➺ s✐♥❤ tè✐ t✐➸✉ ❝õ❛ Bi = C ❈❤♦ B ❧➔ ❦❤æ♥❣ t❤ø❛✳ ❈❤♦ Bi = C + D, ❦❤æ♥❣ ❧➔ ❜➜t ❦❤↔ tê♥❣ t❤➻ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ {ei } →♣ ỵ t õ tố t tr♦♥❣ t➟♣ p✲t❤ù ❝➜♣ Q ❝õ❛ A ❧➔ p✲t❤➔♥❤ ∗ p (A)✳ ❚❛ ♥❤➢❝ ❧↕✐ ❦➳t q✉↔ ❝õ❛ ❨✳❨❛♦ ❬✸✶❪ ❈❤♦ A ❧➔ R✲♠æ✤✉♥ ❆rt✐♥ ✈➔ Att(A) = {p1 , p2 , , pn }✳ ●✐↔ sû r➡♥❣ ✈ỵ✐ ♠é✐ i = 1, 2, , s✱ Qi ❧➔ pi ✲t❤➔♥❤ tự ỵ ỵ ❝➜♣ ❝õ❛ A✱ tù❝ ❧➔ Q ∈ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉✳ ✳ ❑❤✐ ✤â A = Q1 + Q2 + + Qn ❧➔ ♠ët ❜✐➸✉ ∗ pi (A) ❇ê ✤➲ s❛✉ ❧➔ ❦➳t q✉↔ ✤è✐ ♥❣➝✉ ✈ỵ✐ ❦➳t q ỵ A ❧➔ R✲♠æ✤✉♥ ❆rt✐♥ ✈➔ AttR (A) = {p1 , p2 , , pn }✳ ●✐↔ sû A = Q1 + Q2 + + Qn ❧➔ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛ A✱ tr♦♥❣ ✤â Qi ❧➔ pi ✲t❤ù ❝➜♣ ✈ỵ✐ ♠å✐ i = 1, , n✳ ❑❤✐ ✤â irR (A) = irR (Q1 ) + irR (Q2 ) + + irR (Qn ) ♥➳✉ Qi ❧➔ pi ✲t❤➔♥❤ ♣❤➛♥ ♥❤ó♥❣ tè✐ t✐➸✉ ❝õ❛ A✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ A✱ ❧➔ ❝ü❝ t✐➸✉ tr♦♥❣ tr♦♥❣ ✤â ❣✐↔ sû Qi A = Q1 + + Qn Qi = Qi1 + + Qiti ❧➔ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛ ∗ pi (A)✱ i = 1, , n✳ ✣➦t irR (Qi ) = ti ❧➔ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ tê♥❣ tè✐ t✐➸✉ ❝õ❛ Qi ✳ ✈➔ ●✐↔ sû r➡♥❣ t1 + t2 + + tn = irR (Q1 ) + + irR (Qn ) > irR (A) õ tỗ t i {1, 2, , n} ✈➔ j ∈ {1, , ti } s❛♦ ❝❤♦ Q1 + + Qi−1 + Qi + Qi+1 + + Qn ⊇ Qij , tr♦♥❣ ✤â Qi = Qi1 + + Qi(j−1) + Qi(j+1) + + Qiti Qi + Qk = Qi + k=i ❝ô♥❣ ❧➔ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛ t➼♥❤ tè✐ t✐➸✉ ❝õ❛ Qi tr♦♥❣ Qi ✳ ❙✉② r❛ Qk = A k=i A✳ ∗ pi (0)✳ ❉♦ ✈➟② ❉♦ ✤â Qi ∈ ∗ pi (A)✱ ♠➙✉ t❤✉➝♥ irR (A) = irR (Q1 ) + + irR (Qn )✳ ❈❤♦ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✿ irR N ỵ irR(N ) irR DR(M/N ) = irR DR(M/N ) = irRDR(M /N ) ✹✾ ✭✐✐✐✮ ◆➳✉ R (M/N )