❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ P❍❆◆ ❚❍➚ ✣➱ ◗❯❨➊◆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ❈❻P ❈❆❖ ◆❍❻❚ ❱⑨ ❚➑◆❍ ❈❆❚❊◆❆❘❨ ❈Õ❆ ●■⑩ ❑❍➷◆● ❚❘❐◆ ▲❼◆ ❈Õ❆ ▼➷✣❯◆ ❍Ú❯ ❍❸◆ ❙■◆❍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✶✾ download by : skknchat@gmail.com ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ P❍❆◆ ❚❍➚ ✣➱ ◗❯❨➊◆ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ❈❻P ❈❆❖ ◆❍❻❚ ❱⑨ ❚➑◆❍ ❈❆❚❊◆❆❘❨ ❈Õ❆ ●■⑩ ❑❍➷◆● ❚❘❐◆ ▲❼◆ ❈Õ❆ ▼➷✣❯◆ ❍Ú❯ ❍❸◆ ❙■◆❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ✈➔ ❧➼ t❤✉②➳t sè ▼➣ sè✿ ✽✹✻✵✶✵✹ ữớ ữợ download by : skknchat@gmail.com ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✹ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ✶✳✼ ✶✳✽ ✶✳✾ ✶✳✶✵ ✣➛② ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣à❛ ♣❤÷ì♥❣ ❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤✐➲✉ ❑r✉❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼æ✤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤✐➲✉ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ổ ố ỗ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ ✳ ❚➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ✈➔♥❤ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✼ ✾ ✶✶ ✶✹ ✶✺ ✶✼ ✶✾ ✷✶ ✷✷ ✷ ▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆● ❈❻P ❈❆❖ ◆❍❻❚ ✷✺ ✷✳✶ ❚➼♥❤ ❝❤➜t✿ ❆♥♥(0 :❆ p) = p, ∀ p ∈ V (❆♥♥R(A)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ t ố ợ ổ ố ỗ ♣❤÷ì♥❣ ❝➜♣ ❝❛♦ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ❚➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ❑➌❚ ▲❯❾◆ ✹✺ ❉❆◆❍ ▼Ö❈ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✹✻ download by : skknchat@gmail.com ◗❯❨➌❚ ✣➚◆❍ ●■❆❖ ✣➋ ❚⑨■ ▲❯❾◆ ❱❿◆ download by : skknchat@gmail.com ✹✾ ✶ ▼Ð ✣❺❯ ❱➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✻✵✱ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✤➣ ợ t ỵ tt ố ỗ ữỡ ỵ tt tr t ổ ổ t❤➸ t❤✐➳✉ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❚♦→♥ ❤å❝ ♥❤÷ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✱ ❍➻♥❤ ❤å❝ ✣↕✐ sè✱ ✣↕✐ sè tê ❤ñ♣✳✳✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❦➳t q✉↔ q✉❛♥ trồ ổ ố ỗ ữỡ t tr✐➺t t✐➯✉ ✈➔ t➼♥❤ ❆rt✐♥✳ ❈❤♦ (R, m) ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ M ❧➔ R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ Hmd (M ) = ✈➔ ❦❤æ♥❣ ❤ú✉ ❤↕♥ s✐♥❤✱ tr♦♥❣ ✤â d = ❞✐♠M ✈➔ Hmi (M ) ❧➔ ♠ỉ✤✉♥ ❆rt✐♥ ✈ỵ✐ ♠å✐ i ≥ ❚❤❡♦ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❬✶✷❪✱ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ R✲♠æ✤✉♥ ❆rt✐♥ A✱ ❦➼ ❤✐➺✉ ❧➔ ❆ttR(A) ❝â ✈❛✐ trá ♥❤÷ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✤è✐ ✈ỵ✐ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ◆❤✐➲✉ ♥❤➔ ❚♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tố t ổ ố ỗ ữỡ ❝➜♣ ❜➜t ❦➻ ✈ỵ✐ ❣✐→ ❝ü❝ ✤↕✐ Hmi (M ) ổ ố ỗ ữỡ t HId(M ) ✈ỵ✐ ❣✐→ ❧➔ ♠ët ✐✤➯❛♥ I ❝õ❛ R✱ ✤➸ tø ✤â ❧➔♠ rã ❝➜✉ tró❝ ♠ỉ✤✉♥ M ✈➔ ỡ s R rữợ t ỹ ❈÷í♥❣✱ ▲➯ ❚❤❛♥❤ ◆❤➔♥ ✈➔ ◆❣✉②➵♥ ❚❤à ❉✉♥❣ ❬✼❪ ✤➣ ✤➲ ①✉➜t ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t✿ download by : skknchat@gmail.com ✷ ❱ỵ✐ ♠é✐ R✲♠ỉ✤✉♥ ❆rt✐♥ A✱ ❆♥♥R (0 :❆ p) = p ✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝❤ù❛ ❆♥♥R(❆) ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤➸ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✱ ❝❤ó♥❣ tỉ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿✧▼ỉ✤✉♥ ✤è✐ ỗ ữỡ t t tr ❝õ❛ ❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥ ❝õ❛ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✧✳ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ t ỗ õ ữỡ ữỡ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➢♥ t➢t ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ✤➛② ✤õ❀ ✤à❛ ♣❤÷ì♥❣ ❤â❛❀ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì❀ ❝❤✐➲✉ ❑r✉❧❧❀ ♠ỉ✤✉♥ ❆rt✐♥❀ tự tr ổ ố ỗ ữỡ t tr ỗ ữỡ ổ ố ỗ ữỡ ♥❤➜t ✣➙② ❧➔ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤ó♥❣ tæ✐ s➩ t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ❝❤➜t✿ ❆♥♥(0 :❆ p) = p ✈ỵ✐ ♠å✐ p ∈ V (❆♥♥R(A))❀ t t ố ợ ổ ố ỗ ♣❤÷ì♥❣ ❝➜♣ ❝❛♦ ♥❤➜t ✈➔ t➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ❣✐→ ❦❤ỉ♥❣ trë♥ ❧➝♥✳ ❚➜t ❝↔ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ tr➼❝❤ ❞➝♥ ❝❤õ ②➳✉ tø t➔✐ ❧✐➺✉ ❬✼❪✳ ◆❤✐➺♠ ✈ư ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔ ❧➔♠ rã ❧↕✐ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ✤â ✈➔ ❤➺ t❤è♥❣ ❧↕✐ t❤❡♦ ♠ët ❜è ủ ỵ ữủ t ữợ sỹ ữợ ỏ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ✈➔ t➙♠ ❤✉②➳t tr♦♥❣ ✈✐➺❝ ✤à♥❤ ữợ õ ỵ ú ù tổ tr s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛②✳ download by : skknchat@gmail.com ✸ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❍å❝ ◗✉② ◆❤ì♥ ❝ị♥❣ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ✈➔ ♥❣♦➔✐ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ❦➯ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ ❤å❝ ✈✐➯♥ tr♦♥❣ ❧ỵ♣ ❝❛♦ ❤å❝ ❚♦→♥ ❦❤â❛ ✷✵✱ ❜↕♥ ❜➧ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ▼➦❝ ❞ò ❜↔♥ t❤➙♥ ✤➣ ❤➳t sù❝ ❝è ❣➢♥❣ ✈➔ ♥ê ❧ü❝ ❧➔♠ ✈✐➺❝ r➜t ♥❤✐➲✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔② ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥✱ t❤í✐ ❣✐❛♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟②✱ tổ rt ữủ sỹ õ ỵ t ổ ❣✐→♦ ✈➔ ❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ◆❣➔② ✷✾ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✾ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ P❤❛♥ ❚❤à ✣é ◗✉②➯♥ download by : skknchat@gmail.com ✹ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✤➣ ❜✐➳t ✈➲ ✤➛② ✤õ❀ ✤à❛ ♣❤÷ì♥❣ ❤â❛❀ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì❀ ❝❤✐➲✉ ❑r✉❧❧❀ ♠æ✤✉♥ ❆rt✐♥❀ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣❀ ❝❤✐➲✉ ◆♦❡t❤❡r❀ ♠æ✤✉♥ ✤è✐ ỗ ữỡ t tr ỗ ❝➜✉ ♣❤➥♥❣ ♥❤➡♠ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ t❤❡♦ ❞ã✐ ❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ s❛✉✳ ✶✳✶ ✣➛② ✤õ ◆ë✐ ❞✉♥❣ ❝õ❛ t✐➳t ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② t❤❡♦ t➔✐ ❧✐➺✉ ❬✶✹❪✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼ët ✈➔♥❤ ❧å❝ ❧➔ ♠ët ✈➔♥❤ R ❝ò♥❣ ✈ỵ✐ ♠ët ❤å (Rn )n ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ R t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ (i) R0 = R❀ (ii) Rn+1 ⊂ Rn ✈ỵ✐ ♠å✐ n 0❀ (iii) Rn Rm ⊂ Rn+m ✈ỵ✐ ♠å✐ n, m 0✳ R download by : skknchat@gmail.com ✺ ❱➼ ❞ö ✶✳✶✳✷✳ (i) ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤✳ ▲➜② R0 = R ✈➔ Rn = ✈ỵ✐ ♠å✐ ❑❤✐ ✤â (Rn)n ❧➔ ♠ët ❧å❝ ❝õ❛ R ✈➔ ❣å✐ ❧➔ ♠ët ❧å❝ t➛♠ t❤÷í♥❣✳ (ii) ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â (I n )n ❧➔ ♠ët ❧å❝ ❝õ❛ R✱ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❧å❝ I ✲❛❞✐❝✳ (iii) ❈❤♦ (Rn )n ❧➔ ♠ët ❧å❝ ❝õ❛ R ✈➔ S ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ R✳ ❑❤✐ ✤â (Rn ∩ S)n ❧➔ ♠ët ❧å❝ ❝õ❛ S ✱ ♥â ✤÷đ❝ ❣å✐ ❧➔ ❧å❝ ❝↔♠ s✐♥❤ tr➯♥ S ✳ n 1✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❧å❝ ✈ỵ✐ ❧å❝ (Rn)n 0✳ ▼ët R✲♠æ✤✉♥ ▼ ❧å❝ ❧➔ ♠ët R✲♠æ✤✉♥ M ❝ị♥❣ ✈ỵ✐ ♠ët ❤å (Mn)n ❝→❝ R✲♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿ (i) M0 = M ❀ (ii) Mn+1 ⊂ Mn ✈ỵ✐ ♠å✐ n 0❀ (iii) Rn Mm ⊂ Mn+m ✈ỵ✐ ♠å✐ n, m 0✳ ❱➼ ❞ư ✶✳✶✳✹✳ (i) ❈❤♦ M ❧➔ ♠ët R✲♠ỉ✤✉♥ ✈➔ R ❝â ❧å❝ t➛♠ t❤÷í♥❣✳ ❑❤✐ ✤â M ❝ơ♥❣ ❝â ♠ët ❧å❝ t➛♠ t❤÷í♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ M0 = M ✈➔ Mn = ✈ỵ✐ ♠å✐ n 1✳ (ii) ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ ①➨t ❧å❝ I ✲❛❞✐❝ ❝õ❛ R✳ ✣à♥❤ ♥❣❤➽❛ ❧å❝ I ✲❛❞✐❝ ❝õ❛ M ❜➡♥❣ ❝→❝❤ ❧➜② Mn = I n M ✳ ❑❤✐ ✤â M ❧➔ ♠ët R✲♠æ✤✉♥ ❧å❝✳ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥ ❧å❝✳ ▲å❝ (Mn)n tr➯♥ M ①→❝ ✤à♥❤ ♠ët tổổ tr M tữỡ t ợ trú õ ❛❜❡❧ ❝õ❛ M ♠➔ (Mn)n ❧➔ ♠ët ❝ì sð ❧➙♥ ❝➟♥ ❝➟♥ ❝õ❛ ✵✳ ❚ỉ♣ỉ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ tæ♣æ ❝↔♠ s✐♥❤ ❜ð✐ ❧å❝ (Mn)n 0✳ ❈❤♦ M ❧➔ ởt Rổ ợ (Mn)n tổổ ữủ (Mn)n0 rữợ t ú tổ ❦❤→✐ ♥✐➺♠ ❞➣② ❈❛✉❝❤②✿ ▼ët ❞➣② (xn) ❝→❝ ♣❤➛♥ tû tr M ữủ ởt ợ ộ k N tỗ t n0 s xm − xn ∈ Mk , ✈ỵ✐ ♠å✐ m, n n0✳ download by : skknchat@gmail.com ✻ ●å✐ T ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❞➣② ❈❛✉❝❤② tr♦♥❣ M ✳ ❚r➯♥ T q✉❛♥ ổ ữủ ợ (xn) (yn) ∈ T, (xn) ∼ (yn) ⇔ ✈ỵ✐ ♠é✐ m N, tỗ t n0 s xn yn ∈ Mm , ✈ỵ✐ ♠å✐ n ≥ n0 ❑❤✐ ✤â q✉❛♥ ❤➺ tr➯♥ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✳ ❑➼ ❤✐➺✉ M = T /∼ = {(xn )|(xn ) ∈ T } ❚÷ì♥❣ tü✱ ❣å✐ S ❧➔ t➟♣ ❝→❝ ❞➣② ❈❛✉❝❤② tr♦♥❣ R ù♥❣ ✈ỵ✐ ❧å❝ (Rn)n≥0✳ ❑➼ ❤✐➺✉ R = S/∼ = {(an)n≥0 | (an) ∈ S} ❑❤✐ ✤â (R, +, ) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥✿ ❱ỵ✐ ♠å✐ (an)n≥0, (bn)n≥0 ∈ R, (an ) + (bn ) = (an + bn )n≥0 ; (an ).(bn ) = (an bn )n≥0 ❚✐➳♣ t t ổ ữợ tr M ữủ ợ (xn), (yn) ∈ M , (xn) + (yn) = (xn + yn)n≥0 ❱ỵ✐ ♠å✐ (an) ∈ R✱ ✈ỵ✐ ♠å✐ (xn) ∈ M ✱ (an).(xn) = (anxn) ❑❤✐ ✤â M ❧➔ R✲♠æ✤✉♥✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✱ tỉ♣ỉ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ M ❜ð✐ ❧å❝ I ✲❛❞✐❝ ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ I ✲❛❞✐❝ ✈➔ ❜❛♦ ✤➛② ✤õ M ✤÷đ❝ ❣å✐ ❧➔ ❜❛♦ ✤➛② ✤õ I ✲❛❞✐❝✳ download by : skknchat@gmail.com ✸✺ ✭✐✐✐✮ =⇒ ✭✐✮ ❈❤♦ p ∈ ❱(❆♥♥Hmd (M )) ●✐↔ sû r➡♥❣ ◆✲❞✐♠(0 :H (M ) p) = d r õ t tỗ t↕✐ ①1 , , ①r ∈ p ❧➟♣ t❤➔♥❤ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝õ❛ Hmd (M ) tr♦♥❣ p✳ ❘ã r➔♥❣✱ ♣❤➛♥ ❤➺ t❤❛♠ sè ♥➔② ❧➔ ❝ü❝ ✤↕✐ tr♦♥❣ p✳ ✣➦t d m :Hmd (M ) (①1 , , ①r )R = ❆1 + + ❆♥ ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t❤✐➸✉ ❝õ❛ :H (M ) (①1, , ①r )R, tr♦♥❣ ✤â ❆✐ qtự ợ ộ m ú ỵ r➡♥❣ ② ❧➔ ♠ët ♣❤➛♥ tû t❤❛♠ sè ❝õ❛ :H (M ) (①1, , ①r )R ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ② ∈/ q✐ ✈ỵ✐ ♠å✐ ✐ t❤ä❛ ♠➣♥ ◆✲❞✐♠ ❆✐ = d − r ✭❳❡♠ ❬✷✶❪✱ ❇ê ✤➲ ✷✳✶✹✮✳ ❱➻ (①1, , ①r ) ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝ü❝ ✤↕✐ ❝õ❛ Hmd (M ) tr♦♥❣ p ♥➯♥ t❛ ❝â d m d m p⊆ ◆✲❞✐♠ ❆ =d−r q✐ ✐ ✈➔ ❞♦ ✤â p ⊆ q✐ ✈ỵ✐ ✐ ♥➔♦ ✤â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ◆✲❞✐♠ ❆✐ = d − r ❚ø ❣✐↔ t❤✐➳t ✭✐✐✐✮✱ t❛ ❝â t❤➸ ❦✐➸♠ tr❛ r➡♥❣ (①1, , ①r ) ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝ü❝ ✤↕✐ ❝õ❛ M/❯M (0) tr♦♥❣ p ❱➻ t tỗ t ởt tố q ss M/❯M (0)/(①1 , , ①r )M/❯M (0) s❛♦ ❝❤♦ ❞✐♠ R/q = d − r ✈➔ p ⊆ q ❍ì♥ ♥ú❛ ✈➻ p ∈ ❙✉♣♣ M/❯M (0)/(①1, , ①r )M/❯M (0) ♥➯♥ t❛ s✉② r❛ p = q ❉♦ ✤â✱ ❞✐♠ R/p = d − r ❱➻ ❆✐ ❧➔ q✐ ✲t❤ù ❝➜♣✱ ♥➯♥ t❤❡♦ ❇ê ✤➲ ✷✳✷✳✹ ✭✐✮ t❛ ✤÷đ❝ ◆✲❞✐♠ ❆✐ ≤ ❞✐♠ R/q✐ ▼➦t ❦❤→❝✱ ✈➻ p ⊆ q✐ ♥➯♥ t❛ ❝â d − r = ◆✲❞✐♠ ❆✐ ≤ ❞✐♠ R/q✐ ≤ ❞✐♠ R/p = d − r ❉♦ ✤â p = q✐ ♥➯♥ s✉② r❛ p ∈ ❆tt(0 :Hmd (M ) (①1 , , ①r )R) download by : skknchat@gmail.com õ tỗ t tè p ∈ ❆ttR(0 :H p ∩ R = p ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d m (M ) (①1 , , ①r )R) s❛♦ ❝❤♦ p ⊆ ❆♥♥(0 :Hmd (M ) p) ⊆ ❆♥♥R (0 :Hmd (M ) p) ∩ R = p ∩ R = p ❉♦ ✤â ❆♥♥(0 :H d m (M ) p) = p ✷✳✸ ❚➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥✳ ◆❤➢❝ ❧↕✐ r➡♥❣ ✈➔♥❤ R ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈ỵ✐ ♠é✐ ❝➦♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè p, q ❝õ❛ R s❛♦ ❝❤♦ p ⊂ q, ♠å✐ ❞➣② ❜➣♦ ❤á❛ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❜➢t ✤➛✉ tø p ❦➳t t❤ó❝ t↕✐ q ✤➲✉ ❝â ❝ị♥❣ ✤ë ❞➔✐ ỳ ú ỵ r R ♥❣❤➽❛ ❧➔ ❞✐♠R/q = ❞✐♠R ✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t❤✐➸✉ q ∈ ❆ssR t❤➻ R ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❞✐♠R/p + ❤tp = ❞✐♠R ✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R ❚❛ ♥â✐ r➡♥❣ ❙✉♣♣ M ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈ỵ✐ ♠é✐ ❝➦♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè p✱ q ∈ ❙✉♣♣ M ✈ỵ✐ p ⊂ q, ♠å✐ ❞➣② ❜➣♦ ❤á❛ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ①✉➜t ♣❤→t tø p ✈➔ ❦➳t t❤ó❝ t↕✐ q ❝â ❝❤✉♥❣ ✤ë ❞➔✐✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ✈➲ t➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ❙✉♣♣ M t❛ t❤➜② r➡♥❣ ❙✉♣♣ M ❧➔ ❝❛t❡♥❛r② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈➔♥❤ R/❆♥♥ M ❧➔ ❝❛t❡♥❛r②✳ ❉♦ ✤â ❙✉♣♣ M ❧➔ ❝❛t❡♥❛r② ✈➔ ❞✐♠ R/p = d ✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ∈ ❆ss M ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❞✐♠ R/p + ❞✐♠ Mp = d ✈ỵ✐ ♠å✐ p ∈ ❙✉♣♣ M ✣➦❝ ❜✐➺t✱ ✈➻ ❞✐♠ R/p = d ✈ỵ✐ ♠å✐ p ∈ ❆ssM/❯M (0) ♥➯♥ ❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥ ❯s✉♣♣ M = ❙✉♣♣ M/❯M (0) ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❞✐♠ R/p + ❞✐♠ Mp = d ✈ỵ✐ ♠å✐ p s M ỵ s t q✉↔ ❝❤➼♥❤ ❝õ❛ t✐➳t ♥➔②✱ ♥â ❝❤➾ r❛ r➡♥❣ t➼♥❤ t Hmd (M ) t tữỡ ữỡ ợ t tr s M rữợ ự ỵ t