Tính minimax và tính cofinite của môđun đối đồng điều địa phương

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❚❘❺◆ ❚❍➚ ❚❍❯ ❍❖⑨■ ❚➑◆❍ ▼■◆■▼❆❳ ❱⑨ ❚➑◆❍ ❈❖❋■◆■❚❊ ❈Õ❆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❚❘❺◆ ❚❍➚ ❚❍❯ ❍❖⑨■ ❚➑◆❍ ▼■◆■▼❆❳ ❱⑨ ❚➑◆❍ ❈❖❋■◆■❚❊ ❈Õ❆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ Pì số ỵ tt số sè✿ ✽ ✹✻ ✵✶ ✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ữợ P ❍♦➔♥❣ ❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✶✽ ✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trò♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr ữủ ró ỗ ố ◆❣✉②➯♥✱ ♥❣➔② ✶✻ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❚r➛♥ ❚❤à ❚❤✉ ❍♦➔✐ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❝❤✉②➯♥ ♠ỉ♥ ữợ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ✈➔♦ t❤→♥❣ ữợ sỹ ữợ P ❍♦➔♥❣✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ t ỡ s s tợ t ỳ qỵ ❣✐→ tø tr❛♥❣ ❣✐➜② ✈➔ ❝↔ ♥❤ú♥❣ ❜➔✐ ❤å❝ tr♦♥❣ ❝✉ë❝ sè♥❣ t❤➛② ❞↕② ❣✐ó♣ tỉ✐ tü t✐♥ ❤ì♥ ✈➔ tr÷ð♥❣ t❤➔♥❤ ❤ì♥ ♥❤✐➲✉✳ ❚ỉ✐ ①✐♥ ❝↔♠ ì♥ P❤á♥❣ ✣➔♦ ❚↕♦ ✲ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ❚❤→✐ ♥❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ sỵ♠ ❦❤â❛ ❤å❝✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tỵ✐ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ð ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✈➔ ❝→❝ t❤➛② ð ❱✐➺♥ t♦→♥ ✈ỵ✐ ♥❤ú♥❣ ❜➔✐ ❣✐↔♥❣ ✤➛② ♥❤✐➺t t❤➔♥❤ ✈➔ t➙♠ ❤✉②➳t✱ ①✐♥ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ✤➣ ❧✉ỉ♥ q✉❛♥ t➙♠ ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ t❤❛♠ ❣✐❛ sr ợ ữỡ tr ❚ỉ✐ ①✐♥ ❝↔♠ ì♥ t➜t ❝↔ ❝→❝ ❛♥❤✱ ❡♠ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï tỉ✐ ♥❤✐➺t t➻♥❤ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ữủ ỷ ỡ tợ tt t tr ❣✐❛ ✤➻♥❤ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ✤÷đ❝ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ✐✐✐ ▼ö❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ▼æ✤✉♥ ◆♦❡t❤❡r ✈➔ ▼æ✤✉♥ ❆rt✐♥ ✳ ✶✳✸ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ▼æ✤✉♥ Ext ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ố ỗ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ✐✐✐ ✶ ✺ ✳ ✺ ✳ ✻ ✳ ✽ ✳ ✶✵ ✳ ✶✷ ❈❤÷ì♥❣ ✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ t ổ ố ỗ ✤à❛ ♣❤÷ì♥❣ ✶✺ ✷✳✶ ▼ỉ✤✉♥ ♠✐♥✐♠❛① ✈➔ ♠ỉ✤✉♥ ❝♦❢✐♥✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ♠ët ✈➔ t➼♥❤ ❝❤➜t ♠✐♥✐♠❛① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ❈❤÷ì♥❣ ✸ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✷ ✈➔ t➼♥❤ ▲❛s❦❡r ②➳✉ ✷✼ ✸✳✶ ▼æ✤✉♥ ▲❛s❦❡r ②➳✉ ✈➔ ♠æ✤✉♥ ❝♦❢✐♥✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ❤❛✐ ✈➔ t➼♥❤ ❝❤➜t ▲❛s❦❡r ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✐✈ ▼ð ✤➛✉ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✭❝â ✤ì♥ ✈à✮✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R ✲ ♠æ✤✉♥ ❦❤→❝ 0✳ ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠ i ❝❤♦ trữợ t õ ổ ố ỗ ữỡ tự i M ố ợ I ữủ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✭①❡♠ ❬✶✶❪ ❤♦➦❝ ❬✽❪✮ ♥❤÷ s❛✉✿ i n HIi (M ) = − lim → ExtR (R/I , M ) n≥1 ❈→❝ t➼♥❤ ❝❤➜t ❝ì ợ ổ ố ỗ ữỡ õ t t tr ố s ởt ỵ q trồ tr ố ỗ ữỡ ỵ ữỡ t ỳ ổ ố ỗ ữỡ ỵ ts t ợ ởt i số ữỡ r ❝→❝ Rp✲♠ỉ✤✉♥ HIR (Mp ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i ≤ r ✈➔ ♠å✐ p ∈ Spec R ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ R✲♠æ✤✉♥ HIi (M ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i ≤ r✧✳ ❈â ♠ët tr t ỵ ✤à❛ ♣❤÷ì♥❣ ✲ t♦➔♥ ❝ư❝ ❝õ❛ ❋❛❧t✐♥❣s ♠➔ t❛ q✉❛♥ t➙♠ ð ✤➙②✱ ❧✐➯♥ q✉❛♥ ✤➳♥ sü ❦❤→✐ q✉→t ❤â❛ ❝❤✐➲✉ ❤ú✉ ❤↕♥ fI (M ) ❝õ❛ M ✤è✐ ✈ỵ✐ I ✱ tr♦♥❣ ✤â p fI (M ) := inf{i ∈ N | HIi (M ) ❦❤æ♥❣ ❧➔ ❤ú✉ ❤↕♥ s}, t q ữợ r inf() = ❑❤✐ ✤â :R HIi (M ) } fI (M ) := inf{i ∈ N | I = inf{i ∈ N | I n HIi (M ) = ✶ ợ n N}; ỗ tớ ú õ ỵ ữỡ t ts ữủ ❝❤♦ ð ❝æ♥❣ t❤ù❝ s❛✉ ✤➙②✿ fI (M ) = inf{fIRp (Mp ) | p ∈ Spec R} = inf{fIRp (Mp ) | p ∈ Supp(M/IM ) ✈➔ dim R/ p 0}, ỵ r❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤➾ sè ✤➛✉ t✐➯♥ ♠➔ ổ ố ỗ ữỡ ợ ✐✤➯❛♥ ❜➜t ❦➻ ❦❤æ♥❣ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ❝❤➾ sè õ ổ ố ỗ q ✤à❛ ♣❤÷ì♥❣ ❤â❛ t↕✐ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè tr➯♥ ✈➔♥❤ ❝ì sð✳ ◆➠♠ ✷✵✶✸✱ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲❙❡❞❣❤✐ ✭①❡♠ ❬✹❪✮ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ❝❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ n ❝õ❛ M ✤è✐ ợ I fIn(M ) ữủ ❜ð✐ ❝æ♥❣ t❤ù❝✿ fIn (M ) = inf{fIRp (Mp ) | p ∈ Supp(M/IM ) ✈➔ dim(R/ p) ≥ n} ú ỵ r fIn(M ) số ữỡ ❤♦➦❝ ❧➔ ∞ ✈➔ t❛ ❝â fI0(M ) = fI (M )✳ ❚ø ✤â ♠ët ❝➙✉ ❤ä✐ tü ♥❤✐➯♥ ✤÷đ❝ ✤➦t r❛ ❧➔ t➻♠ ❤✐➸✉ t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ✤è✐ ỗ ữỡ ợ ỳ ❜➟❝ ✷ ❝õ❛ M ✤è✐ ✈ỵ✐ I ✳ ❈❤➥♥❣ ❤↕♥ ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ✤➙② fI1 (M ) = inf{i ∈ N | HIi (M ) ❦❤æ♥❣ ❧➔ ♠✐♥✐♠❛①} ✈➔ fI2 (M ) = inf{i ∈ N | HIi (M ) ❦❤ỉ♥❣ ❧➔ ▲❛s❦❡r ②➳✉} ❝â ✤ó♥❣ ❤❛② ❦❤ỉ♥❣❄ ❑➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲❙❡❞❣❤✐ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪ ❧➔ tr↔ ❧í✐ ❝❤♦ ❤❛✐ ❝➙✉ ❤ä✐ tr➯♥✳ ❈ư t❤➸ ❦➳t q✉↔ t❤ù ♥❤➜t ❝õ❛ ❤å ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ sè ♥❣✉②➯♥ i ♥❤ä ♥❤➜t ✤➸ HIi (M ) ❦❤æ♥❣ ❧➔ ổ ợ số fI1(M ) ỵ ✷✳✷✳✽✮❀ ❦➳t q✉↔ ❝❤➼♥❤ t❤ù ❤❛✐ ❝õ❛ ❤å ✷ ❧➔ ❝❤➾ r❛ r➡♥❣ sè ♥❣✉②➯♥ i ♥❤ä ♥❤➜t s❛♦ ❝❤♦ HIi (M ) ❦❤ỉ♥❣ ❧➔ ♠ỉ✤✉♥ ▲❛s❦❡r ②➳✉ ❜➡♥❣ ✈ỵ✐ fI2(M ) ❦❤✐ R ❧➔ ✈➔♥❤ ♥û❛ ✤à❛ ♣❤÷ì♥❣ ✭①❡♠ ỵ ổ ự t q tự t tr ỵ s ỵ ỵ ✈➔♥❤ ◆♦❡t❤❡r✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â R✲♠æ✤✉♥ HIi (M ) ❧➔ ♠✐♥✐♠❛① ✈➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < fI1(M ) ✈➔ HIf (M )(M ) ❦❤æ♥❣ ❧➔ ♠✐♥✐♠❛①✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ♠é✐ ♠ỉ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① N ❝õ❛ HIf (M )(M )✱ t❤➻ R✲♠æ✤✉♥ f (M ) HomR (R/I, HI (M )/N ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ R I I I ❑❤→✐ ♥✐➺♠ ♠æ✤✉♥ I t tr ỵ tr ữủ ợ t ❍❛rts❤♦r♥❡ ♥➠♠ ✶✾✼✵ ✭①❡♠ ❬✶✷❪✮ ✈➔ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ R✲♠ỉ✤✉♥ M ✤÷đ❝ ❣å✐ ❧➔ I ✲❝♦❢✐♥✐t❡ ♥➳✉ Supp(M ) ⊆ V (I) ✈➔ ExtiR(R/I, M ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i ≥ 0✳ ▼ët tr♦♥❣ ❝→❝ ❝ỉ♥❣ ❝ư ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ t❤ù ❤❛✐ rr ỵ ữợ ỵ ỵ R tr I ❧➔ ✐✤➯❛♥ ❝õ❛ R✱ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ t ≥ ❧➔ ♠ët sè ♥❣✉②➯♥ s❛♦ ❝❤♦ ❝→❝ R✲♠æ✤✉♥ HI0 (M ), , HIt−1 (M ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✤à❛ ♣❤÷ì♥❣ ✈ỵ✐ ♠å✐ p ∈ Supp(M/IM ) ♠➔ dim(R/p) > 1✳ ❑❤✐ ✤â✱ ❝→❝ R✲♠æ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i ≤ t ✈➔ R✲♠ỉ✤✉♥ HomR(R/I, HIt (M )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❚ø ♥❤ú♥❣ ❦➳t q✉↔ tr➯♥ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲❙❡❞❣❤✐ ❬✹❪ ✤➣ ✤÷❛ r❛ ❝→❝ ❤➺ q✉↔ ❝õ❛ ✤à♥❤ ỵ õ ởt số rở ❦➳t q✉↔ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲ ◆❛❣❤✐♣♦✉r tr♦♥❣ ❬✼❪✱ ❉❡❧❢✐♥♦✲▼❛r❧❡② tr♦♥❣ ❬✾❪ ✈➔ ❑✳ ■✳ ❨♦s❤✐❞❛ tr♦♥❣ ❬✶✾❪ ✤è✐ ✈ỵ✐ ♠ët ✈➔♥❤ tr tũ ỵ ỵ ỵ ✶✳✸❪ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r✱ ■ ❧➔ ✐✤➯❛♥ ❝õ❛ R✱ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ s❛♦ ❝❤♦ dim(M/IM ) ≤ 1✳ ❑❤✐ ✤â R✲♠æ✤✉♥ HIt (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥✳ ▼ët ❦➳t q✉↔ ❝❤➼♥❤ ❦❤→❝ ♥ú❛ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪ ✤â ❧➔✿ ◆➳✉ (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r ✤➛② ✤õ✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ R✲♠æ✤✉♥ ExtjR(R/I, HIi (M )) ❧➔ ▲❛s❦❡r ②➳✉ ✈ỵ✐ ♠å✐ i < fI3 (M ) ✈➔ ✈ỵ✐ ♠å✐ j ≥ 0✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ♠é✐ ♠æ✤✉♥ ❝♦♥ ▲❛s❦❡r ②➳✉ N ❝õ❛ f (M ) f (M ) HI (M )✱ t❤➻ t❛ ❝â R✲♠æ✤✉♥ HomR (R/I, HI (M )/N ) ❝ô♥❣ ❧➔ ▲❛s❦❡r ②➳✉ ✭①❡♠ ỵ ứ t q ự t❤✉ ✤÷đ❝ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲ ❙❡❞❣❤✐ ♥❤÷ tr➯♥ ✤➙②✱ ✤➲✉ ✤÷❛ ✤➳♥ ❜➔✐ t♦→♥ ①❡♠ ①➨t ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤➸ ❝❤♦ t➟♣ ❤ñ♣ AssR(HIi (M )) ❧➔ ❤ú✉ ❤↕♥ ❦❤✐ i = fIj (M ) ✭❝❤➥♥❣ ❤↕♥ ✈ỵ✐ j = 1, 2, 3✮✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ♥❤÷ ✤➣ ♥➯✉ tr➯♥✱ ❝→❝ ❦✐➳♥ t❤ù❝ ♥➔② ❞ü❛ tr➯♥ ❜➔✐ ❜→♦ ❝❤➼♥❤ ❧➔ ❜➔✐ ❜→♦ ❬✹❪✿ ❑✳ ❇❛❤♠❛♥♣♦✉r✱ ❘✳ ◆❛❣❤✐♣♦✉r ❛♥❞ ▼✳ ❙❡❞❣❤✐✱ ▼✐♥✐♠❛①♥❡ss ❛♥❞ ❈♦❢✐♥✐t❡ ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ❱♦❧✳ ✹✶ ✭✷✵✶✸✮✱ P♣✳ ✷✼✾✾✲✷✽✶✹✳ ✭❉❖■✿ ✶✵✳ ✶✵✽✵✴✵✵✾✷✼✽✼✷✳✷✵✶✷✳✻✻✷✼✵✾✮✳ ❇➯♥ ❝↕♥❤ ✤â ✤➸ ✈✐➺❝ tr➻♥❤ ❜➔② ✤÷đ❝ ró ỵ ỡ t t❤➯♠ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ð ❜➔✐ ❜→♦ ❬✺❪✱ ❬✻❪✱ ❬✼❪✱ ❬✶✼❪✱✳ ✳ ✳ ❀ ✈➔ ❝→❝ ❝✉è♥ s→❝❤ ❬✽❪ ✈➔ ❬✶✺❪✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❜è ❝ư❝ ❧➔♠ ❜❛ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ t❤✐➳t ✤➸ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❝❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✶ ❝õ❛ ♠ỉ✤✉♥ M ✤è✐ ✈ỵ✐ ✐✤➯❛♥ I tr♦♥❣ ố ợ t t ổ ữỡ ✸ ❝õ❛ ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ✈➲ ❝❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✷ ❝õ❛ M ✤è✐ ✈ỵ✐ ✐✤➯❛♥ I ✈➔ t➼♥❤ ❝❤➜t ▲❛s❦❡r ②➳✉ ❝õ❛ ♠æ✤✉♥✳ I I ✹ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à Ð ❝❤÷ì♥❣ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❈→❝ ❦✐➳♥ t❤ù❝ ð ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ ✈➔♦ ❝→❝ ❝✉è♥ s→❝❤ ❬✽❪ ✈➔ ❬✶✺❪✳ ✶✳✶ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ✭■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✮✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥✱ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â p ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ♥➳✉ tỗ t ởt tỷ = x M s❛♦ ❝❤♦ AnnR(x) = p✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ AssR(M ) ❤♦➦❝ Ass(M )✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭✣❛ t↕♣ ❝õ❛ ✐✤➯❛♥✮✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✱ ❦❤✐ ✤â ✤❛ t↕♣ ❝õ❛ I ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ V (I) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ V (I) = {p ∈ Spec(R) | I ⊆ p} ▼➺♥❤ ✤➲ ✶✳✶✳✸✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥ ✈➔ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â t❛ ❝â ✐✮ AssR(0 :M I) = AssR(M ) ∩ V (I)✳ ✐✐✮ AssR(M/(0 :M I)) ⊆ AssR(M )✳ ✺ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû a ❝❤➾ ❝❤ù❛ tỷ ữợ ổ tr M ❑❤✐ ✤â a ⊆ p∈Ass M p ✈➔ ✈➻ M ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ♥➯♥ Ass(M ) ❤ú✉ ❤↕♥✳ ❍ì♥ ỳ t ỵ tr tố tỗ t p ∈ Ass(M ) s❛♦ ❝❤♦ a ⊆ p✳ ❱➻ M ❝â ♠ët ♠æ✤✉♥ ❝♦♥ ❝â ❧✐♥❤ ❤♦→♥ tû ❧➔ p ♥➯♥ s✉② r❛ (0 :M a) = 0✳ ◆❤÷ ✈➟②✱ Γa(M ) = ✭♠➙✉ t❤✉➝♥✮✳ ❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❚✐➳♣ t❤❡♦ t❛ ♥❤➢❝ ❧↕✐ ♠ët ❦➳t q✉↔ ❝õ❛ ❚✳ ❑❛✇❛s❛❦✐✳ ▼➺♥❤ ✤➲ ✸✳✶✳✸✳ ✭❬✶✸✱ ❇ê ✤➲ ✶❪✮ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✱ ✈➔ p ❧➔ ♠ët sè ♥❣✉②➯♥ ❦❤ỉ♥❣ ➙♠✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ R✲♠æ✤✉♥ T ✱ t❛ ❝â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✐✮ ExtiR(R/I, T ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i ≤ p✳ ✐✐✮ ❱ỵ✐ ♠é✐ P ∈ min(R/I) t❤➻ ExtiR(R/P, T ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i ≤ p✳ ✐✐✐✮ ❱ỵ✐ ♠å✐ R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ N ❝â Supp(N ) ⊆ V (I)✱ t❤➻ ExtiR(N, T ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i ≤ p✳ ✭tr♦♥❣ ✤â min(R/I) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t✐➸✉ ❝õ❛ I ✮✳ ▼➺♥❤ ✤➲ ✸✳✶✳✹✳ ✭❬✼✱ ❇ê ✤➲ ✷✳✹✱ ✷✳✺❪✮ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ A ❧➔ ♠ët R✲♠ỉ✤✉♥ ❆rt✐♥✳ ✐✮ ●✐↔ sû x ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ m s❛♦ ❝❤♦ V (Rx) ∩ AttR A ⊆ {m}✳ ❑❤✐ ✤â R✲♠æ✤✉♥ A/xA ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳ ✐✐✮ ●✐↔ sû I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R s❛♦ ❝❤♦ R✲♠æ✤✉♥ HomR(R/I, A) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â V (I) AttR A V (m) ữợ ởt ♠➺♥❤ ✤➲ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲❙❡❞❣❤✐✳ ▼➺♥❤ ✤➲ ✸✳✶✳✺✳ ✭❬✹✱ ▼➺♥❤ ✤➲ ✸✳✶❪✮ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✈➔ I ❧➔ ✷✽ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ s❛♦ ❝❤♦ (HIi (M ))p ❧➔ Rp✲ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i < t ✈➔ ♠å✐ p ∈ Supp(M/IM ) ✈ỵ✐ dim R/p > 1✱ tr♦♥❣ ✤â t ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â HomR(R/I, HIt (M )) ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ HIi (M ) ❧➔ R✲♠ỉ✤✉♥ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ t✳ ◆➳✉ t = 1✱ ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ ❞♦ ▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ●✐↔ sû ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ ✈ỵ✐ t − ✭✈ỵ✐ t > 1✮✳ ❚❤❛② M ❜ð✐ M/ΓI (M )✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ M ✭❦❤→❝ ❦❤æ♥❣ ✈➔ ❤ú✉ ❤↕♥ s✐♥❤✮ ❧➔ ♠ët R✲♠æ✤✉♥ I ✲①♦➢♥ tü ❞♦✳ ❑❤✐ ✤â t❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✳✷ s✉② r❛ I p∈Ass M p✳ ❚✐➳♣ t❤❡♦ ✈ỵ✐ ♠å✐ n ∈ N ✈➔ ♠å✐ ≤ i < t✱ t❛ ✤➦t R Hi,n = (0 :HIi (M ) I n ) ❑❤✐ ✤â tø ❣✐↔ t❤✐➳t q✉② ♥↕♣ ✈➔ ▼➺♥❤ ✤➲ ✸✳✶✳✸ t❛ ❝â R✲♠æ✤✉♥ Hi,n ỳ s õ tỗ t số k s❛♦ ❝❤♦ Supp(Hi,n+1 /Hi,n ) = Supp(Hi,k+1 /Hi,k ) ✈ỵ✐ ♠å✐ ✈➔ ♠å✐ dim(R/p) > 1✳ ❍ì♥ ♥ú❛✱ ✈➻ n ≥ k+1 i < t✳ ❚❛ ❧➜② p ∈ Supp(Hi,k+1 /Hi,k ) s❛♦ ❝❤♦ Supp(Hi,k+1 /Hi,k ) ⊆ Supp(M/IM ) ♥➯♥ s✉② r❛ p ∈ Supp(M/IM ) ✈➔ t❤❡♦ ❣✐↔ t❤✐➳t t❛ t❤✉ ✤÷đ❝ (HIi (M ))p ❧➔ Rp ổ ỳ s õ tỗ t ởt ổ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ L ❝õ❛ HIi (M ) s❛♦ ❝❤♦ (HIi (M ))p = Lp ✳ ▼➦t ❦❤→❝ ✈➻ I (L) = L s r tỗ t số ♥❣✉②➯♥ n ≥ k + s❛♦ ❝❤♦ I nL = 0✳ ❑❤✐ ✤â Lp = (HIi (M ))p ⊇ (Hi,n+1 )p ⊇ (Hi,n )p ⊇ Lp ✷✾ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ (Hi,n+1/Hi,n)p = 0✱ tù❝ ❧➔ p ∈/ Supp(Hi,n+1/Hi,n) ✭♠➙✉ t❤✉➝♥✮✳ ❉♦ ✤â ✈ỵ✐ ♠å✐ p ∈ Supp(Hi,k+1/Hi,k ) t❛ ❝â dim R/p ≤ 1✳ ❇ð✐ ✈➻ i ∪∞ n=1 Hi,n = HI (M ) ♥➯♥ t❛ ❞➵ ❞➔♥❣ t❤➜② Supp(HIi (M )/Hi,k ) = Supp(Hi,k+1 /Hi,k ) ❑➨♦ t❤❡♦ Supp(HIi (M )/Hi,k ) ⊆ {p ∈ Spec(R) | dim(R/p) ≤ 1} ❚❛ ❧↕✐ ❝â R✲♠æ✤✉♥ HomR(R/I, HIi (M )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i < t ♥➯♥ tø ❞➣② ❦❤ỵ♣ HomR (R/I, HIi (M )) → HomR (R/I, HIi (M )/Hi,k ) → Ext1R (R/I, Hi,k ) s✉② r❛ R✲♠æ✤✉♥ HomR(R/I, HIi (M )/Hi,k ) ❝ơ♥❣ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i < t✳ ❚✐➳♣ t❤❡♦ t❛ ❝â ΓI (HIi (M )/Hi,k ) = HIi (M )/Hi,k ♥➯♥ s✉② r❛ t➟♣ AssR HIi (M )/Hi,k ❧➔ ❤ú✉ ❤↕♥✳ ❚❛ ✤➦t Ti = p ∈ Supp(HIi (M )/Hi,k ) | dim R/p = ❑❤✐ ✤â Ti ⊆ AssR HIi (M )/Hi,k ✈➔ t➟♣ T = ∪t−1 i=0 Ti ❧➔ ❤ú✉ ❤↕♥✳ ●✐↔ sû T = {p1, , pl }✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✳✻✱ t❛ t❤➜② Rp ✲♠æ✤✉♥ (HIi (M ))p ❧➔ ♠✐♥✐♠❛① ✈ỵ✐ ♠å✐ j = 1, , l ✈➔ ♠å✐ i = 0, , t õ tỗ t ♠ët ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ Li,j ❝õ❛ HIi (M ) s❛♦ ❝❤♦ Rp ✲♠æ✤✉♥ (HIi (M )/Li,j )p ❧➔ ❆rt✐♥✳ ❚✐➳♣ t❤❡♦ t❛ ✤➦t j j j j Li = Li,1 + + Li,l + Li,k ❱ỵ✐ ♠å✐ p ∈ T ✱ t❛ t❤➜② Rp✲♠ỉ✤✉♥ (HIi (M )/Li)p ❧➔ ❆rt✐♥✳ ▼➦t ❦❤→❝✱ tø ❞➣② ❦❤ỵ♣ HomR (R/I, HIi (M )) → HomR (R/I, HIi (M )/Li ) → Ext1R (R/I, Li ) ✸✵ ❦➨♦ t❤❡♦ R✲♠æ✤✉♥ HomR(R/I, HIi (M )/Li) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ❞♦ ✤â s✉② r❛ ✈ỵ✐ ♠å✐ p ∈ T t❤➻ Rp✲♠ỉ✤✉♥ HomR (Rp/IRp, (HIi (M )/Li)p) ❝ơ♥❣ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✸✳✶✳✹✱ t❛ ❝â p AttRp ((HIi (M )/Li )p ) ∩ V (IRp ) ⊆ {pRp } ✣➦t l i S = ∪t−1 i=0 ∪j=1 q ∈ Spec R | qRpj ∈ AttRpj (HI (M )/Li )pj ❑❤✐ ✤â S ∩ V (I) ⊆ T t tỗ t tỷ x I s❛♦ ❝❤♦ x∈ / (∪q∈S\V (I) q) ∪ (∪p∈AssR M p) ❚❛ ①➨t ❞➣② ❦❤ỵ♣ x 0→M → − M → M/xM → 0, ❦❤✐ ✤â ♥â ❝↔♠ s✐♥❤ r❛ ♠ët ❞➣② ❦❤ỵ♣ ❞➔✐ x x → HIi (M ) → − HIi (M ) → HIi (M/xM ) → HIi+1 (M ) → − HIi+1 (M ) → ❉♦ ✤â ✈ỵ✐ ♠å✐ i ≥ 0✱ t❛ s✉② r❛ ❞➣② ❦❤ỵ♣ ♥❣➢♥ s❛✉ → HIi (M )/xHIi (M ) → HIi (M/xM ) → (0 :HIi+1 (M ) x) → ✭✯✮ ❱➻ Supp(M/xM/I(M/xM )) = Supp(M/IM ) ♥➯♥ tø ❞➣② ❦❤ỵ♣ tr➯♥ ✈➔ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ s✉② r❛ r➡♥❣ ❝→❝ R✲♠æ✤✉♥ HI0 (M/xM ), HI1 (M/xM ), , HIt−2 (M/xM ) ❧➔ ■✲❝♦❢✐♥✐t❡ ✈➔ R✲♠ỉ✤✉♥ HomR(R/I, HIt−1(M/xM )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❱ỵ✐ ♠å✐ i < t✱ ✈➻ (xHIi (M ) + Li)/xHIi (M ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝õ❛ HIi (M )/xHIi (M ) t tỗ t ởt ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ Ni ❝õ❛ HIi (M/xM ) s❛♦ ❝❤♦ ❞➣② s❛✉ ❧➔ ❦❤ỵ♣ → HIi (M )/(Li + xHIi (M )) → HIi (M/xM )/Ni → (0 :HIi+1 (M ) x) → ✸✶ ❚❛ ✤➦t Ui = HIi (M )/(Li + xHIi (M )) ✈➔ V (i) = HIi (M/xM )/Ni✳ ❑❤✐ ✤â✱ tø ▼➺♥❤ ✤➲ ✸✳✶✳✹ t❛ ❞➵ ❞➔♥❣ t❤➜② r➡♥❣ Rp ✲♠æ✤✉♥ (Ui)p ❧➔ ❤ú✉ ❤↕♥ ✈ỵ✐ ♠å✐ j = 1, , l ú ỵ r x / qS\V (I) q õ tỗ t ởt Rổ Bij ❝õ❛ Ui s❛♦ ❝❤♦ (Ui)p = (Bij )p ✳ ✣➦t Bi = Bi1 + + Bil ❑❤✐ ✤â Bi ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝õ❛ Ui s❛♦ ❝❤♦ j j j j SuppR Ui /Bi ⊆ Supp(HIi (M )/Ki )\T ⊆ Max R ▼➦t ❦❤→❝ ❞➣② ❦❤ỵ♣ → Ni → HIi (M/xM ) → Vi → ❝↔♠ s✐♥❤ r❛ ❞➣② ❦❤ỵ♣ s❛✉ HomR (R/I, HIi (M/xM )) → HomR (R/I, Vi ) → Ext1R (R/I, Ni ) ✈ỵ✐ ♠å✐ i < t✳ ❚ø ✤â s✉② r❛ R✲♠æ✤✉♥ HomR(R/I, Vi) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i < t✳ ❉♦ ✤â✱ ✈➻ ❞➣② → HomR (R/I, Ui ) → HomR (R/I, Vi ) ❧➔ ❞➣② ❦❤ỵ♣✱ ♥➯♥ R✲♠ỉ✤✉♥ HomR(R/I, Ui) ❝ơ♥❣ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❚ø ✤â s✉② r❛ R✲♠æ✤✉♥ HomR(R/I, Ui/Bi) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❍ì♥ ♥ú❛ ✈➻ Supp(Ui/Bi) ⊆ Max R ♥➯♥ R✲♠æ✤✉♥ HomR (R/I, Ui /Bi ) ❧➔ ❆rt✐♥ ✈ỵ✐ ♠å✐ i < t✳ ❱➻ Ui /Bi ❧➔ I ✲①♦➢♥✱ ♥➯♥ t❤❡♦ ▼❡❧❦❡rss♦♥ ✭❇ê ✤➲ ✷✳✷✳✶✮ t❛ s✉② r❛ Ui /Bi ❧➔ R✲♠æ✤✉♥ ❆rt✐♥✳ ❚ù❝ ❧➔ Ui ❧➔ R✲♠æ✤✉♥ ♠✐♥✐♠❛① ✈ỵ✐ ♠å✐ i < t✳ ▼➦t ❦❤→❝✱ ✈➻ xHIi (M ) + Li/xHIi (M ) ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝õ❛ HIi (M )/xHIi (M ) ✈➔ Ui ∼ = (HIi (M )/xHIi (M ))/((xHIi (M ) + Li )/xHIi (M )) ♥➯♥ t❛ s✉② r❛ R✲♠æ✤✉♥ HIi (M )/xHIi (M ) ❧➔ ♠✐♥✐♠❛① ✈ỵ✐ ♠å✐ i < t✳ ❍ì♥ ♥ú❛ t❤❡♦ ❞➣② ❦❤ỵ♣ ✭✯✮✱ R✲♠ỉ✤✉♥ HomR(R/I, HIi (M )/xHIi (M )) ❝ô♥❣ ❧➔ ❤ú✉ ❤↕♥ ✸✷ s✐♥❤ ✈ỵ✐ ♠å✐ i < t✳ ❉♦ ✈➟② t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✻ t❤➻ R✲♠æ✤✉♥ HIi (M )/xHIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡✳ ❚ø ❞➣② ❦❤ỵ♣ ✭✯✮ ❝ơ♥❣ t❛ s✉② r❛ R✲♠ỉ✤✉♥ (0 :H (M ) x) ❝ơ♥❣ ❧➔ I ✲ ❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < t✳ ✣➦❝ ❜✐➺t✱ t❛ s✉② r❛ r➡♥❣ R✲♠æ✤✉♥ HIt−1(M )/xHIt−1(M ) ❧➔ ♠✐♥✐♠❛① ✈➔ I ✲❝♦❢✐♥✐t❡✳ ▼➦t ❦❤→❝ tø ❞➣② ❦❤ỵ♣ ✭✯✮ ❝ơ♥❣ s✉② r❛ r➡♥❣ HomR (R/I, HIt (M )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❈✉è✐ ❝ò♥❣ t❛ t❤➜② r➡♥❣ ✈➻ R✲♠ỉ✤✉♥ (0 :H (M ) x) ✈➔ HIi (M )/xHIi (M ) ✤➲✉ ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < t ♥➯♥ tø ▼➺♥❤ ✤➲ ✷✳✶✳✺ t❛ s✉② r❛ ✤÷đ❝ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < t✳ i+1 I i I ỵ R ◆♦❡t❤❡r ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛ R✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔ ✤ó♥❣✿ ✐✮ R✲♠ỉ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < fI2(M )✳ ✐✐✮ ❱ỵ✐ ♠å✐ ♠æ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① N ❝õ❛ HIf (M )(M )✱ t❛ ❝â ❝→❝ R✲♠æ✤✉♥ I f (M ) HomR (R/I, HI I (M )/N ) ✈➔ f (M ) Ext1R (R/I, HI I (M )/N ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✱ ♠é✐ ❦❤✐ fI2(M ) ❧➔ sè ❤ú✉ ❤↕♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ fI2(M ) ✭①❡♠ ❧↕✐ ❝æ♥❣ t❤ù❝ ( ) ð ♣❤➛♥ ♠ð ✤➛✉✮✱ t❛ ❝â fI2 (M ) = inf{fIRp (Mp ) | p ∈ Supp(M/IM ), dim R/p ≥ 2} ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ i < fI2(M ) t❤➻ HIi (Mp) ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ p ∈ Supp(M/IM ) ♠➔ dim R/p ≥ 2✳ ⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✸✳✶✳✺ t❛ ❝â HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < fI2 (M ) ✈➔ Hom(R/I, HIt (M )) ❤ú✉ ❤↕♥ s✐♥❤ ợ t = fI2 (M ) ữỡ tỹ ự ỵ q ✸✳✶✳✼✳ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛ R✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ✤➦t s = inf depth(IRp , Mp ) | p ∈ Supp(M/IM ) ✈➔ dim R/p > ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ✤✐➺♥ s❛✉ ❧➔ ✤ó♥❣ ✐✮ R✲♠ỉ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < s✳ ✐✐✮ ❱ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① N ❝õ❛ HIs(M )✱ t❛ ❝â ❝→❝ R✲♠æ✤✉♥ HomR (R/I, HIs (M )/N ) ✈➔ Ext1R (R/I, HIs (M )/N ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✱ ♠é✐ ❦❤✐ s ❧➔ ❤ú✉ ❤↕♥✳ ❈❤ù♥❣ s fI2(M ) ỵ ✸✳✶✳✻ t❛ s✉② r❛ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❚✐➳♣ t❤❡♦ t❛ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✱ ✤â ❧➔ ♠ët ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❇❛❤♠❛♥♣♦✉r ✲ ◆❛❣❤✐♣♦✉r tr♦♥❣ ❬✼❪✱ ✈➔ ✤â ❝ô♥❣ ❧➔ ❦➳t q✉↔ ❝õ❛ ❉❡❧❢✐♥♦ ✲ ▼❛r❧❡② tr♦♥❣ ❬✾❪ ✈➔ ❨♦s❤✐❞❛ tr♦♥❣ ❬✶✾❪ ố ợ tr tũ ỵ q ✸✳✶✳✽✳ ❈❤♦ R✱ I ✱ M ♥❤÷ tr♦♥❣ ❍➺ q✉↔ ✸✳✶✳✼✳ ❈❤♦ t ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦ dim Supp HIi (M ) ≤ ✈ỵ✐ ♠å✐ i < t✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤ó♥❣ ✐✮ R✲♠ỉ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i < t✳ ✐✐✮ ❱ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① N ❝õ❛ HIt (M )✱ t❛ ❝â ❝→❝ R✲♠æ✤✉♥ HomR (R/I, HIt (M )/N ) ✈➔ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ✸✹ Ext1R (R/I, HIt (M )/N ) ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t s = inf depth(IRp , Mp ) | p ∈ Supp(M/IM ) ✈➔ dim R/p > ❱ỵ✐ t ≤ s✱ t❤❡♦ ❦➳t q✉↔ ❝õ❛ ❍➺ q✉↔ ✸✳✶✳✼ t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✸✳✶✳✾✳ ❈❤♦ R✱ I ✱ M ♥❤÷ tr♦♥❣ ❍➺ q✉↔ ✸✳✶✳✼✳ ●✐↔ sû dim(M/IM ) ≤ 1✳ ❑❤✐ ✤â R✲♠æ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ i✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ Supp HIi (M ) ⊆ Supp(M/IM ) ✈➔ dim(M/IM ) ≤ ♥➯♥ dim Supp HIi (M ) ≤ ✈ỵ✐ ♠å✐ i ≥ 0✳ ❉♦ ✤â ❦➳t q✉↔ ✤÷đ❝ s✉② r❛ trü❝ t✐➳♣ tø ❍➺ q✉↔ ✸✳✶✳✽✳ ✸✳✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ t t sr rữợ tr ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ♠ö❝ ♥➔② t❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ♠æ✤✉♥ ❋❙❋ ❝õ❛ P✳ ❍✳ ◗✉② ❬✶✽❪✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✶✳ ✭❬✶✽✱ ✣à♥❤ ♥❣❤➽❛ ✷✳✶❪✮ ▼ët R✲♠ỉ✤✉♥ M ✤÷đ❝ ❣å✐ F SF tỗ t ởt ổ ỳ ❤↕♥ s✐♥❤ N ❝õ❛ M s❛♦ ❝❤♦ t➟♣ ❣✐→ ❝õ❛ t➟♣ M/N ❧➔ ❤ú✉ ❤↕♥ ✭tù❝ ❧➔ Supp(M/N ) ❧➔ t➟♣ ❤ú✉ ❤↕♥✮✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② ❝❤➾ r❛ r➡♥❣ ❦❤→✐ ♥✐➺♠ ❦❤→✐ ♥✐➺♠ ♠ỉ✤✉♥ ❋❙❋ trò♥❣ ✈ỵ✐ ❦❤→✐ ♥✐➺♠ sr ỵ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✈➔ M ❧➔ R✲♠æ✤✉♥✳ ❑❤✐ ✤â M ❧➔ ♠æ✤✉♥ ▲❛s❦❡r ②➳✉ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ M ❧➔ ♠æ✤✉♥ ❋❙❋✳ ❚✐➳♣ t❤❡♦ ❧➔ ♠ët ♠➺♥❤ ✤➲ ❝➛♥ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ ð ♠ö❝ ♥➔②✳ ◆❤➢❝ ❧↕✐ r➡♥❣ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ♥û❛ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ Max R t ỳ ỵ ✤➲ ✸✳✼❪✮ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ♥û❛ ✤à❛ ♣❤÷ì♥❣✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â fI2 (M ) = inf i ∈ N | HIi (M ) ❦❤æ♥❣ ❧➔ ▲❛s❦❡r ②➳✉ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t t = inf i ∈ N | HIi (M ) ❦❤æ♥❣ ❧➔ ▲❛s❦❡r ②➳✉ ❚❛ ❜✐➳t r➡♥❣ ♠ët ♠æ✤✉♥ ❜➜t ❦➻ ❝â ❣✐→ ❤ú✉ ❤↕♥ ✤➲✉ ❝â ❝❤✐➲✉ ❣✐→ ❦❤ỉ♥❣ q✉→ 1✳ ❉♦ ✤â ✈ỵ✐ ♠å✐ i < t✱ t❛ ❝â HIi (M ) ❧➔ ▲❛s❦❡r ②➳✉✳ ❙✉② r❛ HIi (M ) ❧➔ ❋❙❋ t❤❡♦ ❇ê ✤➲ ✸✳✷✳✷ ✈ỵ✐ ♠å✐ i < t✳ ❍ì♥ ♥ú❛ HIi (M )p ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i < t✱ ♠å✐ p ∈ Supp(M/IM ) ♠➔ dim R/p ≥ 2✳ ❈❤ù♥❣ tä t ≤ fI2 (M )✱ tù❝ ❧➔ fI2 (M ) ≥ inf i ∈ N | HIi (M ) ❦❤ỉ♥❣ ❧➔ ▲❛s❦❡r ②➳✉ ◆❣÷đ❝ ❧↕✐✱ tø ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ▼➺♥❤ ✤➲ ✸✳✶✳✺ t❛ s✉② r❛ ợ ộ i < fI2(M ) tỗ t ♠æ✤✉♥ ❝♦♥ ❤ú✉ ❤↕♥ s✐♥❤ (0 :H (M ) I k ) ✭✈ỵ✐ k ♥➔♦ ✤â✮ t❤ä❛ ♠➣♥ i I dim Supp(HIi (M )/(0 :HIi (M ) I k )) õ t ỵ t s r❛ R✲♠æ✤✉♥ HIi (M )/(0 :H (M ) I k ) ❧➔ I ✲❝♦❢✐♥✐t❡✳ ❱➻ ✈➟② t➟♣ ❤ñ♣ i I (Supp(HIi (M )/0 :HIi (M ) I k ))\ Max(R)) ⊆ AsshR (HIi (M )/0 :HIi (M ) I k )) ❧➔ ❤ú✉ ❤↕♥✳ ❱➻ t➟♣ Max(R) ❧➔ ❤ú✉ ❤↕♥ ♥➯♥ tø ✤â t❛ s✉② r❛ t➟♣ Supp(HIi (M )/(0 :HIi (M ) I k )) ❧➔ ❤ú✉ ❤↕♥✳ ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✸✳✷✳✷✱ t❛ s✉② r❛ ❝→❝ R✲♠æ✤✉♥ HIi (M ) ❧➔ ❋❙❋ ✭❤❛② ❧➔ ▲❛s❦❡r ②➳✉✮ ✈ỵ✐ ♠å✐ i < fI2(M )✳ ❱➻ t❤➳ fI2 (M ) ≤ inf i ∈ N | HIi (M ) ❚❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✸✻ ❦❤æ♥❣ ❧➔ ▲❛s❦❡r ②➳✉ ❚✐➳♣ t❤❡♦ tr➻♥❤ ❜➔② t❤➯♠ ♠ët sè ❦➳t q✉↔ rr ỵ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❈❤♦ t ❧➔ ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦ dim Supp HIi (M ) ≤ ✈ỵ✐ ♠å✐ i < t✳ ❑❤✐ ✤â ❝→❝ R✲♠æ✤✉♥ ExtjR (R/I, HI0 (M )), , ExtjR (R/I, HIt−1 (M )) ✈➔ HomR (R/I, HIt (M )) ❧➔ ▲❛s❦❡r ②➳✉ ✈ỵ✐ ♠å✐ j ≥ 0✳ ✣➦❝ ❜✐➺t✱ t❛ s✉② r❛ t➟♣ AssR HIi (M ) ❧➔ ❤ú✉ ❤↕♥ ✈ỵ✐ ♠å✐ i ≤ t✳ ▼➺♥❤ ✤➲ ✸✳✷✳✺✳ ✭❬✹✱ ▼➺♥❤ ✤➲ ✸✳✽❪✮ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❈❤♦ t ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦ dim Supp HIi (M ) ≤ ✈ỵ✐ ♠å✐ i < t✳ ❑❤✐ ✤â ✐✮ ExtjR(R/I, HIi (M )) ❧➔ R✲♠ỉ✤✉♥ ▲❛s❦❡r ②➳✉ ✈ỵ✐ ♠å✐ i < t ✈➔ ♠å✐ j ≥ 0✳ ✐✐✮ HomR(R/I, HIt (M )) ✈➔ Ext1R(R/I, HIt (M )) ❧➔ R✲♠ỉ✤✉♥ ▲❛s❦❡r ②➳✉✳ ✐✐✐✮ ❱ỵ✐ ♠å✐ ♠æ✤✉♥ ❝♦♥ ▲❛s❦❡r ②➳✉ N ❝õ❛ HIt (M )✱ t❛ ❝â R✲♠æ✤✉♥ HomR (R/I, HIt (M )/N ) ✈➔ Ext1R (R/I, HIt (M )/N ) ❧➔ ▲❛s❦❡r ②➳✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤➦t Φ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♠æ✤✉♥ ExtjR(R/I, HIi (M ) ✈➔ ExtsR (R/I, HIt (M ) tr♦♥❣ ✤â j = 0, 1, 2, ; i = 0, 1, , t − ✈➔ s = 0, 1✳ ❑❤✐ ✤â ❦➳t q✉↔ ❝õ❛ P❤➛♥ ✐✮ ✈➔ ✐✐✮ ✤÷đ❝ s✉② r❛ tø ❝→❝❤ ❝❤ù♥❣ t÷ì♥❣ tü ❝❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✸✳✷✳✹✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✐✐✐✮ t❛ sû ❞ư♥❣ ❞➣② ❦❤ỵ♣ s❛✉ ✈➔ ✐✐✮ HomR (R/I, HIt (M )) → HomR (R/I, HIt (M )/N ) → Ext1I (R/I, N ) → Ext1R (R/I, HIt (M )) → Ext1R (R/I, HIt (M )/N ) Ext2I (R/I, N ) ỵ ỵ (R, m) ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ ❧➔ sè ❤ú✉ ❤↕♥✳ ❑❤✐ ✤â R ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ●✐↔ sû fI3 (M ) ✐✮ ExtjR(R/I, HIi (M )) ❧➔ ▲❛s❦❡r ②➳✉ ✈ỵ✐ ♠å✐ i < fI3(M ) ✈➔ ♠å✐ j ≥ 0✳ ✐✐✮ ❱ỵ✐ ♠å✐ ♠ỉ✤✉♥ ❝♦♥ ▲❛s❦❡r ②➳✉ N ❝õ❛ HIf (M )(M )✱ t❛ ❝â ❝→❝ R✲♠æ✤✉♥ I f (M ) Hom(R/I, HI I (M )/N ) ✈➔ f (M ) Ext1R (R/I, HI I (M )/N ) ❧➔ ▲❛s❦❡r ②➳✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣è✐ ✈ỵ✐ ❝❤ù♥❣ ♠✐♥❤ ✐✮✱ t❛ ❧➜② t = fI3(M ) ✈➔ ❧➜② Φ ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ♠æ✤✉♥ ExtjR(R/I, HIi (M )) ✈➔ Hom(R/I, HIt (M )) ✈ỵ✐ j ≥ ✈➔ i = 0, 1, 2, , t − 1✳ ▲➜② L ∈ Φ ✈➔ L ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ L✳ ●✐↔ sû tr r ổ ú õ tỗ t L ∈ Φ s❛♦ ❝❤♦ L ❦❤æ♥❣ ❧➔ ▲❛s❦❡r ②➳✉✳ ❚ø ✤â s✉② r❛ t➟♣ {q ∈ Ass(L) | dim R/q 1} ổ õ tỗ t t➟♣ ❝♦♥ ✈ỉ ❤↕♥ ✤➳♠ ✤÷đ❝ {qk }∞k=1 ❝õ❛ T s❛♦ ❝❤♦ ❦❤æ♥❣ ♣❤➛♥ tû ❧➔ m✳ ❚❤❡♦ ❬✶✹✱ ❇ê ✤➲ ✸✳✷❪✱ s✉② r❛ −1 ∞ m ⊆ ∪∞ k=1 qk ✳ ▲➜② S ❧➔ t➟♣ ✤â♥❣ ♥❤➙♥ R \ ∪k=1 qk ✳ ❑❤✐ ✤â t❛ t❤➜② r➡♥❣ S R✲ ♠æ✤✉♥ S −1L ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✱ ✈➔ ❞♦ ✤â AssS R(S −1L) ❧➔ ❤ú✉ ❤↕♥✳ ◆❤÷♥❣ S −1 q ∈ AssS R (S −1 L) ✈ỵ✐ ♠å✐ k = 1, 2, 3, ✱ ✤✐➲✉ ♥➔② ❧➔ ✈æ ỵ 1 t tr ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ◆❤➢❝ ❧↕✐ ❝→❝ ❦✐➳♥ t❤ù❝ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ❧✉➟♥ ✈➠♥✿ ❚➟♣ ❣✐→✱ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✱ ♠æ✤✉♥ ◆♦t❤❡r✱ ♠æ✤✉♥ ♥ë✐ ①↕✱ ♠æ✤✉♥ ①↕ ↔♥❤✱ ♠æ✤✉♥ t tự ổ ố ỗ ♣❤÷ì♥❣ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ ♠ỉ✤✉♥ ♥➔②✳ ✷✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ ❝❤➼♥❤ tr♦♥❣ ❝❤÷ì♥❣ ✷✿ ❈❤♦ ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛ R✳ ❈❤♦ M ❧➔ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤ó♥❣✿ ✐✮ ✐✐✮ R✲♠ỉ✤✉♥ HIi (M ) f (M ) R✲♠æ✤✉♥ HI I ❧➔ ♠✐♥✐♠❛① ✈➔ (M ) f (M ) HomR (R/I, HI I R✲♠æ✤✉♥ N R ❝õ❛ fI1 (M ) HI (M )/N ) fI1 (M ) ✸✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿ ❈❤♦ ✈➔ < fI1 (M )✳ fI1 (M ) (M )✱ ❧➔ ❤ú✉ ❤↕♥✳ ❦❤✐ ✤â ❝→❝ R✲♠æ✤✉♥ f (M ) ExtR (R/I, HI I (M )/N ) ❧➔ ❤ú✉ ❤↕♥✳ ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ I ❧➔ ♠ë✐ ✐✤➯❛♥ ❝õ❛ ✹ ✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿ ❈❤♦ R ❦❤ỉ♥❣ ❧➔ ♠✐♥✐♠❛① ✈➔ M ❧➔ ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛ ❈❤♦ M ❧➔ R✳ ✲ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔ ✤ó♥❣✿ ✐✮ R ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â fI1 (M ) = inf i ∈ N0 | HIi (M ) R ✈ỵ✐ ♠å✐ ❦❤ỉ♥❣ ❧➔ ♠✐♥✐♠❛① ❦❤✐ ✐✐✐✮ ▼å✐ ♠ỉ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤✐ I ✲❝♦❢✐♥✐t❡ R✲♠æ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈ỵ✐ ♠å✐ ✸✾ i < fI2 (M )✳ ✐✐✮ ▼å✐ ♠æ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① f (M ) HomR (R/I, HI I ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤✐ N ❝õ❛ (M )/N ) fI2 (M ) f (M ) HI I ✈➔ ❦❤✐ ✤â ❝→❝ R✲♠æ✤✉♥ f (M ) Ext1R (R/I, HI I ❧➔ ❤ú✉ ❤↕♥✳ ✹✵ (M )✱ (M )/N ) ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❆❜❛③❛r✐ ❘✳ ❛♥❞ ❇❛❤♠❛♥♦✉r ❑✳ ✭✷✵✶✶✮✱ ✧❈♦❢✐♥✐t❡♥❡ss ♦❢ ❡①t❡♥s✐♦♥ ❢✉♥❝t♦rs ♦❢ ❝♦❢✐♥✐t❡ ♠♦❞✉❧❡s✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✸✸✵✱ ✺✵✼✲✺✶✻✳ ❬✷❪ ❆③❛♠✐ ❏✳✱ ◆❛❣❤✐♣♦✉r ❘✳ ❛♥❞ ❱❛❦✐❧✐ ❇✳ ✭✷✵✵✾✮✱ ✧❋✐♥✐t❡♥❡ss ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ❢♦r a ✲ ♠✐♥✐♠❛① ♠♦❞✉❧❡s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✸✼✱ ✹✸✾✲✹✹✽✳ ❬✸❪ ❇❛❤♠❛♥♣♦✉r ❑✳ ❛♥❞ ❑❤♦❥❛❧✐ ❆✳ ✭✷✵✶✶✮✱ ✧❖♥ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ❋❙❋ ❛♥❞ ✇❡❛❦❧② ▲❛s❦❡r✐❛♥ ❝❧❛ss❡s✧✱ ♣r❡♣r✐♥t✳ ❬✹❪ ❇❛❤♠❛♥♣♦✉r ❑✳✱ ◆❛❣❤✐♣♦✉r ❘✳ ❛♥❞ ❙❡❞❣❤✐ ▼✳ ✭✷✵✶✸✮✱ ✧▼✐♥✐♠❛①♥❡ss ❛♥❞ ❈♦❢✐♥✐t❡ ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡✲ ❜r❛✱ ❱♦❧ ✹✶✱ ✷✼✾✾✲✷✽✹✶✳ ❬✺❪ ❇❛❤♠❛♥♣♦✉r ❑✳ ❛♥❞ ◆❛❣❤✐♣♦✉r ❘✳ ✭✷✵✵✽✮✱ ✧❖♥ t❤❡ ❝♦❢✐♥✐t❡♥❡ss ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✸✻✱ ✷✸✺✾✲✷✸✻✸ ❬✻❪ ❇❛❤♠❛♥♣♦✉r ❑✳ ❛♥❞ ◆❛❣❤✐♣♦✉r ❘✳ ✭✷✵✵✽✮✱ ✧❆ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ❛♥❞ ▼❛t❧✐s ❞✉❛❧✐t②✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✸✷✵✱ ✷✻✸✷✲✷✻✹✶✳ ❬✼❪ ❇❛❤♠❛♥♣♦✉r ❑✳ ❛♥❞ ◆❛❣❤✐♣♦✉r ❘✳ ✭✷✵✵✾✮✱ ✧❈♦❢✐♥✐t❡♥❡ss ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧s ❢♦r ✐❞❡❛♥❧s ♦❢ s♠❛❧❧ ❞✐♠❡♥s✐♦♥✧✱ ❏✳ ❆❧❣❡r❛✱ ✸✷✶✱ ✶✾✾✼✲✷✵✶✶✳ ❬✽❪ ❇r♦❞♠❛♥ ▼✳ P✳ ❛♥❞ ❙❤❛r♣ ❘✳ ❨✳ ✭✶✾✾✽✮✱ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②❀ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✾❪ ❉❡❧❢✐♥♦ ❉✳ ❛♥❞ ▼❛r❧❡② ❚✳ ✭✶✾✾✼✮✱ ✧❈♦❢✐♥✐t❡ ♠♦❞✉❧❡s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✧✱ ❏✳ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣❡❜r❛✱ ✶✷✶✱ ✹✺✲✺✷✳ ✹✶ ❬✶✵❪ ❋❛❧t✐♥❣s ●✳ ✭✶✾✽✶✮✱ ✧❉❡r ❡♥❞❧✐❝❤❦❡✐tss❛t③ ✐♥ ❞❡r ❧♦❦❛❧❡♥ ❦♦❤♦♠♦❧♦❣✐❝✑✱ ▼❛t❤✳ ❆♥♥✳ ✷✺✺✱ ✹✺✲✺✻✳ ❬✶✶❪ ●r♦t❤❡♥❞✐❡❝❦ ❆✳ ✭✶✾✻✻✮✱ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✱ ◆♦t❡s ❜② ❘✳ ❍❛rts❤♦r♥❡✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✽✻✷✳ ❬✶✷❪ ❍❛rts❤♦r♥❡ ❘✳ ✭✶✾✼✵✮✱ ✧❆❢❢✐♥❡ ❞✉❛❧✐t② ❛♥❞ ❝♦❢✐♥✐t❡♥❡ss✧✱ ■♥✈❡♥t✳ ▼❛t❤✳ ✾✱ ✶✹✺✲✶✻✹✳ ❬✶✸❪ ❑❛✇❛s❛❦✐ ❑✳ ■✳ ✭✶✾✾✻✮✱ ✧❖♥ t❤❡ ❢✐♥✐t❡♥❡ss ♦❢ ❇❛ss ♥✉♠❜❡rs ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✷✹✱ ✸✷✼✺✲✸✷✼✾✳ ❬✶✹❪ ❑❤❛s❤❛②❛r♠❛♥❡s❤ ❑✳ ✭✷✵✵✼✮✱ ✧❖♥ t❤❡ ❢✐♥✐t❡♥❡ss ♣r♦♣❡rt✐❡s ♦❢ ❡①t❡♥s✐♦♥ ❛♥❞ t♦rs✐♦♥ ❢✉♥❝t♦rs ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✸✺✱ ✶✸✶✾✲✶✸✷✼✳ ❬✶✺❪ ▼❛ts✉♠✉r❛ ❍✳ ✭✶✾✽✻✮✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r② ✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❬✶✻❪ ▼❡❧❦❡rss♦♥ ▲✳ ✭✶✾✾✵✮✱ ✧❖♥ ❛s②♠♣t♦t✐❝ st❛❜✐❧✐t② ❢♦r s❡ts ♦❢ ♣r✐♠❡s ✐❞❡❛❧s ❝♦♥♥❡❝t❡❞ ✇✐t❤ t❤❡ ♣♦✇❡rs ♦❢ ❛♥ ✐❞❡❛❧s✧✱ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳ ✶✵✼✱ ✷✻✼✲✷✾✷✳ ❬✶✼❪ ▼❡❧❦❡rss♦♥ ▲✳ ✭✷✵✵✺✮✱ ✧▼♦❞✉❧❡s ❝♦❢✐♥✐t❡ ✇✐t❤ r❡s♣❡❝t t♦ ❛♥ ✐❞❡❛❧✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✽✺✱ ✻✹✾✲✻✻✽✳ ❬✶✽❪ ◗✉② P✳ ❍✳ ✭✷✵✶✵✮✱ ✧❖♥ t❤❡ ❢✐♥✐t❡♥❡ss ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✸✽✱ ✶✾✻✺✲✶✾✻✽✳ ❬✶✾❪ ❨♦s❤✐❞❛ ❑✳ ■✳ ✭✶✾✾✼✮✱ ✧❈♦❢✐♥✐t❡♥❡ss ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ❢♦r ✐❞❡❛❧s ♦❢ s t oă s❝❤✐♥❣❡r ❍✳ ✭✶✾✽✻✮✱ ✧▼✐♥✐♠❛① ♠♦❞✉❧❡s✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✶✵✷✱✶✲✸✷✳ ✹✷