www.VNMATH.com ✶ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖ ◆●❯❨➍◆ ❚❘❯◆● ❉Ô◆● ❚➑◆❍ ✃◆ ✣➚◆❍ ❚■➏▼ ❈❾◆ ❈Õ❆ ❚❾P ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ▲■➊◆ ❑➌❚ ❈Õ❆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✵ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✷ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖ ◆●❯❨➍◆ ❚❘❯◆● ❉Ô◆● ❚➑◆❍ ✃◆ ✣➚◆❍ ❚■➏▼ ❈❾◆ ❈Õ❆ ❚❾P ■✣➊❆◆ ◆●❯❨➊◆ ❚➮ ▲■➊◆ ❑➌❚ ❈Õ❆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ Pì số ỵ tt số sè✿ ✻✵✳ ✹✻✳ ✵✺ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ữớ ữợ ◆❣✉②➯♥ ✲ ✷✵✶✵ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✸ ▼ö❝ ❧ö❝ ❚r❛♥❣ ▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▲í✐ ❝↔♠ ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝ð sð ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳ ▼æ✤✉♥ Ext ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳ ▼æ✤✉♥ ố ỗ ữỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✹✳ ❈❤✐➲✉ ✈➔ ✤ë s➙✉ ❝õ❛ ♠æ✤✉♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺✳ ❱➔♥❤ ✈➔ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❈❤÷ì♥❣ ✷✳ ❚➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ♠ët sè ♠ð rë♥❣ ❝õ❛ ✤ë s➙✉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✻ ✷✳✶✳ M −❞➣② tø ❝❤✐➲✉ >k ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳ ❈❤ù♥❣ ♠✐♥❤ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳ ▼ët sè t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ ✤➳♥ depthk ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ❈❤÷ì♥❣ ✸✳ ❚➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ự ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ự ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✳ ự ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ Số hóa Trung tâm Học liệu – Đại học Thái Ngun http://www.lrc-tnu.edu.vn www.VNMATH.com ✹ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ s❛✉ ❤❛✐ ♥➠♠ ❤å❝ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ s÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✈➔ ữợ sỹ ữợ t t ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❍♦➔♥❣✳ ◆❤➙♥ ❞à♣ ♥➔② tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ✈➔ ❣✐❛ ✤➻♥❤✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tỵ✐ ●❙✳❚❙❑❍ ◆❣✉②➵♥ ❚ü ❈÷í♥❣✱ P●❙✳❚❙ ◆❣✉②➵♥ ◗✉è❝ ❚❤➢♥❣✱ P●❙✳❚❙ ▲➯ ❚❤❛♥❤ ◆❤➔♥ ✈➔ ❚❙✳ ◆❣✉②➵♥ ❚❤à ❉✉♥❣❀ ❝→❝ t❤➛② ❝æ ð ❑❤♦❛ ❚♦→♥ ✈➔ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✣↕✐ ❤å❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣ tỉ✐ ①✐♥ ❜➔② tä ỏ t ỡ tợ ữớ t tt ❝↔ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✵ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚r✉♥❣ ❉ơ♥❣ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✺ ▼ð ✤➛✉ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ ■✱ ❏ ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ ❘ ✈➔ ▼ ❧➔ ♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ◆➠♠ ✶✾✼✾✱ ▼✳ ❇r♦❞♠❛♥♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❝→❝ t➟♣ AssR (M/J n M ) ❧➔ ê♥ ✤à♥❤ ❦❤✐ n ❦❤✐ n M −❞➣② k ≥ −1✱ tø ❝❤✐➲✉ ♠ët ❞➣② >k ♥❣✉②➯♥ i x1 , , xr ♠➔ ❚❛ ❦➼ ❤✐➺✉ ✤ë ❞➔✐ ❝❤✉♥❣ ♥➔② ❧➔ tr♦♥❣ depth(I, M ) I ✱ depth0 (I, M ) ❝õ❛ I > k ❝→❝ ♣❤➛♥ tû ❝õ❛ i ∈ {1, , r} ✈➔ ♥❤÷ s❛✉✿ ❝❤♦ m ✤÷đ❝ ❣å✐ ❧➔ xi ∈ / p t❛ ❝â ✈ỵ✐ ♠å✐ ❍å ✤➣ ❝❤➾ r❛ r➡♥❣ ♠å✐ ✤➲✉ ❝â ✤ë ❞➔✐ ♥❤÷ ♥❤❛✉ ✈➔ ❜➡♥❣ sè p ∈ Supp(HIi (M )) depthk (I, M )✳ tr♦♥❣ s u ă ❚❛♥❣ ✈➔ I M tø ❝❤✐➲✉ dim(R/p) > k ✳ tố tr t s tỗ t ✤ë s➙✉ M −❞➣② ♥➳✉ ✈ỵ✐ ♠é✐ p ∈ AssR (M/(x1 , , xi−1 )M ) M −❞➣② tø ❝❤✐➲✉ > k depth(I, J n M/J n+1 M ) ✤õ ❧ỵ♥✳ ●➛♥ ✤➙②✱ ▼✳ ❇r♦❞♠❛♥♥ ✈➔ ▲✳❚✳ ◆❤➔♥ ✤➣ ✤à♥❤ ♥❣❤➽❛ ❦❤→✐ ♥✐➺♠ sè ♥❣✉②➯♥ ✈➔ ✤õ ❧ỵ♥✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ tr➯♥✱ æ♥❣ ✤➣ ❞ü❛ ✈➔♦ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ depth(I, M/J n M ) AssR (J n M/J n+1 M ) I ❝â dim(R/p) > k ✳ ✣➦❝ ❜✐➺t✱ depth−1 (I, M ) ❧➔ ✤ë ❞➔✐ ❝õ❛ f-depth(I, M ) ❝õ❛ M −❞➣② tè✐ ✤↕✐ M I tr♦♥❣ depth1 (I, M ) ❧➔ ✤ë s➙✉ s✉② rë♥❣ ❝õ❛ M ✤÷đ❝ tr♦♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ▲✳ ❚✳ ◆❤➔♥✳ ❚ø ✤â t❛ ❝â ♠ët ❝➙✉ ❤ä✐ ♠ð ✤➦t r❛ ❧➔✿ ❈➙✉ ❤ä✐ ✶✿ ▲✐➺✉ r➡♥❣ ❝→❝ sè depthk (I, J n M/J n+1 M ) ✈➔ depthk (I, M/J n M ) ❝â trð ♥➯♥ ê♥ ✤à♥❤ ❤❛② ❦❤ỉ♥❣ ❦❤✐ n ✤õ ❧ỵ♥❄ ◆➠♠ ✷✵✵✽✱ tr♦♥❣ ♠ët ❜➔✐ ❜→♦ ❝õ❛ ◆✳ ❚✳ ❈÷í♥❣✱ ◆✳ ❱✳ ❍♦➔♥❣ ✈➔ P✳ ❍✳ ❑❤→♥❤ ✭①❡♠ ❬✽❪✮✱ ❤å ✤➣ tr↔ ❧í✐ ❦❤➥♥❣ ✤à♥❤ ❝❤♦ ❝➙✉ ❤ä✐ tr➯♥✱ ✤â ❝ô♥❣ ❧➔ ♠ët ❦➳t q✉↔ rở ởt ỵ r t ỵ s ỵ ỵ ✶✳✶❪ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✱ I, J ⊆ R ❧➔ ❝→❝ ✐✤➯❛♥ ✈➔ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ sè Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✻ ♥❣✉②➯♥ k ≥ −1✱ ❝→❝ sè depthk (I, J n M/J n+1 M ) ✈➔ depthk (I, M/J n M ) trð t❤➔♥❤ ❝→❝ ❤➡♥❣ sè rk ✈➔ sk ✈ỵ✐ n ✤õ ❧ỵ♥✳ ▼➦t ❦❤→❝✱ ♥➠♠ ✶✾✾✵✱ ❈✳ ❍✉♥❡❦❡ ✤➣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t r➡♥❣ ✏t➟♣ AssR (HIj (M )) I✱ ✈➔ ♠å✐ j ❧➔ ❤ú✉ ❤↕♥ ✈ỵ✐ ♠å✐ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ M✱ ♠å✐ ✐✤➯❛♥ ✑✳ ❈➙✉ tr↔ ❧í✐ ❦❤➥♥❣ ✤à♥❤ ❝❤♦ ❝➙✉ ❣✐↔ t❤✉②➳t ✤â ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❍✉♥❡❦❡✲❘✳❨✳ ❙❤❛r♣✱ ●✳ ▲②✉❜❡③♥✐❦ ❝❤♦ ❝→❝ ✈➔♥❤ ❝❤➼♥❤ q✉② ✤à❛ ♣❤÷ì♥❣ ❝❤ù❛ ♠ët tr÷í♥❣✳ ▼➦❝ ❞ị✱ s❛✉ ✤â ❆✳ ❙✐♥❣❤✱ ▼✳ ❑❛t③♠❛♥ ✤➣ ❝❤➾ r❛ ❝→❝ ♣❤↔♥ ✈➼ ❞ư ❝❤♦ ❣✐↔ t❤✉②➳t ♥➔②✱ ♥❤÷♥❣ ❣✐↔ t❤✉②➳t ✤â ✈➝♥ ❝á♥ ✤ó♥❣ tr♦♥❣ ♥❤✐➲✉ tr÷í♥❣ ❤đ♣✳ ❈❤➥♥❣ ❤↕♥✱ ❑✳ ❑❤❛s❤②❛r♠❛♥❡s❤✲❙❤✳ ❙❛❧❛r✐❛♥✱ ▲✳❚✳ ◆❤➔♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ j ≤ depth1 (I, M )✳ ❚ø ✤â ✈➔ tø ỵ t t r j r1 = depth1 (I, J n M/J n+1 M ) ❝→❝ t➟♣ n AssR (HIj (M )) ❧➔ t➟♣ ❤ú✉ ❤↕♥ ✈➔ i ≤ s1 = depth1 (I, M/J n M ) t❤➻ AssR (HIj (J n M/J n+1 M )) ✈➔ AssR (HIi (M/J n M )) ❧➔ ❤ú✉ ❤↕♥ ợ ợ t ữ ởt tỹ ♥❣÷í✐ t❛ ❤ä✐ r➡♥❣ ❈➙✉ ❤ä✐ ✷✳ ❈❤♦ ❝→❝ sè ♥❣✉②➯♥ j ≤ r1 ✈➔ i ≤ s1 ✱ ❧✐➺✉ r➡♥❣ ❝→❝ t➟♣ AssR (HIj (J n M/J n+1 M )) ✈➔ AssR (HIi (M/J n M )) ❝â trð ♥➯♥ ê♥ ✤à♥❤ ❤❛② ❦❤ỉ♥❣ ❦❤✐ n ✤õ ❧ỵ♥❄ ❈ơ♥❣ tr♦♥❣ ❜➔✐ ❜→♦ ♥➯✉ tr➯♥ ❝õ❛ ◆✳ ❚✳ ❈÷í♥❣✱ ◆✳ ❱✳ ❍♦➔♥❣ ✈➔ P✳ ❍✳ ❑❤→♥❤ ✭①❡♠ ❬✽❪✮✱ ❤å ✤➣ tr↔ ❧í✐ ❦❤➥♥❣ ✤à♥❤ ❝❤♦ ♠ët ❝➙✉ ❤ä✐ ②➳✉ ❤ì♥ ọ tr t t ữủ ỵ s ỵ ỵ (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✱ I, J ⊆ R ❧➔ ❝→❝ ✐✤➯❛♥ ✈➔ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ▲➜② rk = depthk (I, J n M/J n+1 M ) ✈➔ sk = depthk (I, M/J n M ) ❦❤✐ n ợ ữ tr ỵ õ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣✿ r s ✭✐✮ AssR (HI −1 (J n M/J n+1 M )) ✈➔ AssR (HI −1 (M/J n M )) ❧➔ ❝→❝ t➟♣ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ✭✐✐✮ j≤r0 AssR HIj (J n M/J n+1 M )) ✈➔ i≤s0 AssR HIi (M/J n M )) ❧➔ ❝→❝ t➟♣ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✼ ✭✐✐✐✮ t≤j AssR HIt (J n M/J n+1 M ))∪{m} ✈➔ t≤i AssR HIt (M/J n M ))∪{m} ✈ỵ✐ ♠å✐ j ≤ r1 ✈➔ i ≤ s1 ❧➔ ❝→❝ t➟♣ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ◆❤ú♥❣ ✈➜♥ ✤➲ ♥➯✉ tr➯♥ ❝â ♠ët þ ♥❣❤➽❛ q✉❛♥ trå♥❣ ❝❤✉②➯♥ ♥❣➔♥❤ ✤↕✐ sè✱ ✤↕✐ sè số ỗ t õ ✤➣ t❤✉ ❤ót sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ tr t ợ tr ữợ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❤➺ t❤è♥❣ ♠ët sè ❦✐➳♥ t❤ù❝ tt số số ỗ ✤✐➲✉ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ❝➙✉ ❤ä✐ ✶✱ ✷❀ ❙❛✉ ✤â tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝❤ù♥❣ ỵ ỵ ởt số q ú ỗ ✸ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ❞➔♥❤ ✤➸ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ✤↕✐ sè ❣✐❛♦ ❤♦→♥✱ ✤è✐ ỗ ữỡ ổ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❝→❝ ❝❤÷ì♥❣ t✐➳♣ s r ữỡ ú tổ ợ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ tø ❝❤✐➲✉ > k✱ ✤ë ❞➔✐ ❝õ❛ M −❞➣② tø ❝❤✐➲✉ > k tr♦♥❣ I✳ M −❞➣② t ú tổ ự ỵ ❤➺ q✉↔ ❝õ❛ ♥â✳ P❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ①➨t ♠ët sè t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❝❤✐➲✉ >k M −❞➣② tø ✈➔ ♠ð rë♥❣ ❝õ❛ ✤ë s➙✉✳ ❈❤÷ì♥❣ ❝✉è✐ ❝ị♥❣✱ ❝❤ó♥❣ tỉ✐ ❞➔♥❤ t♦➔♥ ❜ë ❝❤♦ ✈✐➺❝ ự ỵ r õ trữợ ộ ❝❤ù♥❣ ♠✐♥❤ ❝❤ó♥❣ tỉ✐ ✤➲✉ ✤÷❛ r❛ ❝→❝ t➼♥❤ ❝❤➜t ❝â ❧✐➯♥ q✉❛♥✳ Số hóa Trung tâm Học liệu – Đại học Thái Ngun http://www.lrc-tnu.edu.vn www.VNMATH.com ✽ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝ì sð ❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ ổ (R, m) ữỡ tr ợ ❝ü❝ ✤↕✐ ❞✉② ♥❤➜t ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ m; ✈➔ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ✶✳✶ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ Ann(x) = p✳ AssR (M ) M p R tỗ t ởt tỷ ❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ ❤♦➦❝ ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ M x ∈ M s❛♦ ❝❤♦ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ Ass(M )✳ ❙❛✉ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ ▼➺♥❤ ✣➲ ✶✳✶✳✷✳ ✭❛✮ ❈❤♦ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✳ ❑❤✐ ✤â p ∈ AssR (M ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ M ❝❤ù❛ ♠ët ♠æ✤✉♥ ❝♦♥ ✤➥♥❣ ❝➜✉ ✈ỵ✐ R/p✳ ✭❜✮ ❈❤♦ p ❧➔ ♣❤➛♥ tû tè✐ ✤↕✐ ❝õ❛ t➟♣ ❝→❝ ✐✤➯❛♥ ❝â ❞↕♥❣ Ann(x) tr♦♥❣ ✤â = x ∈ M ✳ ❑❤✐ ✤â p ∈ AssR (M )✳ ❱➻ t❤➳✱ M = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AssR (M ) = 0✳ ❍ì♥ ♥ú❛✱ t ZD(M ) ữợ ổ M Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✾ ❧➔ ❤ñ♣ ❝õ❛ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✭❝✮ ❈❤♦ −→ M −→ M −→ M −→ ❧➔ ❞➣② ❦❤ỵ♣ ❝→❝ R−♠æ✤✉♥✳ ❑❤✐ ✤â AssR M ⊆ AssR M ⊆ AssR M ∪ AssR M ✭❞✮ AssR (M ) ⊆ SuppR (M ) ✈➔ ♠é✐ ♣❤➛♥ tû tè✐ t❤✐➸✉ ❝õ❛ SuppR (M ) ✤➲✉ t❤✉ë❝ AssR (M )✳ ✭❡✮ ◆➳✉ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ AssR (M ) ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ❍ì♥ ♥ú❛✱ AssR (M ) ⊆ V (Ann M ) ✈➔ ♠é✐ ♣❤➛♥ tû tè✐ t❤✐➸✉ ❝õ❛ V (Ann M ) ✤➲✉ t❤✉ë❝ AssR (M )✳ ❱➻ t❤➳ Ann(M ) ❧➔ ❣✐❛♦ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳ ✭❢✮ ◆➳✉ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M t❤➻ AssR (N ) ⊆ AssR (M ) ⊆ AssR (M/N ) ∪ AssR (N ) ✭❤✮ AssRp (Mp ) = {qRp |q ∈ AssR (M ), q p} ữợ ởt t q r➜t q✉❛♥ trå♥❣ ❝õ❛ ▼✳ ❇r♦❞♠❛♥♥ ✈➲ t➼♥❤ ê♥ ✤à♥❤ t tố t ỵ ✶✳✶✳✸✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ t➟♣ AssR (M/I n M ) ✈➔ AssR (I n−1 M/I n M ) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ n ❦❤✐ n ✤õ ❧ỵ♥✳ ✶✳✷ ▼ỉ✤✉♥ ❊①t ✣➸ t✐➺♥ t❤❡♦ ❞ã✐✱ tr♦♥❣ ♠ư❝ ♥➔②✱ t❛ ♥❤➢❝ ♥❣➢♥ ❣å♥ ❝→❝ ❦❤→✐ ♥✐➺♠ ♠æ✤✉♥ Ext ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥â✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ▼ët ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ M ❧➔ ♠ët ❞➣② ❦❤ỵ♣ −→ P2 −→ P1 −→ P0 −→ M −→ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✶✵ tr♦♥❣ õ ộ Pi ú ỵ sỷ Y P1 M ổ tỗ t t P0 = yY Ry ✱ tü ❞♦ tr➯♥ t➟♣ ❨✳ ❑❤✐ ✤â t❛ ❝â t♦➔♥ ❝➜✉ ϕ(ay )y∈Y = Σy∈Y ay y ✳ ❧➔ M ●✐↔✐ ①↕ ↔♥❤ ❝õ❛ ♠ët ♠æ✤✉♥ ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ R−♠æ✤✉♥ ❜ð✐ ❧➔ ♠æ✤✉♥ ①↕ ↔♥❤✳ ✣➦t K1 = Ker ϕ✳ ▲➜② Y1 ✈ỵ✐ Ry = R✱ ϕ : P0 −→ M ❧➔ ❤➺ s✐♥❤ ❝õ❛ ❧➔ ❝❤♦ K1 ✈➔ R−♠æ✤✉♥ tü ❞♦ s✐♥❤ ❜ð✐ Y1 ✳ ❑❤✐ ✤â t❛ ❝â ♠ët t♦➔♥ ❝➜✉ tü ♥❤✐➯♥ f1 : P1 −→ K1 ✳ tü ♥❤✐➯♥ tø K1 ✣➦t ✈➔♦ µ1 = j1 f1 ✱ tr♦♥❣ ✤â j : K → P0 P0 ✳ ❉➵ t❤➜② Im µ1 = Ker ϕ✳ ✣➦t K2 = Ker µ1 ✳ ❇➡♥❣ ❝→❝❤ ❧➟♣ ❧✉➟♥ t÷ì♥❣ tü✱ t❛ ❝â ♠ët t♦➔♥ ❝➜✉ ❧➔ ♠ỉ✤✉♥ tü ❞♦ ✈➔ ❧➔ ♣❤➨♣ ♥❤ó♥❣ f2 : P2 −→ K2 Im µ2 = Ker µ1 ✱ tr õ à2 = j2 f2 ợ s P2 j : K → P1 ❧➔ ♣❤➨♣ ♥❤ó♥❣ tü ♥❤✐➯♥✳ ❈ù t✐➳♣ tö❝ q✉→ tr➻♥❤ tr➯♥ t❛ t❤✉ ữủ ởt ợ à2 à1 −→ P1 −→ P0 −→ M −→ tr♦♥❣ ✤â ♠é✐ Pi ❧➔ ♠æ✤✉♥ tü ❞♦✳ ❱➻ ♠é✐ ♠æ✤✉♥ tü ❞♦ ❧➔ ①↕ ↔♥❤ ♥➯♥ ❞➣② ❦❤ỵ♣ tr➯♥ ❧➔ ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ❈❤♦ N ♣❤↔♥ ❜✐➳♥✱ ❦❤ỵ♣ tr→✐✳ ❈❤♦ f2 ❧➔ M M✳ R−♠ỉ✤✉♥✳ ❧➔ ❳➨t ❤➔♠ tû R−♠æ✤✉♥✳ f1 f0 Hom(−, N ) ▲➜② ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ ❧➔ M✳ µ −→ P2 −→ P1 −→ P0 −→ M −→ ❚→❝ ✤ë♥❣ ❤➔♠ tû Hom(−, N ) ✈➔♦ ❞➣② ❦❤ỵ♣ tr➯♥ t❛ ❝â ♣❤ù❝ f∗ f∗ f∗ −→ Hom(P0 , N ) −→ Hom(P1 , N ) −→ Hom(P2 , N ) −→ ❑❤✐ ✤â ∗ ExtiR (M, N ) = Ker fi∗ / Im fi−1 ✳ ✈➔♦ ✈✐➺❝ ❝❤å♥ ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ ▼æ✤✉♥ ♥➔② ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ M✳ ❙❛✉ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♠ỉ✤✉♥ Ext✳ ▼➺♥❤ ✣➲ ✶✳✷✳✹✳ Số hóa Trung tâm Học liệu – Đại