❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❚❍➚ ❍❯❨➋◆ ❚❾P ❈⑩❈ ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ❈Õ❆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✷ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ◆●❯❨➍◆ ❚❍➚ ❍❯❨➋◆ ❚❾P ❈⑩❈ ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ❈Õ❆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ❈❤✉②➯♥ ♥❣➔♥❤ ✿ ✣❸■ ❙➮ ❱⑨ ▲Þ ❚❍❯❨➌❚ ❙➮ ▼➣ sè✿ ✻✵ ✹✻ ✵✺ ▲❯❾◆ ❱❿◆ ữớ ữợ ◆●❯❨➍◆ ❚❍➚ ❍➬◆● ▲❖❆◆ ◆❣❤➺ ❆♥ ✲ ✷✵✶✷ ▼Ö❈ ▲Ö❈ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✶ ✹ ✶✳✶✳ ❈❤✐➲✉ ❑✉❧❧ ✈➔ ❣✐→ ❝õ❛ ♠æ✤✉♥ ✶✳✷✳ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸✳ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺✳ ■✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✻✳ ổ ố ỗ ữỡ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ổ ố ỗ ữỡ t ❚r÷í♥❣ ❤đ♣ ❣✐→ ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ❚r÷í♥❣ ❤đ♣ ❣✐→ ❧➔ ✐✤➯❛♥ tũ ỵ ổ ỳ s ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳ ❚r÷í♥❣ ❤đ♣ ❣✐→ ❧➔ tũ ỵ ổ ổ ỳ s ✳ ✷✸ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷✼ ✷✽ ✶ ▼Ð ✣❺❯ ❑❤→✐ ♥✐➺♠ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝❤♦ ❝→❝ ♠ỉ✤✉♥ ❆rt✐♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ❉✳ ●✳ ◆♦rt❤❝♦tt ♥➠♠ ✶✾✼✷✱ ❉✳ ❑✐r❜② ♥➠♠ ✶✾✼✸✳ ❙❛✉ ✤â ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♥➔② ♠ët ❝→❝❤ tê♥❣ q✉→t ổ tũ ỵ ổ õ ❞✐➵♥ t❤ù ❝➜♣ ✤➸ ❦❤ä✐ ♥❤➛♠ ❧➝♥ ✈ỵ✐ ❦❤→✐ ♥✐➺♠ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✤➣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❝❤♦ ❝→❝ ♠æ✤✉♥ ◆♦❡t❤❡r✳ ❈â t❤➸ ♥â✐ ❦❤→✐ ♥✐➺♠ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❧➔ ✤è✐ ♥❣➝✉ ✈ỵ✐ ❦❤→✐ ♥✐➺♠ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì✳ ▼å✐ ♠ỉ✤✉♥ ❆rt✐♥ ✤➲✉ ❝â ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣✳ ●✐↔ sû ❧➔ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✱ ♠ët R M ✲ ♠ỉ✤✉♥ ✤÷đ❝ ❣å✐ t❤ù ❝➜♣ ♥➳✉ M = ✈➔ ✈ỵ✐ ♠é✐ x ∈ R tỹ ỗ x,M : M M, ✤à♥❤ ❜ð✐ p := √ ϕx,M (u) = xu, ∀u ∈ M AnnR M ❈❤♦ M ❧➔ ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè✱ t❛ ❣å✐ R✲♠æ✤✉♥✳ M1 + M2 + + Mn M =0 ❤♦➦❝ ❧➔ ♠ët t♦➔♥ ❝➜✉ ❤♦➦❝ ❧ô② ❧✐♥❤✳ ❑❤✐ ✤â✱ M ▼ët R M ✲ ♠æ✤✉♥ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M p ✲ t❤ù ❝➜♣✳ ❧➔ ♠ët ♣❤➙♥ t➼❝❤ t❤➔♥❤ tê♥❣ ❤ú✉ ❤↕♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ pi M ❧➔ ❝â ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ t❤➻ t❛ ♥â✐ ❧➔ ✲ t❤ù ❝➜♣✳ ◆➳✉ ❜✐➸✉ ❞✐➵♥ ❞✐➵♥ ✤÷đ❝✳ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ♥➔② ❣å✐ ❧➔ tè✐ t❤✐➸✉ ♥➳✉ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè pi ❧➔ ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈➔ ❦❤æ♥❣ ❝â ❤↕♥❣ tû ♠æ✤✉♥ ❝♦♥ p p ✲ t❤ù ❝➜♣ ❝õ❛ ✲ t❤ù ❝➜♣ ❝õ❛ M✳ R ✲ ♠æ✤✉♥ Mi M ♥➔♦ t❤ø❛✳ ◆➳✉ t❤➻ M1 + M2 ❱➻ t❤➳ ♠å✐ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M1 ✈➔ M2 ❧➔ ❝→❝ ❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ ❝♦♥ M ✤➲✉ ❝â t❤➸ q✉② ✈➲ ♠ët ❜✐➸✉ ❞✐➵♥ tè✐ t❤✐➸✉✳ ●✐↔ sû M = M1 + M2 + + Mn ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t❤✐➸✉ ❝õ❛ M✱ tr♦♥❣ ✤â Mi ❧➔ pi ✲ t❤ù ❝➜♣✳ ❚➟♣ ✷ {p1 , , pn } ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ M ❤đ♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♣❤ö t❤✉ë❝ ✈➔♦ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t❤✐➸✉ ❝õ❛ tè p1 , , pn ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝→❝ M✳ ♠➔ ❦❤æ♥❣ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ♠æ✤✉♥ M ✱ t❛ ✈✐➳t AttR M = {p1 , , pn } t❤➔♥❤ ♣❤➛♥ t❤ù ❝➜♣ ❝õ❛ M ✳ ◆➳✉ pi ❧➔ tè✐ t❤✐➸✉ tr♦♥❣ AttR M t❤➻ Mi ✤÷đ❝ ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ t❤ù ổ ú ỵ r t tự ❝➜♣ ❝ỉ ❧➟♣ ❝õ❛ M ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ❈→❝ ❤↕♥❣ tû Mi ✱ i = 1, , n ✈➔♦ ❜✐➸✉ ❞✐➵♥ tè✐ t❤✐➸✉ ❝õ❛ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ M✳ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ✈➔ ỵ tt tự ổ ỳ tr ự ổ ố ỗ ♣❤÷ì♥❣ ❝õ❛ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✳ ❈❤♦ ♣❤÷ì♥❣✱ m ❧➔ ✐✤➯❛♥ tè✐ ✤↕✐ ❞✉② ♥❤➜t ❝õ❛ s✐♥❤ ✈ỵ✐ ❝❤✐➲✉ ❑r✉❧❧ dimM = d✳ R ✈➔ M R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ❧➔ R ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ ◆➠♠ ✶✾✼✷✱ ▼❛❝❞♦♥❛❧❞ ✈➔ ❙❤❛r♣ ❬✽❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r ổ ố ỗ ữỡ t ✈ỵ✐ ❣✐→ ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ Hmd (M ) ❧➔ R ✲ ♠æ✤✉♥ ❆rt✐♥ ✈➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ♥â ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ AttR Hmd (M ) = {p ∈ AssR M |dimR/p = d} ◆➠♠ ✷✵✵✺✱ ▼✳ ■✳ ❉✐❜❛❡✐ ✈➔ ❙✳ ❨❛ss❡♠✐ ❬✻❪ rở t q tr ổ ố ỗ ữỡ t ợ ởt tũ ỵ tr ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✻❪ ❝õ❛ ▼✳ ■✳ ❉✐❜❛❡✐ ✈➔ ❙✳ ❨❛ss❡♠✐✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ỗ ữỡ ữỡ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ✣↕✐ sè ❣✐❛♦ ữ ỵ tt tự ổ ố ỗ ữỡ ỡ sð ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ▲✉➟♥ ✈➠♥ ð ❈❤÷ì♥❣ ✷✳ ◆❣♦➔✐ r❛ ❝❤ó♥❣ ✸ tỉ✐ ❝á♥ tr ởt số t q õ ữợ ♥❤ú♥❣ ♠➺♥❤ ✤➲ ♥❤➡♠ ♣❤ö❝ ✈ö ❝❤♦ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ð ♣❤➛♥ s❛✉✳ ❈❤÷ì♥❣ ✷✿ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè t ổ ố ỗ ữỡ ❝❛♦ ♥❤➜t✳ ▼ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✻❪ ss t trữợ t ú tổ tr ự ỵ t tố t ổ ố ỗ ữỡ t ợ ỹ ỵ t q tr ❜➔✐ ❜→♦ ❬✽❪ ❝õ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✈➔ ❘✳ ❨✳ ❙❤❛r♣✳ ❙❛✉ ✤â ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦➳t q✉↔ ❝õ❛ rở ỵ ố ợ ổ ố ỗ ữỡ t ợ tũ ỵ tr trữớ ủ ♠æ✤✉♥ ❜❛♥ ✤➛✉ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ❦❤æ♥❣ ❤ú✉ ❤↕♥ s✐♥❤✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ữợ sỹ ữợ t t ✈➔ ❤➳t sù❝ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ ❝æ ❣✐→♦ ❚❙✳ ◆❣✉②➵♥ ỗ ữủ tọ ❝↔♠ ì♥ s➙✉ s➢❝ ♥❤➜t ✤➳♥ ❝ỉ ❣✐→♦ ❚❙✳ ◆❣✉②➵♥ ỗ t t ữợ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ✤➲ t➔✐✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❇ë ♠æ♥ ✣↕✐ ❙è✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❑❤♦❛ ❚♦→♥ ✤➣ trü❝ t✐➳♣ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ✣↕✐ ❙è✳ ❚→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ✈➔ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✤↕✐ ❤å❝✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✷ ❚→❝ ❣✐↔ ✹ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët số tự số ữ ỵ tt tự ổ ố ỗ ♣❤÷ì♥❣✱ ✳✳✳ ♥❤➡♠ ♠ư❝ ✤➼❝❤ ❧➔♠ ❝ì sð ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ▲✉➟♥ ✈➠♥ ð ❈❤÷ì♥❣ ✷✳ ◆❣♦➔✐ r❛ ❝❤ó♥❣ tỉ✐ ❝á♥ tr➼❝❤ ❞➝♥ ♠ët sè t q õ ữợ ỳ ♣❤ö❝ ✈ö ❝❤♦ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ð ♣❤➛♥ s❛✉✳ ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ❱➔♥❤ ❧✉ỉ♥ ✤÷đ❝ ❣✐↔ t❤✐➳t ❧➔ ❣✐❛♦ ❤♦→♥ ✈➔ ❝â ✤ì♥ ✈à✳ ✶✳✶ ❈❤✐➲✉ ❑✉❧❧ ✈➔ ❣✐→ ❝õ❛ ♠æ✤✉♥ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ▼ët ❞➣② ❣✐↔♠ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✈➔♥❤ R✿ p0 ⊃ ⊃ pn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❑➼ ❤✐➺✉ ①➼❝❤ ♥❣✉②➯♥ tè ❝â ✤ë ❞➔✐ n✳ SpecR ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ❣å✐ ❧➔ ♣❤ê ❝õ❛ ✈➔♥❤ ❈❤♦ p ∈ SpecR✱ ✈ỵ✐ p0 = p ❝õ❛ R✳ R✳ ❑❤✐ ✤â SpecR ✤÷đ❝ R✳ ❝➟♥ tr➯♥ ❝õ❛ t➜t ❝↔ ❝→❝ ✤ë ❞➔✐ ❝õ❛ ❝→❝ ①➼❝❤ ♥❣✉②➯♥ tè ✤÷đ❝ ❣å✐ ❧➔ ✤ë ❝❛♦ ❝õ❛ ❑❤✐ ✤â ✤ë ❝❛♦ ❝õ❛ I p✱ ❦➼ ❤✐➺✉ ❧➔ ht(p)✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ht(I) = inf{ht(p)|p ∈ SpecR, p ⊇ I} ❈➟♥ tr➯♥ t➜t ❝↔ ❝→❝ ✤ë ❞➔✐ ❝õ❛ ①➼❝❤ ♥❣✉②➯♥ tè tr♦♥❣ ❑✉❧❧ ❝õ❛ ✈➔♥❤ R✱ ❦➼ ❤✐➺✉ ❧➔ dimR✳ R ✤÷đ❝ ❣å✐ ❧➔ ❝❤✐➲✉ ✺ ❈❤♦ M ❧➔ ♠ët R ✲ ♠æ✤✉♥✳ ❑❤✐ ✤â ❑✉❧❧ ❝õ❛ ♠æ✤✉♥ M ✱ ❦➼ ❤✐➺✉ ❧➔ dimM ✳ ✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ s❛♦ ❝❤♦ Mp = 0✳ ❱ỵ✐ ♠é✐ ❑➼ ❤✐➺✉ ❚➟♣ x ∈ M✱ dim(R/AnnR M ) Supp(M ) Supp(M ) ✤÷đ❝ ❣å✐ ❧➔ ❝❤✐➲✉ ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✤÷đ❝ ❣å✐ ❧➔ ❣✐→ ❝õ❛ ♠ỉ✤✉♥ M ✳ R ❦➼ ❤✐➺✉ AnnR (x) = {a ∈ R|ax = 0}; AnnR M = {a ∈ R|aM = 0} = {a ∈ R|ax = 0, ∀x ∈ M }, ❚❛ ❝â AnnR (x) ✈➔ AnnR (M ) ❧➔ ♥❤ú♥❣ ✐✤➯❛♥ ❝õ❛ R✱ AnnR (M ) ❣å✐ ❧➔ ❤â❛ tû ❝õ❛ ♠æ✤✉♥ M ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ M ❧➔ R✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ ❧✐♥❤ Supp(M ) = {p ∈ SpecR|p ⊇ AnnR (M )} ✶✳✷ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ tố tỗ t p ❈❤♦ ❝õ❛ 0=x∈M R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à ✈➔ M R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❧➔ R✲ ♠æ✤✉♥✳ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M s❛♦ ❝❤♦ p = AnnR (x) = {r ∈ R : rx = 0} ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ AssRM ✭❤❛② ❝á♥ ✤÷đ❝ ❦➼ ❤✐➺✉ ✤ì♥ ❣✐↔♥ ❧➔ AssM ✮✳ ✶✳✷✳✷ ✣à♥❤ ❧➼✳ ●✐↔ sû R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ ❝â ✤ì♥ ✈à ✈➔ M ❧➔ R ✲ ♠æ✤✉♥✳ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤ó♥❣✿ ✭✐✮ ■✤➯❛♥ ♥❣✉②➯♥ tè p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M ❝❤ù❛ ♠ët ♠æ✤✉♥ ❝♦♥ M s❛♦ ❝❤♦ M ∼ = R/p ✳ ✭✐✐✮ ❈❤♦ → M → M → M → ❧➔ ❞➣② ❦❤ỵ♣ ❝→❝ R ✲ ♠ỉ✤✉♥✳ ❑❤✐ ✤â✿ AssM ⊆ AssM ⊆ AssM ∪ AssM ✻ ✭✐✐✐✮ P❤➛♥ tû tè✐ ✤↕✐ ❝õ❛ t➟♣ {AnnRx : x = 0, x ∈ M } ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳ ◆➳✉ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M = t❤➻ AssM = ∅✳ ❍ì♥ ♥ú❛✱ ♥➳✉ M ❧➔ R ✲ ♠æ✤✉♥ ◆♦❡t❤❡r t❤➻ AssM ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭✐✮ ■✤➯❛♥ q=R ∈ q p r ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✈➔ M ❧➔ R ✲ ♠ỉ✤✉♥✳ ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ ♥➳✉ ✈ỵ✐ ♠å✐ tr➯♥ r ∈ R✱ R/q ❧➔ ✤ì♥ ❝➜✉ ❤♦➦❝ ❧ơ② ❧✐♥❤✳ ◆❣❤➽❛ ❧➔ ∀a, r ∈ R a ∈ q ❤♦➦❝ ♥➳✉ a ∈ / q t❤➻ ∃n ∈ N s❛♦ ❝❤♦ rn ∈ q✳ ❚r♦♥❣ √ ♥➔② q ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè✱ ❝❤➥♥❣ ❤↕♥ p✱ ✈➔ t❛ ❣å✐ q ✲ ♥❣✉②➯♥ sì✳ ✭✐✐✮ ▼ỉ✤✉♥ ❝♦♥ N = M ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥â✐ R t❤➻ tr÷í♥❣ ❤đ♣ ❧➔ R ❝õ❛ ♣❤➨♣ ♥❤➙♥ ❜ð✐ ♠➔ ❈❤♦ N ✭✐✐✐✮ ❈❤♦ ❧➔ N p p ❝õ❛ R ❝õ❛ M ữủ sỡ tỗ t ởt Ass(M/N ) = {p}✳ s❛♦ ❝❤♦ ❑❤✐ ✤â t❛ ❝ô♥❣ ✲ ♥❣✉②➯♥ sì✳ ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M✳ ▼ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ❝õ❛ N ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ N = M1 ∩ M2 ∩ Mn , tr♦♥❣ ✤â Mi ✤÷đ❝ ❣å✐ ❧➔ ❧➔ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ pi ✲ ♥❣✉②➯♥ ❝õ❛ M✳ P❤➙♥ t➼❝❤ tr➯♥ t❤✉ ❣å♥ ♥➳✉ ❝→❝ pi ❧➔ ✤ỉ✐ ♠ët rí✐ ♥❤❛✉ ✈➔ ❦❤æ♥❣ ❝â Mi ♥➔♦ t❤ø❛✳ ✶✳✷✳✹ ◆❤➟♥ ①➨t✳ ❑❤✐ M =R ✈➔ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r t❤➻ ❦❤→✐ ♥✐➺♠ ♠æ✤✉♥ ❝♦♥ ♥❣✉②➯♥ trị♥❣ ✈ỵ✐ ❦❤→✐ ♥✐➺♠ ✐✤➯❛♥ ♥❣✉②➯♥ sì✳ ỵ s sỹ tỗ t t➼❝❤ ♥❣✉②➯♥ ❝õ❛ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ ♠ỉ✤✉♥ ◆♦❡t❤❡r ✈➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ t❤ỉ♥❣ q✉❛ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥✳ ✶✳✷✳✺ ✣à♥❤ ❧➼✳ ❈❤♦ M ❧➔ ♠ët R ✲ ♠æ✤✉♥ ◆♦❡t❤❡r ✈➔ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M✳ ❑❤✐ ✤â t❛ ❝â✿ ✼ ✭✐✮ N ❝â sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥✳ ✭✐✐✮ ◆➳✉ N = N1∩N2∩ .∩Nn❀ ✈➔ N = N 1∩N 2∩ .