ự ỳ s download by : skknchat@gmail.com ✸✼ ❇ê ✤➲ s❛✉ ♥â✐ r➡♥❣ ♥➳✉ R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✤➛② ✤õ ✈➔ M ❧➔ ❦❤ỉ♥❣ trë♥ ❧➝♥ t❤➻ ✈ỵ✐ ♠é✐ ♣❤➛♥ ❤➺ t❤❛♠ sè (x1, , xr ) ❝õ❛ M, ♠æ✤✉♥ M/(x1, , xr )M ❧➔ ✤➥♥❣ ❝❤✐➲✉✳ ❇ê ✤➲ ✷✳✸✳✶✳ ●✐↔ t❤✐➳t r➡♥❣ R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✤➛② ✤õ t❤❡♦ tỉ♣ỉ ✈➔ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ s❛♦ ❝❤♦ ❞✐♠ R/p = d ✈ỵ✐ ♠å✐ p ∈ ❆ssM ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ♣❤➛♥ ❤➺ t❤❛♠ sè (①1, , ①r ) ❝õ❛ M ✈➔ ♠é✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t❤✐➸✉ p ❝õ❛ M/(①1, , ①r )M, t❛ ❝â ❞✐♠R/p = d − r✳ m✲❛❞✐❝ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ (①1, , ①r ) ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝õ❛ M ✈➔ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t tè✐ t❤✐➸✉ ❝õ❛ M/(x1, , xr )M ❚r÷í♥❣ ❤đ♣ r = d ❧➔ t➛♠ t❤÷í♥❣✳ ●✐↔ sû r < d✳ ❱➻ (①1, , ①r ) ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝õ❛ M ♥➯♥ t❛ ❝â ❞✐♠(R/❆♥♥ M + (①1, , ①r )R) = ❞✐♠(M/(①1, , ①r )M ) = d − r ❍ì♥ ♥ú❛✱ ✈➻ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t❤✐➸✉ ❝õ❛ ❆♥♥ M + (①1, , ①r )R ♥➯♥ t ỵ t õ R/p d r õ tỗ t tố tố t❤✐➸✉ q ❝õ❛ ❆♥♥ M s❛♦ ❝❤♦ q ⊆ p ❱➻ q ∈ ❆ssRM ♥➯♥ t❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â ❞✐♠ R/q = d ❍ì♥ ♥ú❛✱ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t✐➸✉ ❝õ❛ q + (①1, , ①r )R ❉♦ ✤â ❤t(p/q) ❦❤ỉ♥❣ ✈÷đt q✉→ r✳ ❱➻ R/q ❧➔ ❝❛t❡♥❛r② ❝❤✐➲✉ ❧➔ d ♥➯♥ t❛ ❝â ❤t (p/q) + ❞✐♠R/p = d ❙✉② r❛ ❞✐♠ R/p = d − ❤t(p/q) download by : skknchat@gmail.com ✸✽ ❉♦ ❤t(p/q) ≤ r ♥➯♥ ❞✐♠R/p = d − ❤t (p/q) ≥ d − r ▼➦t ❦❤→❝ ❞✐♠R/p ≤ d − r ❝❤♦ ♥➯♥ ❞✐♠R/p = d − r ❇ê ✤➲ ✷✳✸✳✷✳ ❈❤♦ p ∈ ❱(❆♥♥ Hmd (M )) s❛♦ ❝❤♦ ❞✐♠ Mp + ❞✐♠R/p = d ❑❤✐ ✤â ❆♥♥(0 :H d m (M ) p) = p ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② p ⊇ ❆♥♥ Hmd (M ) ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R s❛♦ ❝❤♦ ❞✐♠ Mp + ❞✐♠R/p = d ✣➦t ❞✐♠R/p = d − r ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ s✉② r❛ Mp = r t tỗ t ởt ♥❣✉②➯♥ tè ❝õ❛ q ∈ ❆ss M s❛♦ ❝❤♦ q ⊆ p ✈➔ ❤t (p/q) = r ❱➻ ❞✐♠R/q ≥ ❞✐♠R/p + ❤t(p/q) = d, ♥➯♥ t❛ s✉② r❛ ❞✐♠R/q = d ú ỵ r R/pR = R/p = d r t tỗ t ởt ♥❣✉②➯♥ tè p ∈ ❆ssRR/pR s❛♦ ❝❤♦ ❞✐♠ R/p = d−r ❱➻ p ∈ ❆ssRR/pR ♥➯♥ t❛ ❝â p ∩ R ∈ ❆ss R/p, tù❝ ❧➔ p ∩ R = p ú ỵ r tỹ R R ỗ t õ ỗ tọ ỵ ỵ tỗ t ởt tố q ∈ ❙♣❡❝ R s❛♦ ❝❤♦ q ∩ R = q, q ⊆ p ✈➔ download by : skknchat@gmail.com ✸✾ ❤t (p/q) ≥ r ❱➻ d = ❞✐♠ R/q = ❞✐♠ R/qR ≥ ❞✐♠ R/q = ❞✐♠ R/p + ❤t (p/q) ≥d−r+r = d ❙✉② r❛ ❞✐♠ R/q = d ỡ ỳ ỗ s Rq Rq ❧➔ ❤♦➔♥ t♦➔♥ ♣❤➥♥❣ ✈➔ Mq = ♥➯♥ t❛ ❝â Mq ⊗Rq Rq ∼ = Mq = ❉♦ ✤â q ∈ ❙✉♣♣RM ❱➻ ❞✐♠ R/q = d ✈➔ p ⊇ q ♥➯♥ t❛ ❝â ❆♥♥RHmd (M ) ❉♦ ✤â tø t➼♥❤ ❝❤➜t ✤è✐ ♥❣➝✉ ▼❛t❧✐s t❛ s✉② r❛ ❆♥♥R(0 :H d m (M ) p ⊇ p) = p ❈✉è✐ ❝ò♥❣ t❛ ❝â p ⊆ ❆♥♥(0 :Hmd (M ) p) ⊆ ❆♥♥R (0 :Hmd (M ) p) ∩ R = p ∩ R = p ❉♦ ✤â ❆♥♥(0 :H d m (M ) p) = p ✣à♥❤ ỵ s tữỡ ữỡ ❯s✉♣♣ M ❧➔ ❝❛t❡♥❛r②✳ ✭✐✐✮ H d (M ) t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ✭✯✮✳ ♠ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ =⇒ ✭✐✐✮ ❈❤♦ p ∈ V (❆♥♥Hmd (M )) ❱➻ ❯s✉♣♣M ❧➔ ❝❛t❡♥❛r② ♥➯♥ ❞✐♠ R/p + ❞✐♠ Mp = d ❉♦ ✤â t❤❡♦ ❇ê ✤➲ ✷✳✸✳✷ t❛ ❝â✿ download by : skknchat@gmail.com ✹✵ ❆♥♥(0 :H (M ) p) = p ❱➟② Hmd (M ) t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t ✭✯✮✳ ✭✐✐✮ =⇒ ✭✐✮ ❈❤♦ p ∈ ❯s✉♣♣ M ✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❞✐♠ R/p + ❞✐♠ Mp = d ◆➳✉ p = m t❤➻ rã r➔♥❣ d m ❞✐♠ R/m + ❞✐♠ Mm = + ❞✐♠ M = d ●✐↔ sû p = m✳ ✣➦t ❞✐♠ R/p = d − r ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❞✐♠ Mp = r ❱➻ p ⊇ ❆♥♥ M/❯M (0) ♥➯♥ t❛ ❝â ❞✐♠ M/❯M (0)/p(M/❯M (0)) = ❞✐♠ R/p = d − r t tỗ t ởt t số ỹ ✤↕✐ (①1, , ①r ) ❝õ❛ M/❯M (0) tr♦♥❣ p p sM t tỗ t↕✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ∈ ❯s✉♣♣R M s❛♦ ❝❤♦ p ∩ R = p ✣➦t M1 = M /❯M (0) ❱➻ (①1, , ①r ) ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ sè ❝õ❛ M/❯M (0) ♥➯♥ ♥â ❝ô♥❣ ❧➔ ♠ët ♣❤➛♥ t❤❛♠ sè ❝õ❛ ♠æ✤✉♥ ✤➛② ✤õ m✲❛❞✐❝ M/❯M (0) ❝õ❛ M/❯M (0) ❱➻ M1 ❧➔ ♠ỉ✤✉♥ t❤÷ì♥❣ ❝õ❛ ♠ỉ✤✉♥ M/❯M (0) ✈➔ ❞✐♠ M1 = ❞✐♠ M/❯M (0) ♥➯♥ (①1 , , ①r ) ❧➔ ♠ët ♣❤➛♥ ❤➺ t❤❛♠ số M1 ú ỵ r p R M1 /(①1 , , ①r−1 )M1 ♥➯♥ p ⊇ p1 ✈ỵ✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t✐➸✉ p1 ∈ ❙✉♣♣RM1/(①1, , ①r−1)M1 ❱➻ ①r ❧➔ ♠ët ♣❤➛♥ tû t❤❛♠ sè ❝õ❛ M1/(①1, , ①r−1)M1 ♥➯♥ t❤❡♦ ❇ê ✤➲ ✷✳✸✳✶ t❛ s✉② r❛ ①r ∈/ p1 ✣➦t p1 = p1 ∩ R ❑❤✐ ✤â ①r ∈/ p1 ❱➻ ①r ∈ p ♥➯♥ t❛ ❝â p ⊃ p1 ✈➔ p = p1 tữỡ tỹ ữ tr tỗ t tè tè✐ t✐➸✉ p2 ∈ ❙✉♣♣R M1 /(①1 , , ①r−2 )M1 download by : skknchat@gmail.com ✹✶ s❛♦ ❝❤♦ p1 ⊇ p2✳ ✣➦t p2 = p2 ∩R ❑❤✐ ✤â p1 ⊃ p2 ✈➔ p1 = p2 ✈➻ ①r−1 ∈ p1 \p2 tử q tr tr s r ữợ t❛ ♥❤➟♥ ✤÷đ❝ ♠ët ❞➣② ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝❤ù❛ ❆♥♥ M p ⊃ p1 ⊃ p2 ⊃ ⊃ pr s❛♦ ❝❤♦ p✐ = p✐+1 ✈ỵ✐ ♠å✐ ✐ = 1, 2, , r − 1✳ ❉♦ ✤â ❞✐♠ Mp = r ❑❤✐ ✤â ❞✐♠R/p + ❞✐♠ Mp = d − r + r = d ✈ỵ✐ ♠å✐ p ∈ ❯s✉♣♣ M ❱➟② ❯s✉♣♣ M ❧➔ ❝❛t❡♥❛r②✳ ❍➺ q✉↔ s❛✉ ữủ s r tứ ỵ ởt ✤➦❝ tr÷♥❣ ✈➲ t➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ❝→❝ ♠✐➲♥ ♥❣✉②➯♥ ◆♦❡t❤❡r q✉❛ t➼♥❤ ❝❤➜t ✭✯✮✳ ❍➺ q✉↔ ✷✳✸✳✹✳ ●✐↔ sû (R, m) ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r ❝❤✐➲✉ d✳ ❑❤✐ ✤â R ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ Hmd (R) tọ t t ỵ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✭✐✮ ❆♥♥(0 :H (M ) p) = p ✈ỵ✐ ♠å✐ p ∈ ❱(❆♥♥ Hmd (M )) ✭✐✐✮ ❯s✉♣♣ ▼ ❧➔ ❝❛t❡♥❛r②✳ ✭✐✐✐✮ ❯s✉♣♣ M = {p ∩ R : p ∈ ❯s✉♣♣RM } ✭✐✈✮ ❱ỵ✐ ♠é✐ ❞➣② ①1, , ①d ❝→❝ ♣❤➛♥ tû tr♦♥❣ m✱ x = (①1, , ①d) ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ Hmd (M ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ M/❯M (0) d m ự ứ ỵ t s r ỡ ỳ tứ ỵ t❛ ✤÷đ❝ ✭✐✮ ⇔ ✭✐✐✮ ❱➟② t❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤♦ = M0 ⊂ M1 ⊂ ⊂ Mt = M ❧➔ ♠ët ❧å❝ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M, tr♦♥❣ ✤â M✐−1 ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❧ỵ♥ ♥❤➜t ❝õ❛ M✐ ✈ỵ✐ ❞✐♠ M✐−1 < ❞✐♠ M✐ ✈ỵ✐ download by : skknchat@gmail.com ✹✷ ♠å✐ ✐ = 1, , t ổ ổ tỗ t t ❣å✐ ❧å❝ ♥➔② ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ M ✭❳❡♠ ❬✻❪✮✳ ❈❤♦ ❞✐♠ M✐ = d✐ ✈ỵ✐ ✐ = 1, , t ❑❤✐ ✤â ❞➵ ❞➔♥❣ t❛ ❝â t❤➸ ❦✐➸♠ tr❛ r➡♥❣ ❙✉♣♣ M = ✐=1, ,t ❙✉♣♣ M✐/M✐−1 ❱ỵ✐ ♠é✐ ✐ = 1, , t t❛ ❦➼ ❤✐➺✉ ❞✐♠ R/p = d✐ ✈ỵ✐ ♠å✐ p ∈ ❆ss M✐/M✐−1 ❍➺ q✉↔ ✷✳✸✳✻✳ ❙✉♣♣ M ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❍dm (M✐/M✐−1) t❤ä❛ ✐ ♠➣♥ t➼♥❤ ❝❤➜t ✭✯✮ ✈ỵ✐ ♠å✐ ✐ = 1, , t ❇➙② ❣✐í t❛ ♥❣❤✐➯♥ ❝ù✉ ♠ët tr ữỡ ổ tr ú ỵ r➡♥❣ ❜➜t ❦➻ ♠✐➲♥ ♥❣✉②➯♥ ❝❤✐➲✉ ✷ ✤➲✉ ❧➔ ❝❛t❡♥❛r②✱ ữ tỗ t tr ữỡ ổ ❝❛t❡♥❛r② ❝❤✐➲✉ d ✈ỵ✐ d ≥ ✭❳❡♠ ❬✶✱ ✭✽✮❪✮✳ ▼➺♥❤ ✤➲ ✷✳✸✳✼✳ ❈❤♦ R ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ❝❤✐➲✉ ✸ ❦❤ỉ♥❣ ❝❛t❡♥❛r②✳ ✣➦t ❯ = {p ∈ ❙♣❡❝ R : ❞✐♠ R/p + ❤tp = 2}; ❱ = {p ∈ ❙♣❡❝ R : ❞✐♠ R/p + ❤tp = 3} ❑❤✐ ✤â ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣✿ ✭✐✮ ❯s✉♣♣ R = ❙♣❡❝ R = ❯ ∪ ❱ ✈➔ ❯, ❱ = ∅ ✭✐✐✮ ❆♥♥(0 :❍ (R) p) = p ợ p ữ (0 :❍ (R) p) = p ✈ỵ✐ ♠å✐ p ∈ ❯ p t tỗ t p ❙✉♣♣ R/❯R(0) s❛♦ ❝❤♦ p ∩ R = p ◆❤÷♥❣ p t ổ tỗ t p ❙✉♣♣ R/❯R(0) s❛♦ ❝❤♦ p ∩ R = p ✭✐✈✮ ◆✲❞✐♠ ❍2m(R) = ✈➔ ❞✐♠ R/❆♥♥ ❍2m(R) = 3 m m download by : skknchat@gmail.com ✹✸ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❱➻ R ❦❤æ♥❣ ❝❛t❡♥❛r② ♥➯♥ ❯ = ∅ ❉♦ ❞✐♠ R ❱ = ∅ ❘ã r➔♥❣ ❙♣❡❝ R = ỵ t ❝â ❆♥♥(0 :H ❆♥♥(0 :H (M ) p) = p ✈ỵ✐ ♠å✐ p ∈ V m (M ) p) = p, = ♥➯♥ ✈ỵ✐ ♠å✐ p ∈ U ✈➔ m ✭✐✐✐✮ ❙✉② r❛ tø ✭✐✐✮ ✈➔ ✣à♥❤ ỵ p õ R/p = ỗ R R t tỗ t p R s❛♦ ❝❤♦ p ∩ R = p ❱➻ p = m ♥➯♥ p = mR ❙✉② r❛ < ❞✐♠ (R/p) ≤ ❞✐♠ R/p = ❉♦ ✤â ❞✐♠ (R/p) = 1✳ ▼➦t ❦❤→❝ t❤❡♦ ✭✐✐✐✮ t❛ ❝â p ∈/ ❙✉♣♣(R/❯R(0))✳ ❚ø ✤â s✉② r❛ p ❆♥♥R❍3m(R) ❍ì♥ ♥ú❛✱ ỗ tỹ R R tọ ỵ õ = t tỗ t q ❆ss R s❛♦ ❝❤♦ q ⊂ p ✈➔ q = p ❉♦ ✤â ❞✐♠ R/q ≥ ❱➻ p ❆♥♥R❍3m(R) ♥➯♥ ❞✐♠ R/q = ❑❤✐ ✤â✱ t❤❡♦ ✭❬✷❪✱ ❍➺ q✉↔ ✶✶✳✸✳✸✮ t❛ ✤÷đ❝ q ∈ ❆ttR❍2m(R) ✈➔ ❞♦ ✤â q ⊇ ❆♥♥R❍2m(R) ❚ø ✤â s✉② r❛ ◆✲❞✐♠ ❍2m(R) = R/R2m(R) ú ỵ r ỵ t õ 2m(R) 2. t ❍2m(R) = ▼➦t ❦❤→❝ ✈➻ q ∈ ❆ttR❍2m(R) ∩ ❆ss R✱ ♥➯♥ t❛ ❝â q ∩ R ∈ ❆tt ❍2m (R) ∩ ❆ss R download by : skknchat@gmail.com ✹✹ ❉♦ R ❧➔ ♠ët ♠✐➲♥ ♥❣✉②➯♥ ♥➯♥ q ∩ R = ❱➻ t❤➳ = ❆♥♥❍2m(R) ❱➟② ❞✐♠ R/❆♥♥ ❍2m(R) = download by : skknchat@gmail.com ✹✺ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ tr♦♥❣ ❬✼❪✳ ❈❤÷ì♥❣ ✶✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲✿ ✤➛② ✤õ❀ ✤à❛ ♣❤÷ì♥❣ ❤â❛❀ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì❀ ❝❤✐➲✉ ❑r✉❧❧❀ ♠ỉ✤✉♥ ❆rt✐♥❀ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣❀ tr ổ ố ỗ ữỡ t tr ỗ ữỡ ú tổ tr ởt số ổ ố ỗ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ ❝➜♣ ❝❛♦ ♥❤➜t t❤❡♦ ❬✼❪✱ ❝ư t❤➸ ❧➔✿ t➼♥❤ ❝❤➜t✿ ❆♥♥(0 :❆ p) = p ✈ỵ✐ ♠å✐ p ∈ V (❆♥♥R (A))❀ t➼♥❤ ❝❤➜t ✭✯✮ ✤è✐ ✈ỵ✐ ổ ố ỗ ữỡ t t➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥✳ download by : skknchat@gmail.com ✹✻ ❉❆◆❍ ▼Ö❈ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ❇r♦❞♠❛♥♥✱ ▼✳✱ ❆ ♣❛rt✐❝✉❧❛r ❝❧❛ss ♦❢ r❡❣✉❧❛r ❞♦♠❛✐♥s✱ ❏✳ ❆❧❣❡❜r❛✱ ✺✹✱ ✭✶✾✼✽✮✱ ✸✻✻✲✸✼✸✳ ❬✷❪ ❇r♦❞♠❛♥♥✱ ▼✳ ❛♥❞ ❙❤❛r♣✱ ❘✳ ❨✳✱ Local Cohomology : An Algebraic Introduction with Geometric Applications✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✶✾✾✽✳ ❬✸❪ ❈♦❤❡♥✱ ▲✳ ❙✳✱ ❖♥ t❤❡ str✉❝t✉r❡ ❛♥❞ ✐❞❡❛❧ t❤❡♦r② ♦❢ ❝♦♠♣❧❡t❡ ❧♦❝❛❧ r✐♥❣s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✺✾ ✭✶✾✹✻✮✱ ♣♣✳ ✺✹✲✶✵✻✳ ❬✹❪ ❈✉♦♥❣✱ ◆✳ ❚✳ ❛♥❞ ◆❤❛♥✱ ▲✳ ❚✳✱ ❉✐♠❡♥s✐♦♥✱ ♠✉❧t✐♣❧✐❝✐t② ❛♥❞ ❍✐❧❜❡rt ❢✉♥❝t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✱ ❊❛st✲❲❡st ❏✳ ▼❛t❤✳✱ ✶ ✭✷✮✱ ✭✶✾✾✾✮ ♣♣✳ ✶✼✾✲✶✾✻✳ ❬✺❪ ❈✉♦♥❣✱ ◆✳ ❚✳ ❛♥❞ ◆❤❛♥✱ ▲✳ ❚✳✱ ❖♥ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥s✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ✸✵ ✭✷✵✵✷✮✱ ♣♣✳ ✶✷✶✲✶✸✵✳ ❬✻❪ ❈✉♦♥❣✱ ◆✳ ❚✳ ❛♥❞ ◆❤❛♥✱ ▲✳ ❚✳✱ ❖♥ ♣s❡✉❞♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❛♥❞ ♣s❡✉❞♦ ❣❡♥❡r❛❧✐✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ❏✳ ❆❧❣❡❜r❛ ✷✻✼ ✭✶✮ ✭✷✵✵✸✮✱ ♣♣✳ ✶✺✻✲✶✼✼✳ download by : skknchat@gmail.com ✹✼ ❬✼❪ ❈✉♦♥❣✱ ◆✳ ❚✳ ❛♥❞ ❉✉♥❣✱ ◆✳ ❚✳ ❛♥❞ ◆❤❛♥✱ ▲✳ ❚✳✱ ❚♦♣ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ❛♥❞ t❤❡ ❝❛t❡♥❛r✐❝✐t② ♦❢ t❤❡ ✉♥♠✐①❡❞ s✉♣♣♦rt ♦❢ ❛ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠♦❞✉❧❡✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✸✺ ✭✷✵✵✼✮✱ ♣♣✳ ✶✻✾✶✲✶✼✵✶✳ ❬✽❪ ❋❡rr❛♥❞✱ ❉✳ ❛♥❞ ❘❛②♥❛✉❞✱ ▼✳✱❋✐❜r❡s ❢♦r♠❡❧❧❡s ❞✬✉♥ ❛♥♥❡❛✉ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥✱ ❆♥♥✳ ❙❝✐✳❊✬❝♦❧❡ ◆♦r♠✳ ❙✉♣✳ ✸ ✭✹✮ ✭✶✾✼✵✮✱ ♣♣✳ ✷✾✺✲✸✶✶✳ ❬✾❪ ❑✐r❜②✱ ❉✳✱ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ❛♥❞ ❍✐❧❜❡rt ♣♦❧②♥♦♠✐❛❧s✱ ◗✉❛rt✳ ✺✳ ▼❛t❤✳ ❖①❢♦r❞ ✭✷✮ ✷✹ ✭✶✾✼✺✮✱ ♣♣✳ ✹✼✲✺✼✳ ❬✶✵❪ ❑✐r❜②✱ ❉✳✱ ❈♦♣r✐♠❛r② ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✻ ✭✷✮ ✭✶✾✼✸✮✱ ♣♣✳ ✺✼✶✲✺✼✻✳ ❬✶✶❪ ❑✐r❜②✱ ❉✳✱ ❉✐♠❡♥s✐♦♥ ❛♥❞ ❧❡♥❣t❤ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ✭❙❡r✳✷✮ ✹✶ ✭✶✾✾✵✮✱ ♣♣✳ ✹✶✾✲✹✷✾✳ ❬✶✷❪ ▼❛❝❞♦♥❛❧❞✱ ■✳ ●✳✱ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠✲ ♠✉t❛t✐✈❡ r✐♥❣✱ ❙②♠♣♦s✐❛ ▼❛t❤❡♠❛t✐❝❛ ✶✶ ✭✶✾✼✸✮✱ ♣♣✳ ✷✸✲✹✸✳ ❬✶✸❪ ▼❛t❧✐s✱ ❊✳✱ ❚❤❡ ❑♦③✉❧ ❝♦♠♣❧❡① ❛♥❞ ❞✉❛❧✐t②✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✷ ✭✶✮ ✭✶✾✻✵✮✱ ♣♣✳ ✽✼✲✶✹✾✳ ❬✶✹❪ ▼❛ts✉♠✉r❛✱ ❍✳✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✭✶✾✽✻✮✳ ❬✶✺❪ ▼❛ts✉♠✉r❛✱ ❍✳✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛ ✱ ❇❡♥❥❛♠✐♥✱ ❲✳ ❆✳✱ ◆❡✇ ❨♦r❦✱ ✭✶✾✼✵✮✳ ❬✶✻❪ ▼❛t❧✐s✱ ❊✳✱ ■♥❥❡❝t✐✈❡ ♠♦❞✉❧❡s ♦✈❡r ◆♦❡t❤❡r✐❛♥ r✐♥❣s✱ P❛❝✐❢✐❝ ❏✳ ▼❛t❤✳ ✭✶✾✻✽✮✱ ♣♣✳ ✺✶✶✲✺✷✽✳ download by : skknchat@gmail.com ✹✽ ❬✶✼❪ ◆❛❣❛t❛✱ ▼✳✱ ❖♥ t❤❡ ❝❤❛✐♥ ♣r♦❜❧❡♠ ♦❢ ♣r✐♠❡ ✐❞❡❛❧✱ ◆❛❣❛②❛ ▼❛t❤✳ ❏✳ ✽✵ ✭✶✾✽✵✮✱ ♣♣✳ ✶✵✼✲✶✶✻✳ ❬✶✽❪ ❘❛t❧✐❢❢✱ ▲✱ ❏✳✱ ❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❝❛t❡♥❛r② r✐♥❣s✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳✱ ✾✸ ✭✶✾✼✶✮✱ ♣♣✳ ✶✵✼✵✲✶✶✵✽✳ ❬✶✾❪ ❘♦❜❡rts✱ ❘✳ ◆✳✱ ❑r✉❧❧ ❞✐♠❡♥s✐♦♥ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦✈❡r q✉❛s✐✲❧♦❝❛❧ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ✭❙❡r✳✷✮ ✷✻ ✭✶✾✼✺✮✱ ♣♣✳ ✷✻✾✲ ✷✼✸✳ ❬✷✵❪ ❙❤❛r♣✱ ❘✳ ❨✳✱ ❆ ♠❡t❤♦❞ ❢♦r t❤❡ st✉❞② ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✱ ✇✐t❤ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r✱ ✐♥✿ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ▼❛t❤✳ ❙❝✐✳ ❘❡s✳ ■♥st✳ P✉❜❧✳✱ ◆♦✳ ✶✺✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✶✾✽✾✱ ♣♣✳ ✹✹✸✲✹✻✺✳ ❬✷✶❪ ❚❛♥❣✱ ❩✳ ▼✳ ❛♥❞ ❩❛❦❡r✐✱ ❍✳✱❈♦✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❛♥❞ ♠♦❞✉❧❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝✲t✐♦♥s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✷✷ ✭✻✮ ✭✶✾✾✹✮✱ ♣♣✳ ✷✶✼✸✲✷✷✵✹✳ download by : skknchat@gmail.com ✹✾ ◗❯❨➌❚ ✣➚◆❍ ●■❆❖ ✣➋ ❚⑨■ ▲❯❾◆ ❱❿◆ download by : skknchat@gmail.com