học Thái Ngun http://www.lrc-tnu.edu.vn www.VNMATH.com ✸✷ ✈ỵ✐ ♠å✐ n ≥ a✳ ✤à♥❤ ✈ỵ✐ n ✣✐➲✉ ✤â ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ✤õ ❧ỵ♥✱ ✈➻ ❚r÷í♥❣ ❤đ♣ ✸✿ r = s✳ AssR (HIs (M/J n M )) AssR (HIs (M/J a M )) ❧➔ t➟♣ ê♥ ❧➔ ❤ú✉ ❤↕♥ ❜ð✐ ❇ê ✤➲ ✸✳✶✳✶✳ ❇ð✐ ❞➣② ❦❤ỵ♣ ❞➔✐✱ t❛ ❝â ❞➣② ❦❤ỵ♣ s❛✉✿ HIs−1 (M/J n M ) −→HIs (J n M/J n+1 M ) −→ HIs (M/J n+1 M ) −→ HIs (M/J n M ) ✈ỵ✐ ♠å✐ n ≥ a✱ tr♦♥❣ ✤â HIs−1 (M/J n M ) = ❞♦ s − < r✳ ❉♦ ✈➟②✱ t❛ ❝â ❞➣② ❦❤ỵ♣ s❛✉ −→ HIs (J n M/J n+1 M ) −→ HIs (M/J n+1 M ) −→ HIs (M/J n M ) ✈ỵ✐ n a ỵ tỗ t sè ♥❣✉②➯♥ b≥a s❛♦ ❝❤♦ AssR (HIs (J n M/J n+1 M )) = AssR (HIs (J b M/J b+1 M )) ✈ỵ✐ ♠å✐ n ≥ b✳ ✣➦t X := AssR (HIs (J b M/J b+1 M ))✱ ❦❤✐ ✤â t❤❡♦ ❞➣② ❦❤ỵ♣ tr➯♥ t❛ ❝â X ⊆ AssR (HIs (M/J n+1 M )) ⊆ AssR (HIs (M/J n M )) ∪ X ✈ỵ✐ ♠å✐ n ≥ b✳ ❉♦ ✤â✱ ✈ỵ✐ ❜➜t ❦➻ n≥b t❛ ❝â AssR (HIs (M/J n+2 M )) ⊆ AssR (HIs (M/J n+1 M )) ∪ X = AssR (HIs (M/J n+1 M )) ❚ø ✤â✱ ✈➻ AssR (HIs (M/J b+1 M )) AssR (HIs (M/J n M )) ❧➔ ❤ú✉ ❤↕♥ ❜ð✐ t❤❡♦ ❇ê ✤➲ ✸✳✶✳✶✱ ♥➯♥ ❧➔ t➟♣ ê♥ ✤à♥❤ ✈ỵ✐ n ✤õ ❧ỵ♥✳ ự ỵ rữợ t t ♥❤➢❝ ❧↕✐ ♠ët ✤➦❝ tr÷♥❣ ❝õ❛ ✤ë s➙✉ ❧å❝ t❤ỉ♥❣ q t rt ổ ố ỗ ữỡ ỵ f-depth(I, M ) = inf{i|HIi (M ) ❦❤ỉ♥❣ ❧➔ ❆rt✐♥ } Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com t ỵ ự ỵ ự tr ỵ ữợ tữỡ ự ợ N n = Mn tr trữớ ủ ♣❤➙♥ ❜➟❝ M = ⊕n≥0 Mn , ✈➔ Nn = M/J n M ỵ R = ⊕n≥0Rn ❧➔ ♠ët ✤↕✐ sè ♣❤➙♥ ❜➟❝ ❝❤✉➞♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ R0 = R ✈➔ M = ⊕n≥0 Mn ❧➔ ♠ët R− ♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ r0 ❧➔ ❣✐→ trà ê♥ ✤à♥❤ ❝õ❛ f-depth(I, Mn )✳ ❑❤✐ ✤â j≤r0 AssR HIj (Mn )) ❧➔ t➟♣ ê♥ ✤à♥❤ ✈ỵ✐ n ✤õ ❧ỵ♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ r0 = t❛ ❝â AssR HIj (Mn )) = AssR (HI0 (Mn )) j≤r0 = AssR (ΓI (Mn )) = AssR (Mn ) ∩ V (I) ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✺ t❤➻ ❱ỵ✐ r0 = ∞, AssR (ΓI (Mn )) ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ HIi (Mn ) ❧➔ ❆rt✐♥ ✈ỵ✐ ♠å✐ i ♥➯♥ ê♥ ✤à♥❤ ✈ỵ✐ n ✤õ ❧ỵ♥✳ t❤❡♦ ❇ê ✤➲ ✸✳✷✳✶ t❛ ❝â f-depth(I, M ) = inf{i|HIi (Mn ) ♥➯♥ n j≤r0 ❦❤æ♥❣ ❧➔ ❆rt✐♥ AssR HIj (Mn )) = {m} ❞♦ ✤â ♥â ✤õ ❧ỵ♥✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ♣❤➛♥ ❝á♥ ❧↕✐✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ≤ r0 < t ỵ tỗ t số na } a ✤â s❛♦ ❝❤♦ ✈ỵ✐ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② ❧➔ ✤ó♥❣✿ r0 = f-depth(I, Mn )❀ ✭✐✐✮ ❝â ♠ët ❞➣② ❧å❝ ❝❤➼♥❤ q✉② ❝õ❛ x1 , , xr0 Mn tr I s õ ỗ tớ ♠ët ❞➣② ❤♦→♥ ✈à ✤÷đ❝ ✈➔ ❧➔ ❞➣② I−❧å❝ ❝❤➼♥❤ q✉② ❝õ❛ Mn ❤♦→♥ ✈à ✤÷đ❝❀ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✸✹ AssR (Mn /(x1 , , xr0 )Mn ) rữợ t t ự na n ≥ a✳ ❚❛ ❝â ❧➔ t➟♣ ✤ë❝ ❧➟♣ ✈ỵ✐ AssR HIr0 (Mn )) n✳ ❧➔ t➟♣ ❤ú✉ ❤↕♥✳ ▲➜② sè r0 (Mn )) HIr0 (Mn ) ∼ = HI0 (H(x , ,xr ) t❤❡♦ ❇ê ✤➲ ✷✳✸✳✸✱ ✈➔ r0 t t H(x (Mn ) ∼ = lim , ,xr ) −→(Mn /(x1 , , xr0 )Mn ) t t ỵ ✺✳✷✳✾❪✳ ◆❤÷ ✈➟② t❤❡♦ ❬✹✱ ▼➺♥❤ ✤➲ ✷✳✻❪ t❛ t❤✉ ✤÷đ❝ AssR (HIr0 (Mn )) ⊆ AssR Mn /(xt1 , , xtr0 )Mn ) t>0 = AssR Mn /(x1 , , xr0 )Mn ) ❉♦ ✤â t➟♣ n≥a AssR HIr0 (Mn )) ❧➔ t➟♣ ❤ú✉ ❤↕♥ ❜ð✐ ❝→❝❤ ❝❤å♥ ❝õ❛ ❚ø ✤➙② ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ ✈ỵ✐ ❝❤♦ t➟♣ S= n≥a p tè ✈ỵ✐ AssR HIr0 (Mn )) p ∈ AssR HIr0 (Mn )) ❧ỵ♥✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ❜➜t ❦➻ p ∈ AssR HIr0 (Mn )) ◆➯♥ HIrp0 (Mn )p = AssR HIr0 (Mn )) ✈ỵ✐ ♠å✐ n ✤à♥❤ r➡♥❣ ●å✐ t❤➻ n ∈ T n ∈ T✱ ♥➯♥ s❛♦ ❝❤♦ pRp ∈ AssRp HIrp0 (Mn )p ) ❙✉② r❛ ▼➦t ❦❤→❝✱ ✈➻ p ∈ Supp(Mn /IMn ) \ {m}✳ depth(Ip (Mn )p ) ≥ r0 ✳ ❱➟② p ∈ ❉♦ ✤â✱ depth(Ip (Mn )p ) = r0 n T t ỵ ✵✳✵✳✶ t❤➻ depth(Ip (Mn )p ) ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ ✤õ ❧ỵ♥✳ ❉♦ ✤â ❉♦ ✤â AssR HIr0 (Mn ))\{m} ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ n ✤õ depth(Ip (Mn )p ) ≤ r0 ✳ ❞♦ ✈➟② t❤❡♦ ❇ê ✤➲ ✷✳✶✳✸✱ t❛ ❝â n ≥ a p ∈ S, ❝â ♠ët t➟♣ ✈ỉ ❤↕♥ sè ♥❣✉②➯♥ T ✈ỵ✐ ♠å✐ ✈ỵ✐ ♠å✐ a ợ s ỗ tt ♥❣✉②➯♥ ✈ỵ✐ ✈ỉ ❤↕♥ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ a✳ depth(Ip (Mn )p ) = r0 AssRp HIrp0 (Mn )p ) ✈ỵ✐ ♠å✐ ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ AssR HIr0 (Mn )) \ {m} n ❧➔ ê♥ ✤à♥❤ ✈ỵ✐ n ợ õ ợ t ỵ ✸✳✶✳✷✳ n ✤õ ❧ỵ♥✳ r ❧➔ ❣✐→ trà ê♥ ✤à♥❤ ❝õ❛ depth(I, Mn )✳ ❘ã r➔♥❣ r ≤ r0 ✳ ◆➳✉ r = r0 HIj (Mn ) = ✈ỵ✐ ♠å✐ j < r0 ❞♦ ✤â Ass(HIj (Mn )) = AssR HIr0 (Mn )) j≤r0 Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✸✺ n ợ t ỵ r < r0 tỗ t n ợ ✈ỵ✐ = HIr (Mn ) s❛♦ ❝❤♦ ❧➔ ♠ỉ✤✉♥ rt ỵ õ r Ass(HIj (Mn )) = j≤r0 Ass(HIj (Mn )) j 0✳ n ✤õ ❧ỵ♥✳ ❚ø ❞➣② ❦❤ỵ♣ ♥❣➢♥ −→ J n M/J n+1 M −→ M/J n+1 M −→ M/J n M −→ t❛ ❝â ❞➣② ❦❤ỵ♣ ❞➔✐ s❛✉ −→ HIj−1 (M/J n M ) −→ HIj (J n M/J n+1 M ) fj −→ HIj (M/J n+1 M ) −→ HIj (M/J n M ) −→ ✈ỵ✐ ♠å✐ n > 0✳ ❚ø ✤➙② ✈ỵ✐ ❜➜t ❦➻ j≤l ✈➔ n≥a t❛ ❝â ❞➣② ❦❤ỵ♣ s❛✉ −→ Im fj −→ HIj (M/J n+1 M ) −→ HIj (M/J n M ) ♥➯♥ AssR (HIj (M/J n+1 M )) ⊆ AssR (HIj (M/J n M )) ∪ AssR (Im(fj )) t t ỵ ỗ ổ t õ j Im(fj ) ∼ = HI (J n M/J n+1 M )/ Ker fj ♥➯♥ t❤❡♦ ❇ê ✤➲ ✸✳✷✳✸ t❤➻ AssR (Im(fj )) = AssR (HIj (J n M/J n+1 M ))/ Ker fj ) ⊆ AssR (HIj (J n M/J n+1 M )) ∪ SuppR (Ker fj ) ❱➻ j − < s1 ♥➯♥ SuppR (HIj−1 (M/J n M ) ❧➔ t➟♣ ❤ú✉ ❤↕♥ t❤❡♦ ❇ê ✤➲ ✸✳✸✳✶✱ ✈➔ ✈➻ t❤➳ t❛ ❝â SuppR (Ker fj ) ⊆ SuppR (HIj−1 (M/J n M )) ⊆ AssR (HIj−1 (M/J n M )) ∪ {m} Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✹✷ ❉♦ ✤â t❛ ❝â Xl (n + 1) ⊆ Xl (n) ∪ Sl (n) ▼➦t ❦❤→❝✱ t❤❡♦ ❞➣② ❦❤ỵ♣ ❞➔✐ tr➯♥ ✈➔ ❇ê ✤➲ ✸✳✸✳✶✱ t❛ ❝â AssR (HIj (J n M/J n+1 M )) ⊆ AssR (HIj (M/J n+1 M )) ∪ SuppR (HIj−1 (M/J n M )) ⊆ AssR (HIj (M/J n+1 M )) ∪ AssR (HIj−1 (M/J n M )) ∪ {m} ❉♦ ✤â Sl (n) ⊆ Xl (n + 1) ∪ Xl−1 (n) ✈ỵ✐ ♠å✐ n a b a s ỵ t tt q tỗ t số ❝❤♦ Sl (n) = S ✈➔ Xl−1 (n) = X ✈ỵ✐ ♠å✐ n ≥ b✳ ❉♦ ✤â✱ ✈➻ X = Xl−1 (n + 1) ⊆ Xl (n + 1) ♥➯♥ S ⊆ Xl (n + 1) ∪ X = Xl (n + 1) ✈ỵ✐ ♠å✐ n ≥ b✳ ❈✉è✐ ❝ị♥❣✱ t❛ t❤✉ ✤÷đ❝ ❝→❝ ❜❛♦ ❤➔♠ s❛✉ Xl (n + 2) ⊆ Xl (n + 1) ∪ S ⊆ Xl (n + 1) ✈ỵ✐ ♠å✐ n ≥ b✳ ❉♦ ✤â X(n) ❧➔ t➟♣ ê♥ ✤à♥❤ ✈ỵ✐ n ✤õ ❧ỵ♥✱ ✈➻ Xl (b + 1) ỳ t ỵ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ❧➔ www.VNMATH.com ✹✸ ❑➳t ❧✉➟♥ ❚â♠ ❧↕✐✱ tr♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ❧↕✐ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦✿ ✧❆s②♠♣t♦t✐❝ st❛❜✐❧✐t② ♦❢ ❝❡♥t❡r s❡ts ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧ ❝õ❛ ◆✳ ❚✳ ❈÷í♥❣✱ ◆✳ ❱✳ ❍♦➔♥❣ ✈➔ P✳ ❍✳ ❑❤→♥❤✳ ❑➳t q✉↔ ❝❤➼♥❤ ỗ s t❤è♥❣ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❝õ❛ ✐✤➯❛♥ tố t ổ t ổ ố ỗ ✤à❛ ♣❤÷ì♥❣✱ ❝❤✐➲✉ ✈➔ ✤ë s➙✉ ❝õ❛ ♠ỉ✤✉♥✱ ♠ỉ✤✉♥ ✈➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✳ ✷✳ ●✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ M −❞➣② tø ❝❤✐➲✉ > k ✈➔ ❝→❝ t➼♥❤ ❝❤➜t✳ ❈❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ tr➯♥ ✈➲ t➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ♠ët sè ♠ð rë♥❣ ❝õ❛ ✤ë s➙✉✳ ✸✳ ❈❤ù♥❣ ♠✐♥❤ ❧↕✐ ❦➳t q✉↔ ❝õ❛ ❜➔✐ ❜→♦ tr➯♥ ✈➲ sü ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✹✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼✳ ❇r♦❞♠❛♥♥✱ ❆s②♠♣t♦t✐❝ st❛❜✐❧✐t② ▼❛t❤✳ ❙♦❝✳✱ ✭✶✮ ❬✷❪ ▼✳ ❇r♦❞♠❛♥♥✱ ✼✹ ♦❢ AssR (M/I n M )✱ Pr♦❝✳ ❆♠❡r✳ ✭✶✾✼✾✮✱ ✶✻ ✲ ✶✽✳ ❚❤❡ ❛s②♠♣t♦t✐❝ ♥❛t✉r❡ ♦❢ t❤❡ ❛♥❛❧②t✐❝ s♣r❡❛❞✱ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜✳ P❤✐❧✳ ❙♦❝✳✱ ✽✻ ✭✶✾✼✾✮✱ ✸✺ ✲ ✸✾✳ ❬✸❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❆✳▲✳ ❋❛❣❤❛♥✐✱ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ✶✷✽ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✶✵✮ ✭✷✵✵✵✮✱ ✷✽✺✶ ✲ ✷✽✺✸✳ ❬✹❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ▲✳❚✳ ◆❤❛♥✱ ♣r✐♠❡s ♦❢ ❝❡rt❛✐♥ Ext✲♠♦❞✉❧❡s✱ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ t♦ ❛♣♣❡❛r ✐♥ ❈♦♠♠✳ ❆❧❣❡❜r❛✳ ❬✺❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳❨✳ ❙❤❛r♣✱ ✧▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s ✧✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✽✳ ❬✻❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ◆✳ ❱✳ ❍♦❛♥❣✱ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❙♦♠❡ ❢✐♥✐t❡ ♣r♦♣❡rt✐❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❊❛st✲❲❡st ❏✳ ▼❛t❤✳✱ ✭✷✮ ✼ ✭✷✵✵✺✮✱ ✶✵✼ ✲ ✶✶✺✳ ❬✼❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ◆✳ ❱✳ ❍♦❛♥❣✱ ❖♥ t❤❡ ✈❛♥✐s❤✐♥❣ ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ♦❢ s✉♣♣♦rts ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ▼❛t❤✳✱ ✭✶✮ ✶✷✻ ▼❛♥✉s❝r✐♣t❛ ✭✷✵✵✽✮✱ ✺✾ ✲ ✼✷✳ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✹✺ ❬✽❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ◆✳ ❱✳ ❍♦❛♥❣ ❛♥❞ P✳ ❍✳ ❑❤❛♥❤✱ ✧❆s②♠♣t♦t✐❝ st❛✲ ❜✐❧✐t② ♦❢ ❝❡rt❛✐♥ s❡ts ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ✐❞❡❛❧s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ t♦ ❛♣♣❡❛r ✐♥ ❈♦♠♠✳ ❆❧❣❡❜r❛✳ ❬✾❪ ◆✳ ❚✳ ❈✉♦♥❣✱ P✳ ❙❝❤❡♥③❡❧ ❛♥❞ ◆✳ ❱✳ ❚r✉♥❣✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼♦❞✉❧♥✱ ❬✶✵❪ ❈✳ ❍✉♥❡❦❡✱ ▼❛t❤✳ ◆❛❝❤r✳✱ ✽✺ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ✭✶✾✼✽✮✱ ✺✼ ✲ ✼✸✳ Pr♦❜❧❡♠s ♦♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✱ ❋r❡❡ r❡s♦❧✉t✐♦♥s ✐♥ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ✭❙✉♥❞❛♥❝❡✱ ❯t❛❤✱ ✶✾✾✵✮✱ ❘❡s✳ ◆♦t❡s ▼❛t❤✳✱ ✷ ✭✶✾✾✷✮✱ ✾✸ ✲ ✶✵✽✳ ❬✶✶❪ ❈✳ ❍✉♥❡❦❡ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣✱ ♠♦❞✉❧❡s✱ ❇❛ss ♥✉♠❜❡rs ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ❬✶✷❪ ▼✳ ❑❛t③♠❛♥✱ ✸✸✾ ✭✶✾✾✸✮✱ ✼✻✺ ✲ ✼✼✾✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ✐♥❢✐♥✐t❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✺✷ ❬✶✸❪ ❑✳ ❑❤❛s❤②❛r♠❛♥❡s❤ ❛♥❞ ❙❤✳ ❙❛❧❛r✐❛♥✱ ✭✷✵✵✷✮✱ ✶✻✶ ✲ ✶✻✻✳ ❋✐❧t❡r r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✭✽✮ ✷✻ ✭✶✾✾✽✮✱ ✷✹✽✸ ✲ ✷✹✾✵✳ ❬✶✹❪ ❑✳ ❑❤❛s❤②❛r♠❛♥❡s❤ ❛♥❞ ❙❤✳ ❙❛❧❛r✐❛♥✱ ❖♥ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ u ă ✻✶✾✽✳ ❚❤❡ f-depth ♦❢ ❛♥ ✐❞❡❛❧ ♦♥ ❛ ♠♦❞✉❧❡✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✼✮ ❬✶✻❪ ●✳ ▲②✉❜❡③♥✐❦✱ ✷✼ ✶✸✵ ✭✷✵✵✶✮✱ ✶✾✵✺ ✲ ✶✾✶✷✳ ❋✐♥✐t❡♥❡ss ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧② ♠♦❞✉❧❡s ✭❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❉✲♠♦❞✉❧❡s t♦ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✮✱ ✶✶✸ Pr♦❝✳ ■♥✈❡♥t✳ ▼❛t❤✳✱ ✭✶✾✾✸✮✱ ✹✶ ✲ ✺✺✳ ❬✶✼❪ ❚✳ ▼❛r❧❡②✱ ❆ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡ ♦✈❡r r✐♥❣s ♦❢ s♠❛❧❧ ❞✐♠❡♥s✐♦ ♥✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳✱ ✭✹✮ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên ✶✵✹ ✭✷✵✵✶✮✱ ✺✶✾ ✲ ✺✷✺✳ http://www.lrc-tnu.edu.vn www.VNMATH.com ✹✻ ❖♥ ❛s②♠♣t♦t✐❝ st❛❜✐❧✐t② ❢♦r s❡ts ♦❢ ♣r✐♠❡s ✐❞❡❛❧s ❬✶✽❪ ▲✳ ▼❡❧❦❡rss♦♥✱ ❝♦♥♥❡❝t❡❞ ✇✐t❤ t❤❡ ♣♦✇❡rs ♦❢ ❛♥ ✐❞❡❛❧✱ ❙♦❝✳✱ ✶✵✼ ✭✶✾✾✵✮✱ ✷✻✼ ✲ ✷✼✶✳ ❬✶✾❪ ▲✳ ▼❡❧❦❡rss♦♥✱ ♠♦❞✉❧❡✱ ❬✷✵❪ ❯✳ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜✳ P❤✐❧✳ ❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❛ ❝r✐t❡r✐♦♥ ❢♦r ❛rt✐♥✐❛♥♥❡ss ♦❢ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛✱ ◆❛❣❡❧ ❛♥❞ P✳ ❙❝❤❡♥③❡❧✱ ✶✵ ✶ ✭✶✾✾✺✮✱ ✷✾✶ ✲ ✸✵✸✳ ❈♦❤♦♠♦❧♦❣✐❝❛❧ ❛♥♥✐❤✐❧❛t♦rs ❛♥❞ ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ r❡❣✉❧❛r✐t②✱ ■♥✿ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✿ ❙②③②✲ ❣✐❡s✱ ▼✉❧t✐♣❧✐❝✐t✐❡s✱ ❛♥❞ ❇✐r❛t✐♦♥❛❧ ❆❧❣❡❜r❛ ✭❙♦✉t❤ ❍❛❞❧②✱ ✶✾✾✷✮✳ ❈♦♥t❡♠♣✳ ▼❛t❤✳✱ ❬✷✶❪ ▲✳❚✳ ◆❤❛♥✱ ✶✺✾ ✳ ▼❛t❤✳ ❙♦❝✳✱ ✭✶✾✾✹✮✱ ✸✵✼ ✲ ✸✷✽✳ ❖♥ ❣❡♥❡r❛❧✐③❡❞ r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ✸✸ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✭✷✵✵✺✮✱ ✼✾✸ ✲ ✽✵✻✳ ❬✷✷❪ ❆✳ ❙✐♥❣❤✱ p✲t♦rs✐♦♥ ❡❧❡♠❡♥ts ✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ✼ ❘❡s✳ ▲❡tt✳✱ ▼❛t❤✳ ✭✷✵✵✵✮✱ ✶✻✺ ✲ ✶✼✻✳ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ... ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ❱➠♥ ❍♦➔♥❣ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✵ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✸ ▼ö❝ ❧ö❝ ❚r❛♥❣ ▼ö❝ ❧ö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳... ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✹ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ s❛✉ ❤❛✐ ♥➠♠... ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✵ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚r✉♥❣ ❉ô♥❣ Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✺ ▼ð ✤➛✉ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