∩N m ❧➔ ❤❛✐ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ N ✱ tr♦♥❣ ✤â Ni ❧➔ pi ✲ ♥❣✉②➯♥ sì✱ i = 1, n ✈➔ Ni ❧➔ p i ✲ ♥❣✉②➯♥ sì❀ i = 1, m t❤➻ n = m ✈➔ {p1, , pn} = {p 1, , p n}✳ ❱➻ t❤➳ {p1, , pn} ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ N ✳ ❍ì♥ ♥ú❛✱ t❛ ❝â✿ {p1 , , pn } = Ass(M/N ) ✭✐✐✐✮ ❈❤♦ N = N1 ∩ N2 ∩ ∩ Nn✱ tr♦♥❣ ✤â Ni ❧➔ pi ✲ ♥❣✉②➯♥ sì✱ i = 1, n ❧➔ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ N ✳ ◆➳✉ pi ❧➔ ♣❤➛♥ tû tè✐ t❤✐➸✉ tr♦♥❣ t➟♣ Ass(M/N ) t❤➻ ♠æ✤✉♥ ❝♦♥ Ni t÷ì♥❣ ù♥❣ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ t❤✉ ❣å♥ ❝õ❛ N ✳ ✶✳✷✳✻ ▼➺♥❤ ✤➲✳ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ R ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❑❤✐ ✤â N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ a ∈ R✱ ♣❤➨♣ ♥❤➙♥ ❜ð✐ a tr➯♥ M/N ❧➔ ✤ì♥ ❝➜✉ ❤♦➦❝ ❧ơ② ❧✐♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t➟♣ Ann(M/N ) ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè p ✈➔ N ❧➔ p ✲ ♥❣✉②➯♥ sì✳ ✶✳✸ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ sỷ M ởt tỹ ỗ M R ổ ợ ộ tỷ xR ữủ ✤à♥❤ ❜ð✐ ♣❤➨♣ ♥❤➙♥ ❜ð✐ x✱ t❛ ❦➼ ❤✐➺✉ ϕx,M ❧➔ tù❝ ❧➔ ϕx,M (m) : M −→ M −→ xm m ỗ x,M ữủ ụ ú ỵ r t ủ tỷ R ❣å✐ ❧➔ √ (M ) = AnnR M ✳ ✐✤➯❛♥ ❝õ❛ ∃n ∈ N x ∈ R ❝➠♥ ❧ô② ❧✐♥❤ ❝õ❛ M✱ s❛♦ ❝❤♦ s❛♦ ❝❤♦ ❦➼ ❤✐➺✉ ❧➔ ϕxn ,M = 0✳ ϕx,M ❧ô② ❧✐♥❤ ❧➔ ♠ët (M )✳ ❉➵ t❤➜② r➡♥❣ ✶✺ ✤➛✉ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ❦❤ỉ♥❣ ❤ú✉ ❤↕♥ s✐♥❤✳ ✷✳✶ ❚r÷í♥❣ ❤đ♣ ❣✐→ ❧➔ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ✷✳✶✳✶ ❇ê ✤➲✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ tr ữỡ ợ m tố ♥❤➜t ✈➔ M = ❧➔ R ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ❝❤✐➲✉ ❑r✉❧❧ dimM = d✳ ❑❤✐ ✤â✱ t➟♣ ❤đ♣ := {N | N ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M, dim N < d} ❝â ♣❤➛♥ tû ❧ỵ♥ ♥❤➜t N t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠✳ ✣➦t G := M/N ❑❤✐ ✤â t❛ ❝â✿ ✭✐✮ dim G = d❀ ✭✐✐✮ G ❦❤ỉ♥❣ ❝â ♠ỉ✤✉♥ ❝♦♥ ❝â ❝❤✐➲✉ ❜➨ ❤ì♥ d; ✭✐✐✐✮ Hmd (G) ∼ = Hmd (M )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ M ❧➔ R ✲ ♠æ✤✉♥ ◆♦❡t❤❡r ♥➯♥ tr♦♥❣ ❝â ♣❤➛♥ tû ❝ü❝ ✤↕✐ N ❱➻ tê♥❣ ❝õ❛ ❤❛✐ ♣❤➛♥ tû ❜➜t ❦ý ❝õ❛ ❝❤ù❛ ♠é✐ ♣❤➛♥ tû ❝õ❛ ✳ ❉♦ ✤â ✭✐✮ ❚ø ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ R N ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ ❧➔ ♣❤➛♥ tû ❧ỵ♥ ♥❤➜t ❝õ❛ ♥➯♥ N ✳ ✲ ♠æ✤✉♥✿ 0→N →M →G→0 t❛ ❝â ❙✉♣♣(G) ⊆ ❙✉♣♣(M ) ✈➔ ♠é✐ p∈ ❆ssM ✈ỵ✐ ❙✉♣♣(G) ❞♦ ♥â ❦❤æ♥❣ t❤➸ t❤✉ë❝ ❙✉♣♣(N )✳ ❉♦ ✤â dim R/p = d dim G = d ♣❤↔✐ t❤✉ë❝ ✈➔ {p ∈ AssM | dim R/p = d} ⊆ AssG ✭✐✐✮ ●✐↔ sû dim(L/N ) < d L ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ ❚ø ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ R M s❛♦ ❝❤♦ ✲ ♠æ✤✉♥ → N → L → L/N → t❛ ❝â dim L < d ❉♦ ✤â L⊆N ✈➔ L/N = N ⊆ L ⊆ M ✈➔ ✶✻ ✭✐✐✐✮ ❚ø ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ R ✲ ♠ỉ✤✉♥✿ 0→N →M →G→0 t❛ ❝â ❞➣② ❦❤ỵ♣ R ổ ố ỗ ữỡ · · · → Hmd (N ) → Hmd (M ) → Hmd (G) → Hmd+1 (N ) → · · · ❱➻ dim N < d ✶✳✻✳✸✮t❛ ❝â ♥➯♥ t ỵ trt t rt ỵ Hmd (N ) = Hmd+1 (N ) = 0✳ ❉â ✤â tø ❞➣② ❦❤ỵ♣ ❞➔✐ tr➯♥ ✤➙② t❛ ❝â Hmd (G) = Hmd (M ) ỵ s t q tr ỵ ỵ (R, m) tr ữỡ ợ m ✐✤➯❛♥ tè✐ ✤↕✐ ❞✉② ♥❤➜t ✈➔ M = ❧➔ R ✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ❝❤✐➲✉ ❑r✉❧❧ dimM = d✳ ❑❤✐ ✤â Hmd (M ) = ✈➔ AttR Hmd (M ) = {p ∈ AssR M |dimR/p = d} ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ sè tü ♥❤✐➯♥ i✱ ♠æ✤✉♥ Hmi (M ) ❧➔ ❆rt✐♥✱ ✈➻ ✈➟②✱ Hmi (M ) ❧➔ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ✈➔ t❛ ❦➼ ❤✐➺✉ ❦➳t ❝õ❛ AttR Hmi (M ) ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ Hmi (M )✳ ❈❤ó♥❣ t❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ ♠æ✤✉♥ M ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ Hm0 (M ) ∼ = Γm (M ) = M = 0✳ d✳ ❑❤✐ d=0 ❚ø ỵ ỵ t ❝â Att(Hm0 (M )) = AttM = {m} = AssM = {p ∈ AssM : dimR/p = 0} ❉♦ ✤â trữớ ủ ỵ ữủ ự sỷ d>0 ỵ ữủ ự trữớ ❤đ♣ ❝→❝ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤ỉ♥❣ ❝â ❝❤✐➲✉ ❣✐↔ t❤✐➳t r➡♥❣ M R ✲ d − 1✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✶ t❛ ❝â t❤➸ ❦❤æ♥❣ ❝â ♠æ✤✉♥ ❝♦♥ ❝â ❝❤✐➲✉ ❜➨ ❤ì♥ d ✶✼ ❉♦ d>0 m∈ / Ass(M ) ♥➯♥ t❛ ❝â ❦❤→❝ ❝õ❛ ❦❤æ♥❣ tr➯♥ M✳ ✈➔ ✈➻ t❤➳ d=1 r ✈ỵ✐ Hmd (M ) = ●✐↔ sû ♥❣÷đ❝ ❧↕✐ ❧➔ ♠➙✉ t❤✉➝♥✳ ❚❤➟t r m ổ ữợ t s t➻♠ t❤➜② sü t❛ ❝â ≤ gradeM m = depthM ≤ dimM = ❉♦ ✤â✱ d > 1✳ gradeM m = r∈m ❱ỵ✐ ♠é✐ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ❝â ✈➔ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ t❤❡♦ ✣à♥❤ ỵ sỷ ổ ữợ ổ tr dim(M/rM ) = d − 1✳ M✱ M/rM = 0✱ ♠ỉ✤✉♥ ❚ø ❞➣② ❦❤ỵ♣ ♥❣➢♥✿ r 0→M → − M → M/rM → t❛ ❝â ❞➣② ❦❤ỵ♣ ❞➔✐ ❝→❝ ổ ố ỗ ữỡ r Hmd1 (M ) → − Hmd−1 (M ) → Hmd−1 (M/rM ) → ✈ỵ✐ ❣✐↔ t❤✐➳t tr➯♥ M✱ Hmd (M ) = õ ợ ộ rm ổ ữợ ❦❤æ♥❣ t❛ ❝â Hmd−1 (M )/rHmd−1 (M ) ∼ = Hmd−1 (M/rM ) ✈➔ ❝❤ó♥❣ ❦❤→❝ ✵ t❤❡♦ ❣✐↔ t❤✐➳t q r ữợ t t ự ❧↕✐ r➡♥❣ m∈ / Att(Hmd−1 (M )) Hmd−1 (M ) = 0✳ m ∈ Att(Hmd−1 (M ))✳ ❚❛ s➩ ❣✐↔ sû ♥❣÷đ❝ ✈➔ s➩ t➻♠ r❛ sü ♠➙✉ t❤✉➝♥✳ ❑❤✐ õ t ỵ tr tố p) ( ( m q∈Ass(M ) d−1 p∈Att(Hm (M ))) ❱➻ ✈➟②✱ tỗ t r1 Hmd1 (M ) r1 m q) ổ ữợ ổ tr t ✈ỵ✐ Att(Hmd−1 (M ))✳ ❈❤♦ p1 , , pt M Hmd−1 (M/r1 M ) = 0✳ ❧➔ t ổ ụ t ỵ tr tố tỗ t t r2 m\( i=1 ✈➔ pi ) ∪ ( q∈Ass(M ) q) Hmd−1 (M ) = ◆❤÷ ✈➟② m ∈ Att(Hmd−1 (M ))✳ ✶✽ r2 m r2 ổ ữợ ❦❤æ♥❣ tr➯♥ M ♥➯♥ t❛ ❝â Hmd−1 (M )/rHmd−1 (M ) ∼ = Hmd−1 (M/r2 M ), ✈➔ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❤➻ Hmd−1 (M/r2 M ) = ✈➔ Att(Hmd−1 (M/r2 M )) ⊆ {p ∈ Ass(M/r2 M ) : dimR/p = d − 1} ❚❤❡♦ ❬✺✱ ✣à♥❤ ỵ Att(Hmd1 (M/r2 M )) {p Att(Hmd1 (M/r2 M )) : r2 ∈ p}, ✈➔ m ❧➔ t❤➔♥❤ ♣❤➛♥ ❞✉② ♥❤➜t ❝õ❛ t➟♣ ❤ñ♣ ❜➯♥ ♣❤↔✐ ❝õ❛ ❜❛♦ ❤➔♠ t❤ù❝✳ ❉♦ d > 1✱ ♥➯♥ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥✳ ◆❤÷ ✈➟② ❝❤ó♥❣ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ Hmd (M ) = 0✳ ✣➸ ❦➳t tú ữợ q ỡ d ú t ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❚❛ ❜✐➳t r➡♥❣ tr➯♥ M M t❛ ❝â gradeM m ≥ ❦❤æ♥❣ ❝❤ù❛ ♠æ✤✉♥ ❝♦♥ ❝â ❝❤✐➲✉ ❜➨ Att(Hmd (M )) = Ass(M )✳ ✈➔ ✈ỵ✐ ♠é✐ dim(M/rM ) = d − 1✱ r ∈m ✈➻ t ổ ữợ ổ Hmd (M/rM )) = t ỵ trt t rt ứ ợ ổ ố ỗ ữỡ s tø ❞➣② ❦❤ỵ♣ r 0→M → − M → M/rM → t❛ s✉② r❛ Hmd (M ) = rHmd (M ) õ t ỵ m\( p) ⊆ m\( p∈Ass(M ) ❈❤♦ q ∈ Att(Hmd (M ))✳ ♥❣✉②➯♥ tè t❛ ❝â R✲t✉②➳♥ q⊆p q) d (M ))) q∈Att(Hm ❚❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ ♥â✐ tr➯♥ ✈➔ ỵ tr ợ p õ tr Ass(M ) ❉♦ t➼♥❤ ♥➯♥ t❛ ❝â✿ (0 : M ) ⊆ (0 : Hmd (M )) ⊆ q ⊆ p Hmd ❧➔ ♠ët ❤➔♠ tû ✶✾ ❉♦ d = dimR/(0 : M ) = dimR/p ♥➯♥ s✉② r❛ q = p✳ ❉♦ ✤â✿ Att(Hmd (M )) ⊆ Ass(M ) ◆❣÷đ❝ ❧↕✐✱ ❧➜② p ∈ Ass(M ) s❛♦ ❝❤♦ dimR/p = d✳ ỵ sỹ t sỡ tỗ t ởt ổ õ M/Q ú ỵ r ♥❤ä ❤ì♥ ❧➔ R✲ p✲ ♥❣✉②➯♥ ❝õ❛ M ✱ t❛ ❦➼ ❤✐➺✉ ❧➔ Q✳ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤ỉ♥❣ ✈ỵ✐ Ass(M/Q) = {p}✳ M/Q ❦❤ỉ♥❣ t❤➸ ❝â ♠ët ♠æ✤✉♥ ❝♦♥ ♥➔♦ ❦❤→❝ ❦❤æ♥❣ ♠➔ ❝â ❝❤✐➲✉ d✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ tr➯♥ ❝❤♦ M/Q✱ t❛ ❝â✿ Hmd (M/Q) = ✈➔ ∅ = Att(Hmd (M/Q)) ⊆ Ass(M/Q) = {p} ❉♦ ✤â✱ Att(Hmd (M/Q)) = {p}✳ ❝õ❛ ●r♦t❤❡♥❞✐❝❦ t❛ ❝â ❉♦ dimQ < d + 1✱ Hmd+1 (Q) = t ỵ trt t õ tứ ❞➣② ❦❤ỵ♣ → Q → M → M/Q → ❝↔♠ s✐♥❤ ♠ët t♦➔♥ ❝➜✉ Hmd (M ) → Hmd (M/Q) ỵ t õ Ass(M/Q) ⊆ Att(Hmd (M ))✳ Att(Hmd (M ))✳ p = Att(Hmd (M/Q)) ⊆ Att(Hmd (M ))✳ ✣✐➲✉ ♥➔② ❦➳t t❤ó❝ ❝❤ù♥❣ Ass(M/Q) = ỵ ữủ ự ❤♦➔♥ t♦➔♥✳ ✷✳✶✳✸ ❍➺ q✉↔✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ tr ữỡ ợ m tố ♥❤➜t✱ ✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ❝❤✐➲✉ ❑r✉❧❧ dimM = d > 0✳ ❑❤✐ ✤â Hmd (M ) ❦❤æ♥❣ ỳ s ự ỵ ỵ M = ởt R t ổ ố ỗ ♣❤÷ì♥❣ Hmd (M ) ❧➔ ❆tr✐♥ ✈➔ ❦❤→❝ ❦❤ỉ♥❣✱ tr♦♥❣ ✤â ❝â ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❦❤æ♥❣ ❝ü❝ õ t ỵ Hmd (M ) ❦❤ỉ♥❣ ❤ú✉ ❤↕♥ s✐♥❤✳ ✷✵ ✷✳✷ ❚r÷í♥❣ ❤đ♣ ❣✐→ tũ ỵ ổ ỳ s ỵ s ởt tr t q ❝❤➼♥❤ tr♦♥❣ ❬✻❪✱ ♥â ❧➔ ♠ët ♠ð rë♥❣ ❝õ❛ ✣à♥❤ ỵ (R, m) tr ữỡ a ởt tũ ỵ ❝õ❛ R✳ ●✐↔ sû M ❑❤✐ ✤â =0 ❧➔ ♠ët R ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝â ❝❤✐➲✉ d✳ Att(Had (M )) = {p ∈ Ass(M )|cd(a, R/p) = d} tr õ cd(a, K) ố ỗ R✲♠ỉ✤✉♥ K ✤è✐ ✈ỵ✐ ✐✤➯❛♥ a✱ ♥❣❤➽❛ ❧➔ cd(a, K) = sup{i ∈ Z|Hai (K) = 0}✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ d = t❤➻ M ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥ ✈➔ ❞♦ ✤â Att(Ha0 (M )) = Att(M ) = {m} = Ass(M ) ●✐↔ sû d > 0✳ ◆➳✉ ❧➔ t➟♣ ré♥❣ ✈➔ ❞♦ Had (M ) = cd(a, m) < d t❤➻ ✈➳ tr→✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr ỵ tự õ ❝ơ♥❣ ❧➔ t➟♣ ré♥❣✳ ❉♦ ✤â ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ ♠ỉ✤✉♥ ❝♦♥ ❧ỵ♥ ♥❤➜t ❝õ❛ ❝♦♥ M1 , M2 cd(a, M2 ) ❝õ❛ t M M ✈ỵ✐ d>0 cd(a, N ) < d ✈➔ ❜➜t ❦➻ sè ♥❣✉②➯♥ ❞÷ì♥❣ sû ❞ư♥❣ ❬✸✱ ❍➺ q✉↔ ✷✳✸❪✱ t❤➻ ✈➔ Had (M ) = N ú ỵ r➡♥❣ ✈ỵ✐ ❤❛✐ ♠ỉ✤✉♥ t∈Z ♠➔ cd(a, M1 ) cd(a, M1 + M2 ) t✳ t ✈➔ ❱➻ ✈➟② N ữủ ỷ ợ N → M → M/N → t❛ ❝â cd(a, M ) = cd(a, M/N )✳ ❦❤→❝ ❦❤ỉ♥❣ ✈ỵ✐ ❘ã r➔♥❣ r➡♥❣ cd(a, L) < cd(a, M )✳ M/N ❦❤æ♥❣ ❝â ♠æ✤✉♥ ❝♦♥ ❱➻ ✈➟②✱ Ass(M/N ) = {p ∈ Supp(M/N )|cd(a, R/p) = d} L ✷✶ ❍ì♥ ♥ú❛✱ ♥➳✉ p ∈ Supp(M/N ) ✈ỵ✐ d = cd(a, R/p) p ∈ Ass(M/N ) ❉♦ ✤â ✈➔ cd(a, R/p) = d dimR/p t❤➻ dimM/N d Ass(M/N ) = {p ∈ Supp(M/N )|cd(a, R/p) = d}✳ ❉♦ tr♦♥❣ ❞➣② ❦❤ỵ♣ Had (N ) → Had (M ) → Had (M/N ) → Had+1 (N ) ❤❛✐ ♠æ✤✉♥ ❝✉è✐ ❜➡♥❣ M/N t❤❛② t❤➳ ❝❤♦ N cd(a, N ) < d✳ ✈ỵ✐ ◆➳✉ r∈ / 0✱ M✱ Had (M ) ∼ = Had (M/N )✳ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t M ❦❤æ♥❣ ❝â ♠æ✤✉♥ ❝♦♥ ❦❤→❝ ✵ ❇➙② ❣✐í t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ p ❉♦ ✤â ❜➡♥❣ ❝→❝❤ ①➨t Att(Had (M )) = Ass(M )✳ t❤➻ tø ❞➣② ❦❤ỵ♣ p∈Ass(M ) r 0→M → − M → M/rM → t❛ ❝â ❞➣② s❛✉ ❝ơ♥❣ ❦❤ỵ♣ r Had (M ) → − Had (M ) → Had (M/rM ) ❉♦ ✈➔ cd(a, R/p) < d, r∈ / ♥➯♥ p✳ Had (M/rM ) = 0✳ ❤↕♥✱ p ❉♦ ✤â✱ p∈Att(Had (M )) R✲t✉②➳♥ ❉♦ ✤â✱ p = q✳ Ann(M ) dimR/p ✣✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ t❛ ❝â p ✲ ♥❣✉②➯♥ T ❝õ❛ M cd(a, M/T ) = cd(a, R/p) = d✳ ♠æ✤✉♥ ❝♦♥ L=0 Att(Had (M/T )) ✈ỵ✐ cd(a, R/p) < d✳ Ass(M/T )✳ Ann(Had (M )) dimM/N Att(Had (M )) ♠✐♥❤ ❜❛♦ ❤➔♠ t❤ù❝ ♥❣÷đ❝ ❧↕✐✱ t❛ ❣✐↔ sû ♠ỉ✤✉♥ ❝♦♥ Ass(M ) ❉♦ ❧➔ ❤ú✉ q ✈ỵ✐ q ♥➔♦ ✤â tr♦♥❣ Ass(M )✳ ❱➻ Had (−) t➼♥❤ ♥➯♥ t❛ ❝â d = cd(a, R/p) p✳ rHad (M ) = Had (M ) p∈Ass(M ) p∈Att(Had (M )) p ∈ Att(Han (M )) ❦➨♦ t❤❡♦ p ❧➔ ❤➔♠ tû ❚ø ✤â s✉② r❛ ▼æ✤✉♥ ❱➻ ✈➟② d Ass(M )✳ p ∈ Ass(M )✳ s❛♦ ❝❤♦ p✳ ✣➸ ❝❤ù♥❣ ❑❤✐ ✤â ❝â ♠ët Ass(M/T ) = p✳ M/T ❑❤✐ ✤â ❦❤æ♥❣ t❤➸ ❝â ❜➜t ❦➻ ❱➻ ✈➟② t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ t❛ ❝â ❇➙② ❣✐í sû ❞ư♥❣ ❞➣② ❦❤ỵ♣ Att(Had (M )) → ✷✷ Att(Had (M/T )) → t❛ ❝â Att(Had (M/T )) p Att(Had (M )) r ỵ ữủ ự ỵ tr ợ a=m t ữủ ỵ t q tr♦♥❣ ❬✽❪ ♠➔ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ▼ư❝ ✷✳✶✳ ✷✳✷✳✷ ❍➺ q✉↔✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✳ ●✐↔ sû M ❧➔ R ✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤æ♥❣ ❝â ❝❤✐➲✉ d✳ ❑❤✐ ✤â Att(Hmd (M )) = {p ∈ Ass(M )|dim(R/p) = d} ❈❤ù♥❣ ♠✐♥❤✳ ❙û ❞ö♥❣ t➼♥❤ ❝❤➜t cd(m, R/p) = dim(R/p)✳ ●✐↔ sû R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ❝â ❝❤✐➲✉ d a ởt R ỵ trtsr ố ỗ ữỡ ỵ ✶✳✻✳✻✮ t❤➻ tè✐ t❤✐➸✉ q ❝õ❛ R✱ Had (M ) tr✐➺t t✐➯✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠é✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè ✤ë ❝❛♦ ❝õ❛ a(R/q) ❜➨ ❤ì♥ d✳ ❙û ❞ư♥❣ ❦➳t q✉↔ ♥➔② ❙❤❛r♣ ✤➣ ❝❤➾ r❛ r➡♥❣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ♠æ✤✉♥ ❆rt✐♥ ❤➭♣ ❝õ❛ ♥❤ú♥❣ ♥❣✉②➯♥ tè ❜➡♥❣ q ❝õ❛ R ✈ỵ✐ dim(R/q) = d Had (R) ✈➔ ✤ë ❝❛♦ ❝õ❛ ❧➔ t❤✉ a(R/q) d✳ ✷✳✷✳✸ ❍➺ q✉↔✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ❝â ❝❤✐➲✉ d✱ a ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✱ R ❧➔ ❜❛♦ ✤➛② ✤õ m ✲ ❛❞✐❝ ❝õ❛ R ❑❤✐ ✤â Att(Had (R) = {p ∈ Ass(R)|cd(R/p) = d} = {q ∩ R|q ∈ SpecR, dim(R/q) = d, dimR/aR + q = 0} ❈❤ù♥❣ ♠✐♥❤✳ ❙✉② r❛ tø ✣à♥❤ ỵ q s ỵ ❝❤♦ t❤➜② r➡♥❣ t➟♣ ❝→❝ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ R ✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ M ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ Supp(M )✳ ✷✳✷✳✹ ❍➺ q✉↔✳ ❈❤♦ M ✱ N ❧➔ ❝→❝ R ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ s❛♦ ❝❤♦ dim M = dim N = d ✈➔ Supp(M ) = Supp(N )✳ ●✐↔ sû a ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â Att(Had (M )) = Att(Had (N )) ✷✸ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ Ass(R/AnnM ) ❝â ✈ỵ✐ p ∈ Ass(M ) ✈ỵ✐ cd(a, R/p) = d✱ d = cd(a, R/p) cd(a, R/p) = d ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ p ∈ ❜ð✐ ✈➻ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â dimR/p dimR/AnnM = d ❚ø ✤â t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ s ỵ trtsr ổ q✉↔✳ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ a ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R ✈➔ M ❧➔ R ✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ dim(M ) = d✳ ❑❤✐ ✤â Att(Had (M )) = {q ∩ R|q ∈ SpecR, dim(R, q) = d, dimR/aR + q = 0} tr♦♥❣ ✤â✱ S = R/Ann(M ) ✈➔ S ❧➔ ❜❛♦ ✤➛② ✤õ ❝õ❛ S ✤è✐ ✈ỵ✐ ✐✤➯❛♥ ❝ü❝ ✤↕✐✳ ự ú ỵ r ố ợ S ổ T t❛ ❝â AttR (T ) = {p ∩ R | p ∈ AttS (T )} ❙û ❞ö♥❣ ❍➺ q✉↔ ỵ tr ố ỗ ✤à❛ ♣❤÷ì♥❣✱ t❛ ❝â Att(Had (M )) = Att(HadR (S)) ❉♦ ✤â tø ❍➺ q✉↔ ✷✳✷✳✸ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ự rữớ ủ tũ ỵ ✈➔ ♠æ✤✉♥ ❦❤æ♥❣ ❤ú✉ ❤↕♥ s✐♥❤ ✷✳✸✳✶ ✣à♥❤ ❧➼✳ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ a ❧➔ ✐✤➯❛♥ ❝õ❛ R✳ ●✐↔ sû M ❧➔ R ✲ ♠æ✤✉♥ ✭❦❤æ♥❣ ♥❤➜t t❤✐➳t ❤ú✉ ❤↕♥ s✐♥❤✮ s❛♦ ❝❤♦ dimM = dimR = d✳ ❑❤✐ ✤â Had (M ) ❧➔ ♠æ✤✉♥ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ✈➔ =0 Att(Had (M )) {p ∈ Ass(M )|cd(a, R/p) = d} ự rữợ ❤➳t ❝❤ó♥❣ t❛ ①➨t t➟♣ ❤đ♣ ❝→❝ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t✳ ❈❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ Had (M ) = ❈❤♦ X = {p ∈ Ass(M )|cd(a, R/p) = d} N õ tỗ t ổ ❝õ❛ M s❛♦ ❝❤♦ Ass(M/N ) = X, Ass(N ) = Ass(M/X) ❳➨t ❞➣② ❦❤ỵ♣ Had (N ) → Had (M ) → Had (M/N ) → Had+1 (N ) ❉♦ d + > dim(N ) t❛ ❝â Had+1 (N ) = 0✳ ▼➦t ❦❤→❝ d Had (N ) = − lim →Ha (Ni ), tr♦♥❣ ✤â i∈I {Ni |i ∈ I} ❧➔ ❤å ❝→❝ ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝õ❛ t❛ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ N✳ ❱ỵ✐ ❜➜t ❦➻ Had (Ni ) = 0✳ ❚❤➟t ✈➟②✱ ♥➳✉ dim(Ni ) < d t❤➻ ❦❤æ♥❣ ❝â ❣➻ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ◆➳✉ dim(Ni ) = d t t ỵ ú t❛ ❝â Att(Had (Ni )) = {p ∈ Ass(Ni )|cd(a, R/p) = d} AssNi ∩ X = ø✳ ❉♦ ✤â Had (Ni ) = 0✳ d d ❚ø ✤â s✉② r❛ H (N ) = 0✳ ❱➻ ✈➟②✱ H (M ) ∼ = H d (M/N )✳ ◆❤÷ ✈➟② ❜➡♥❣ ❝→❝❤ ▼➦t ❦❤→❝ Ni ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ N M/N a a a ①➨t ♥➯♥ t❤❛② t❤➳ ❝❤♦ M t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ Ass(M ) = {p ∈ Supp(M )|cd(a, R/p) = d} ❇➙② ❣✐í ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tỹ ỵ t õ p p pAtt(Had (M )) ❚ø ✤â s✉② r❛ t➟♣ q ∈ Ass(M ) Ass(M ) p∈Ass(M ) ❧➔ ❤ú✉ ❤↕♥ ✈➔ ♥➳✉ p ∈ Att(Had (M )) ♥➔♦ ✤â✳ ❇➙② ❣✐í ❞♦ d = cd(a, R/q) cd(a, R/q) dimR/q dimR/p dimR = d, dimR t❤➻ p q ✈ỵ✐ ✷✺ t❛ ❝â p = q✳ ✣✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ Att(Had (M )) ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ♠ỉ✤✉♥ sû Ass(M ) = {p1 , , pn }✳ Had (M ) ❧➔ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝✳ ●✐↔ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ d✳ t❤❡♦ ◆➳✉ d=1 t❤➻ d > 1✳ ●✐↔ sû ❝❤♦ Ass(M )✳ Had (M ) ❱ỵ✐ ♠é✐ Ass(Li ) = {pi } ✲ t❤ù ❝➜♣✳ i d ❣✐↔ sû Li ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ Ass(M/Li ) = Ass(M )\{pi }✳ ✈➔ ð tr➯♥ t❛ ❝â Att(Had (Li )) Att(Had (Li )) ❧➔ {pi } p1 ❧➔ {pi } Att(Had (Li )) ✈➔ M s❛♦ ◆❤÷ ✤➣ ❝❤ù♥❣ ♠✐♥❤ Ass(M )\{pi }✳ ❑❤✐ ✤â✱ ✲ t❤ù ❝➜♣ ❤♦➦❝ ❦❤ỉ♥❣✳ ❳➨t ❞➣② ❦❤ỵ♣ ϕi Had (Li ) − → Had (M ) → Had (M/Li ) → 0, ϕi (Had (Li )) ◆➳✉ {pi } ✲ t❤ù ❝➜♣ ❤♦➦❝ ❦❤æ♥❣✳ ϕi (Had (Li )) = ✈ỵ✐ ♠ët sè i ♥➔♦ ✤â t❤➻ Had (N ) = 0✳ ❱➻ ✈➟② Had (M ) ∼ = Had (M/Li ) t❤✐➳t ❧➔ s➩ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✳ ❱➻ ✈➟② t❛ ❝â t❤➸ ❣✐↔ ϕi (Had (Li )) = ✈ỵ✐ ♠å✐ i✳ ❉♦ ✤â t❛ ❝â n Att((Had (M ))/ ϕi (Had (Li )) i=1 n Att(Had (M ))/ϕi (Had (Li ))) i=1 n Att(Had (M/Li )) = Ø = i=1 ❱➻ ✈➟② Had (M ) = ✷✳✸✳✷ ❍➺ q✉↔✳ ●✐↔ sû ❦❤→❝ ❤æ♥❣❦ ✭❦❤æ♥❣ ♥❤➜t t❤✐➳ ❤ú✉ ↕♥❤ s✐♥❤✮ s❛♦ ❝❤♦ ❑❤✐ ✤â Hmd (M ) ❧➔ ✤✉♥ ♠æ ❜✐➸✉ ❞✐➵♥ ❝ ✤÷đ ✈➔ n i=1 ϕi Had (Li ) (R, m) ❧➔ ✈➔♥❤ ❡t❤r ◆♦ ✤à❛ ♣❤÷ì♥❣✱ Att((Hmd (M )) ✳ M ❧➔ R ✲ ✤✉♥ ♠æ dim M = dim R = d✳ {p ∈ Ass(M )|dimR/p = d} ✷✻ ❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ cd(m, R/p) = dim(R/p) ✈➔ ỵ s r r sỹ tr ỵ t ❱➼ ❞ö✳ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ữỡ ợ dimR = d > ởt ♥❣✉②➯♥ tè cd(m, R/p) = d ✈➔ p ❝õ❛ R Hmd (E(R/p)) = 0✳ s❛♦ ❝❤♦ dimR/p = d✳ ❑❤✐ ✤â t❛ ❝â ✷✼ ❑➌❚ ▲❯❾◆ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ▲✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✻❪ ❝õ❛ ▼✳ ■✳ ❉✐❜❛❡✐ ✈➔ ❙✳ ❨❛ss❡♠✐✳ ❉ü❛ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ▲✉➟♥ ✈➠♥ ✤➣ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ♥❤ú♥❣ ✈➜♥ ✤➲ s r ự ỵ t tố t ổ ố ỗ ữỡ t ợ ỹ ỵ ỵ t q ❝❤➼♥❤ tr♦♥❣ ❜➔✐ ❜→♦ ❬✽❪ ❝õ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✈➔ ❘✳ ❨✳ ❙❤❛r♣✳ ✷✳ ❚r➻♥❤ ❜➔② ❦➳t q✉↔ ❝õ❛ ❬✻❪ rở ỵ õ tr t tố t ổ ố ỗ ữỡ t ợ tũ þ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♠ỉ✤✉♥ ❜❛♥ ✤➛✉ ❧➔ ❤ú✉ ❤↕♥ s ổ t tt ỳ s ỵ ỵ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚ü ❈÷í♥❣ ✭✷✵✵✸✮✱ ●✐→♦ tr➻♥❤ ✣↕✐ sè ❤✐➺♥ ✤↕✐✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ữỡ ố t ỡ s ỵ tt ổ ◆❳❇ ✣↕✐ ❤å❝ s÷ ♣❤↕♠✳ ❬✸❪ ❉÷ì♥❣ ◗✉è❝ ❱✐➺t ✭✷✵✵✽✮✱ ỵ tt sữ ❬✹❪ ▼✳ ❋✳ ❆t✐②❛❤ ❛♥❞ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✭✶✾✻✾✮✱ t✐✈❡ ❆❧❣❡❜r❛✱ ❘❡❛❞✐♥❣✱ ▼❛ss✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♠♠✉t❛✲ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ❛♥ ❛♥❣❡✲ ❜r❛✐❝ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡ ✉♥✐✈❡rs✐t② ❬✺❪ ▼✳ P✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✾✽✮✱ ♣r❡❡s✳ ❬✻❪ ■✳ ●✳ ❉✐❜❛❡✐ ❛♥❞ ❙✳ ❨❛ss❡♠✐ ✭✷✵✵✺✮✱ ❆tt❛❝❤❡❞ ♣r✐♠❡s ♦❢ t❤❡r t♦♣ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥ ✐❞❡❛❧✱ ❆r❝❤✳ ▼❛t❤✳ ✽✹✱ ✷✾✷✲ ✷✾✼✳ ❬✼❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✭✶✾✼✸✮✱ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣✱ ▼❛t❤❡♠❛t✐❝❛ ✶✶✱ ✷✸✲✹✸✳ ✷✾ ❬✽❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✼✷✮✱ ❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ ♥♦♥✈❛♥✐s❤✐♥❣ ♦❢ ❝❡rt❛✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❖①❢♦r❞ ✷✸✱ ✶✾✼✲✷✵✹✳ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ... ❝→❝ tố t ổ ố ỗ ✤à❛ ♣❤÷ì♥❣ ❝➜♣ ❝❛♦ ♥❤➜t✳ ▼ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✻❪ ❝õ❛ ▼✳ ■✳ ❉✐❜❛❡✐ ✈➔ ❙✳ ❨❛ss❡♠✐✳ t trữợ t ú tổ tr ự ỵ t tố t... ỵ tr tố p) ( ( m q∈Ass(M ) d−1 p∈Att(Hm (M ))) ❱➻ ✈➟②✱ tỗ t r1 Hmd1 (M ) r1 m q) ổ ữợ ổ tr t ✈ỵ✐ Att(Hmd−1 (M ))✳ ❈❤♦ p1 , , pt M Hmd−1 (M/r1 M ) = 0✳ ❧➔ t ổ ụ t ỵ tr tố tỗ t... s❛✉✿ ✶✳ ❚r➻♥❤ ự ỵ t tố t ổ ố ỗ ữỡ t ợ ỹ ỵ ỵ t q tr ❜→♦ ❬✽❪ ❝õ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✈➔ ❘✳ ❨✳ ❙❤❛r♣✳ ✷✳ ❚r➻♥❤ ❜➔② ❦➳t q✉↔ ❝õ❛ ❬✻❪ ❧➔ ♠ð rë♥❣ ỵ õ tr t tố t ổ ố ỗ ữỡ t ợ