ĐẠI HỌC THÁI NGUYÊN ĐẠI HỌC SƯ PHẠM DƯƠNG THỊ GIANG MÔĐUN COHEN-MACAULAY VỚI CHIỀU > s VÀ MỘT SỐ KẾT QUẢ TRÊN MÔĐUN ĐỐI ĐỒNG ĐIỀU ĐỊA PHƯƠNG 2013 Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ r➺♥❣ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ❤♦➭♥ t♦➭♥ tr✉♥❣ t❤ù❝ ✈➭ ❦❤➠♥❣ trï♥❣ ❧➷♣ ✈í✐ ➤Ị t➭✐ ❦❤➳❝✳ ◆❣✉å♥ t➭✐ ❧✐Ư✉ sư ❞ơ♥❣ ❝❤♦ ✈✐Ư❝ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ sù ➤å♥❣ ý ❝đ❛ ❝➳ ♥❤➞♥ ✈➭ tỉ ❝❤ø❝✳ ❈➳❝ t❤➠♥❣ t✐♥✱ t➭✐ ❧✐Ư✉ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ ➤➢ỵ❝ ❣❤✐ râ ♥❣✉å♥ ❣è❝✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ❉➢➡♥❣ ❚❤Þ ●✐❛♥❣ ❳➳❝ ♥❤❐♥ ❳➳❝ ♥❤❐♥ ❝ñ❛ tr➢ë♥❣ ❦❤♦❛ ❝❤✉②➟♥ ♠➠♥ ❝ñ❛ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ ❉✉♥❣ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ s❛✉ ✷ ♥➝♠ ❤ä❝ t❐♣ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✳ ❱í✐ ❧ß♥❣ ❦Ý♥❤ trä♥❣ ✈➭ ❜✐Õt ➡♥ s➞✉ s➽❝ t➠✐ ①✐♥ ➤➢ỵ❝ ❜➭② tá ❧ê✐ ❝➯♠ ➡♥ ❝❤➞♥ t❤➭♥❤ tí✐ ❚✐Õ♥ sÜ ◆❣✉②Ơ♥ ❚❤Þ ❉✉♥❣✱ ♥❣➢ê✐ ❝➠ ❦Ý♥❤ ♠Õ♥ ➤➲ ❤Õt ❧ß♥❣ ❣✐ó♣ ➤ì✱ ❞➵② ❜➯♦✱ ➤é♥❣ ✈✐➟♥ ✈➭ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➠✐ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❚❤➳✐ ◆❣✉②➟♥✱ ❧➲♥❤ ➤➵♦ ❦❤♦❛ ❚♦➳♥✱ ❧➲♥❤ ➤➵♦ ❦❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝ ❝đ❛ ❚r➢ê♥❣ ➤➲ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❣✐ó♣ ➤ì t➠✐ ❤♦➭♥ t❤➭♥❤ tèt ♥❤✐Ư♠ ✈ơ ❤ä❝ t❐♣ ❝đ❛ ♠×♥❤✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ t❤➬② ❝➠ ➤➲ t❤❛♠ ❣✐❛ ❣✐➯♥❣ ❞➵② ❝❤♦ ❧í♣ ❈❛♦ ❤ä❝ ❝❤✉②➟♥ ♥❣➭♥❤ ❚♦➳♥ ❦❤♦➳ ✶✾✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❝➯♠ ➡♥ ♥❤÷♥❣ ♥❣➢ê✐ t❤➞♥ ②➟✉ tr♦♥❣ ❣✐❛ ➤×♥❤✱ ❜➵♥ ❜❒ ➤➲ ❧✉➠♥ ❝❤♦ t➠✐ ♥✐Ị♠ t✐♥ ✈➭ ➤é♥❣ ❧ù❝ ➤Ó ❤ä❝ t❐♣ tèt✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ❉➢➡♥❣ ❚❤Þ ●✐❛♥❣ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✐✐✐ ▼ơ❝ ❧ơ❝ ❚r❛♥❣ ▲ê✐ ❝❛♠ ➤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲ê✐ ❝➯♠ ➡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ❍Ö t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸✳ ❍➭♠ tö ♠ë ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹✳ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✺✳ ❱Ị ♠ét sè ♠ë ré♥❣ ❝đ❛ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❈❤➢➡♥❣ ✷✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✷✳✶✳ ❉➲② ❝❤Ý♥❤ q✉② ✈í✐ ❝❤✐Ị✉ >s >s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ >s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❈❤➢➡♥❣ ✸✳ ▼ét sè ❦Õt q✉➯ tr➟♥ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✸✵ ✸✳✶✳ a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷✳ ▼ét sè ❦Õt q✉➯ tr➟♥ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✶ ▼ë ➤➬✉ ❈❤♦ (R, m) R ị ữ ❤➵♥ s✐♥❤ ✈í✐ ❑❤➳✐ ♥✐Ư♠ M ✲❞➲② dim M = d ✈í✐ ❝❤✐Ị✉ >s a ❧➭ ♠ét ✐➤➟❛♥ ❝đ❛ R✱ M ❧➭ ✈➭ ❝❤♦ −1 s ❧➭ ♠ét sè ♥❣✉②➟♥✳ ➤➲ ➤➢ỵ❝ ➤➢❛ r❛ ❜ë✐ ❇r♦❞♠❛♥♥✲◆❤➭♥ ❬❇◆❪ ♥❤➢ ❧➭ ♠ét sù ♠ë ré♥❣ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ◆❤➭♥ ❬◆❪ tr➢í❝ ➤ã✳ ❱í✐ ❦❤➳✐ ♥✐Ö♠ ♥➭②✱ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❝❤Ý♥❤ q✉②✱ f ✲❞➲② M ✲❞➲② q✉❡♥ ❜✐Õt ✈➭ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ➤➲ t➢➡♥❣ ø♥❣ trë t❤➭♥❤ ❝➳❝ ✈í✐ ❝❤✐Ị✉ ❦❤➳✐ ♥✐Ư♠ M ✲❞➲② > −1, 0, ✈í✐ ❝❤✐Ị✉ ✭①❡♠ ❬❇◆❪✱ ❬❈❙❚❪✱ ❬◆❪✱✳✳✳✮✳ ◆➝♠ ✷✵✵✾✱ ❞ï♥❣ > s✱ ◆✳ ❩❛♠❛♥✐ ❬◆❩❪ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❧í♣ ♠➠➤✉♥ t❤á❛ ♠➲♥ ♠ä✐ ❤Ư t❤❛♠ sè ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✳ M ✲❞➲② f ✲♠➠➤✉♥ > s ✈➭ ❣ä✐ ❧➭ ❑❤✐ ➤ã✱ ❝➳❝ ❧í♣ ♠➠➤✉♥ q✉❡♥ ❜✐Õt tr♦♥❣ ➤➵✐ sè ❣✐❛♦ ❤♦➳♥ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❙❝❤❡♥③❡❧✲❚r✉♥❣ ❬❈❙❚❪✱ ✈í✐ ❝❤✐Ị✉ f ✲♠➠➤✉♥ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❈➢ê♥❣✲ s✉② ré♥❣ ➤➢❛ r❛ ❜ë✐ ◆❤➭♥✲▼♦r❛❧❡s ❬◆▼❪ t➢➡♥❣ ø♥❣ trë t❤➭♥❤ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > −1, 0, 1✳ ▲✉❐♥ ✈➝♥ ♥❤➺♠ tr×♥❤ ❜➭② ❧➵✐ ❝➳❝ ❦Õt q✉➯ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❜➭✐ ❜➳♦ ❝ñ❛ ◆✳ ❩❛♠❛♥✐ ❬◆❩❪ ✧❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼♦❞✉❧❡s ✐♥ ❉✐♠❡♥s✐♦♥ ♦♥ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✧ >s ❛♥❞ ❘❡s✉❧ts ➤➝♥❣ tr➟♥ t➵♣ ❝❤Ý ❈♦♠♠✉♥✐❝❛t✐♦♥ ✐♥ ❆❧❣❡❜r❛ ♥➝♠ ✷✵✵✾✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ❞➭♥❤ ➤Ĩ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥❤➢ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ❤Ư t❤❛♠ sè✱ ❤➭♠ tư ♠ë ré♥❣✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱✳✳✳ ➜Ĩ t❤❡♦ ❞â✐ ♠ét ❝➳❝❤ t➢➡♥❣ ➤è✐ ❤Ư t❤è♥❣✱ ▼ơ❝ 1.5 ❝đ❛ ❈❤➢➡♥❣ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❝❤Ý♥❤ q✉②✱ ❞➲② ❝❤Ý♥❤ q✉② ❧ä❝✱ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ✈➭ t➢➡♥❣ ø♥❣ ❧➭ ❝➳❝ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ f ✲♠➠➤✉♥✱ f ✲♠➠➤✉♥ s✉② ré♥❣ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ủ ợ trì tr 2 ✈➭ ❈❤➢➡♥❣ ❝đ❛ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ❞➲② ❝❤Ý♥❤ q✉② ✈í✐ ❝❤✐Ị✉ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ >s ✷ tr♦♥❣ ❜➭✐ ❜➳♦ ❝ñ❛ ❇r♦❞♠❛♥♥✲◆❤➭♥ ❬❇◆❪ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ❞➲② ♥➭② t❤➠♥❣ q✉❛ t❐♣ s✉♣♣♦rt ✈➭ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ♠ë ré♥❣ ❧➭ ♠ét sè ♥❣✉②➟♥✱ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ ✈í✐ ❝❤✐Ị✉ > s Ext ❝đ❛ M ✳ s a ❧➭ ✐➤➟❛♥ ❝ñ❛ R✱ ❦❤✐ ➤ã ♥Õ✉ dim M/aM > s tì ỗ >s tr a ó tể ré♥❣ ➤➢ỵ❝ t❤➭♥❤ ♠ét M ✲❞➲② ❝ù❝ ➤➵✐ ✈➭ t✃t ❝➯ ❝➳❝ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s ❝ù❝ ➤➵✐ tr♦♥❣ ➤Ò✉ ❝ã ➤é ❞➭✐ ♥❤➢ ♥❤❛✉ ✈➭ ➤é ❞➭✐ ❝❤✉♥❣ ➤ã ❝❤Ý♥❤ ❜➺♥❣ sè ♥❣✉②➟♥ s❛♦ ❝❤♦ ❈❤♦ dim(Supp(Hai (M ))) > s, ❝ò♥❣ ❝❤Ý♥❤ ❧➭ sè ♥❣✉②➟♥ i a i ♥❤á ♥❤✃t ♥❤á ♥❤✃t s❛♦ ❝❤♦ dim(ExtiR (R/a, M )) > s✳ ➜é ❞➭✐ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é s➞✉ ✈í✐ ❝❤✐Ị✉ > s ❝đ❛ M tr♦♥❣ a✱ ❦Ý ❤✐Ư✉ ❧➭ depth(a, M, > s) ▼ơ❝ t✐Õ♣ t❤❡♦ ❝ñ❛ ❈❤➢➡♥❣ ❧➭ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❧✉❐♥ ✈➝♥✱ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❧➵✐ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ✈Ị ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✈➭ ❝❤Ø ♥Õ✉ > s✿ M ❧➭ ❝❛t❡♥❛r②✱ ➤➻♥❣ ❝❤✐Ò✉ ✭dim M (Supp(M ))>s ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ♠ä✐ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ♥Õ✉ = dim R/p ✈í✐ p ∈ (Supp(M ))>s ✮ ✈➭ Mp ❧➭ Rp ✲♠➠➤✉♥ p ∈ (Supp(M ))>s ❍➡♥ ♥÷❛✱ ♥Õ✉ ❣✐➯ t❤✐Õt R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× ✈í✐ ❝❤✐Ị✉ >s ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ➤➬② ➤đ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② m✲❛❞✐❝ M ❝đ❛ M ❝ị♥❣ ❧➭ ♠➠➤✉♥ > s✳ ❈❤➢➡♥❣ ❝✉è✐ ❝ï♥❣ ❝đ❛ ❧✉❐♥ ✈➝♥ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝ñ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✳ ❈❤ó ý r➺♥❣ ❝➳❝ ❦Õt q t tự ũ ợ s ị ý ✹❪ ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ M = R✱ tr♦♥❣ ➤ã R ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❆s❛❞♦❧❧❛❤✐✲❙❝❤❡♥③❡❧ ❬❆❙✱ ➜Þ♥❤ ❧ý ✶✳✶❪ ♠ë ré♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ✈➭ ◆❤➭♥✲▼♦r❛❧❡s ❬◆▼✱ ➜Þ♥❤ ❧ý ✹✳✶❪ ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ M s✉② ré♥❣✳ P❤➬♥ ❦Õt ❧✉❐♥ ❝đ❛ ❧✉❐♥ ✈➝♥ tỉ♥❣ ❦Õt ❝➳❝ ❦Õt q✉➯ ➤➲ ➤➵t ➤➢ỵ❝✳ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ❧➭ f ✲♠➠➤✉♥ ✸ ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ t❛ ❦Ý ❤✐Ư✉ ♠➠➤✉♥ ❝ã ❝❤✐Ị✉ ❑r✉❧❧ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r ✈➭ M ❧➭ R✲ dim M = d ❈➳❝ ❣✐➯ t❤✐Õt ❦❤➳❝ ✈Ò ✈➭♥❤ ✈➭ ♠➠➤✉♥ ❦❤✐ ❝➬♥ sÏ ➤➢ỵ❝ ♥❤➽❝ ❧➵✐✳ ✶✳✶ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✳ ✭✐✮ ●✐➯ sư ❧➭ ♠ét M R✲♠➠➤✉♥✳ ▼ét ✐➤➟❛♥ ♥❣✉②➟♥ tè p ❝ñ❛ R ➤➢ỵ❝ ❣ä✐ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ M s❛♦ ❝❤♦ ♥Õ✉ tå♥ t➵✐ ♣❤➬♥ tö p = AnnR (x) ✭✐✐✮ ▼➠➤✉♥ ❝♦♥ Q ❝đ❛ M ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➠➤✉♥ ❝♦♥ ♥❣✉②➟♥ s➡ ❝ñ❛ M/Q = ✈➭ ỗ a ZD(M/Q), tồ t n N ❑❤✐ ➤ã p= ✭✐✐✐✮ ❈❤♦ N ♥Õ✉ s❛♦ ❝❤♦ an (M/Q) = M ❧➭ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ R ♠➠➤✉♥ M ♥Õ✉ tå♥ t➵✐ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ♥❣✉②➟♥ s➡ N = Q1 ∩ ∩ Qn s➡✳ ◆Õ✉ M AnnR (M/Q) ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ R, t❛ ♥ã✐ Q ❧➭ ♠ét ♠➠➤✉♥ ❝♦♥ p✲♥❣✉②➟♥ s➡ ❝ñ❛ s➡ =x∈M t❛ ♥ã✐ Qi ✈í✐ N ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ i = 1, , n, t❤➭♥❤ ❣✐❛♦ ❝đ❛ ❤÷✉ ❤➵♥ ❝➳❝ ♠➠➤✉♥ ❝♦♥ pi ✲♥❣✉②➟♥ N = ❤♦➷❝ N = ❝ã ♠ét ♣❤➞♥ tÝ❝❤ s tì t ó N tí ợ s ❧➭ ♣❤➞♥ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ tè✐ t❤✐Ó✉ ✭t❤✉ ❣ä♥✮ ♥Õ✉ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè pi ❧➭ ➤➠✐ ♠ét ❦❤➳❝ ♥❤❛✉ ✈➭ ❦❤➠♥❣ ❝ã ❤➵♥❣ tö Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ Qi ♥➭♦ ❧➭ t❤õ❛✱ ✹ ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ i = 1, , n n Qj ⊆ Qi i=1;i=j ✭✐✈✮ ❉Ô t❤✃② r➺♥❣ ♠ä✐ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝đ❛ ❞➵♥❣ t❤✉ ❣ä♥✳ ❑❤✐ ➤ã t❐♣ ❤ỵ♣ ♥❣✉②➟♥ s➡ tè✐ t❤✐Ĩ✉ ❝đ❛ ❝đ❛ M/N ✱ ❦Ý ❤✐Ư✉ ❜ë✐ N ➤Ị✉ ❝ã t❤Ĩ ➤➢❛ ➤➢ỵ❝ ✈Ị {p1 , , pn } ❧➭ ➤é❝ ❧❐♣ ✈í✐ ✈✐Ư❝ ❝❤ä♥ ♣❤➞♥ tÝ❝❤ ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt AssR M/N ✳ ❧➭ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝đ❛ ❈➳❝ ❤➵♥❣ tư N ✳ ◆Õ✉ pi Qi , i = 1, , n✱ ❧➭ tè✐ t❤✐Ĩ✉ tr♦♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ t❤➭♥❤ ♣❤➬♥ ợ tì Qi N Qi ợ ọ AssR M/N tì ợ ọ t ú ệ ➤Ị ✶✳✶✳✷✳ ✭✐✮ p ❬▼❛t✱ ➜Þ♥❤ ❧ý ✻✳✶✱ ➜Þ♥❤ ❧ý ✻✳✸✱ ➜Þ♥❤ ❧ý ✻✳✺❪✳ ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ R✲♠➠➤✉♥ ❝♦♥ ❝ñ❛ M ✭✐✐✮ ◆Õ✉ s❛♦ ❝❤♦ N M ➤➻♥❣ ❝✃✉ ✈í✐ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ N ❧➭ R/p p ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ ✈➭♥❤ R t❤× Ass(R/p) = {p} ✭✐✐✐✮ ❈❤♦ p ❧➭ ♣❤➬♥ tư tè✐ ➤➵✐ ❝đ❛ t❐♣ ✐➤➟❛♥ ❝ã ❞➵♥❣ Ann(x), tr♦♥❣ ➤ã = x ∈ M ❑❤✐ ➤ã p ∈ Ass(M ) ❱× t❤Õ✱ M = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ Ass(M ) = ∅ ❍➡♥ ♥÷❛✱ t❐♣ ZD(M ) ❝➳❝ ➢í❝ ❝đ❛ ❦❤➠♥❣ ❝đ❛ M ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ✭✐✈✮ AssR M = ❝❤Ý♥❤ ❧➭ ❤ỵ♣ ❝ñ❛ ❝➳❝ ✐➤➟❛♥ M p∈Ass M AssR M /pM ✭✈✮ ❈❤♦ ❞➲② ❦❤í♣ ♥❣➽♥ ❝➳❝ R✲♠➠➤✉♥ −→ M −→ M −→ M −→ ❑❤✐ ➤ã ✭❛✮ AssR (M ) ⊆ AssR (M ) ⊆ AssR (M ) ∪ AssR (M )❀ ✭❜✮ SuppR (M ) = SuppR (M ) ∪ SuppR (M ) Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✺ ✭✈✐✮ ❈❤♦ M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ❑❤✐ ➤ã AssR (M ) ⊆ SuppR (M ) AssR (M ) ❧➭ t❐♣ ❤÷✉ ❤➵♥ ✈➭ ❍➡♥ ♥÷❛✱ ❝➳❝ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ AssR (M ) ✈➭ SuppR (M ) ❧➭ ♥❤➢ ♥❤❛✉✳ ✶✳✷ ❍Ư t❤❛♠ sè ▼ơ❝ ♥➭② ❞➭♥❤ ➤Ĩ ♥❤➽❝ ❧➵✐ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ ✈Ị ❤Ư t❤❛♠ sè tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r, ➤Þ❛ ♣❤➢➡♥❣ (R, m) ✈➭ M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✭①❡♠ ❬▼❛t❪✮✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✶✳ ▼ét ❤Ư ❣å♠ R (M/(x1 , , xd )M ) ột ỗ d ♣❤➬♥ tư < ∞ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M (xn ) ⊆ R ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❞➲② ❈❛✉❝❤② t❤❡♦ t➠♣➠ m✲❛❞✐❝ ♥Õ✉ ✈í✐ k ∈ N ❝❤♦ tr➢í❝✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ n0 n, m ≥ n0 t❤♦➯ ♠➲♥ x1 , , xd ∈ m ❉➲② tr➢í❝✱ tå♥ t➵✐ sè (xn ) ⊆ R n0 s❛♦ ❝❤♦ s❛♦ ❝❤♦ xn − xm ∈ mk ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❦❤➠♥❣ ♥Õ✉ ỗ xn mk ọ n n0 k ∈N (xn − yn ) q✉② t➽❝ ♥❤➞♥ (xn )(yn ) ❞✉② ♥❤✃t ❧➭ R R ❧➭ t❐♣ (xn ) + (yn ) = (xn + yn ) ✈➭ = (xn yn ) ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ❝➳❝❤ ❝❤ä♥ ❝➳❝ ➤➵✐ ❞✐Ư♥ ❝đ❛ ❝➳❝ ❧í♣ t➢➡♥❣ ➤➢➡♥❣✳ ❱× t❤Õ ♥ã ❧➭ ❝➳❝ ♣❤Ð♣ t♦➳♥ tr➟♥ ♣❤Ð♣ t♦➳♥ ♥➭②✱ (xn ), (yn ) ❧➭ ❞➲② ❦❤➠♥❣✳ ❑Ý ❤✐Ö✉ ❝➳❝ ❧í♣ t➢➡♥❣ ➤➢➡♥❣✳ ❈❤ó ý r➺♥❣ q✉② t➽❝ ❝é♥❣ ❝❤♦ ❚❛ tr❛♥❣ ❜Þ q✉❛♥ ❤Ư t➢➡♥❣ ➤➢➡♥❣ tr➟♥ t❐♣ ❝➳❝ ❞➲② ❈❛✉❝❤② ♥❤➢ s❛✉✿ ❍❛✐ ❞➲② ❈❛✉❝❤② ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ♥Õ✉ ❞➲② ✈í✐ ♠ä✐ R ✈➭ ❝ï♥❣ ✈í✐ ❤❛✐ ❧➭♠ t❤➭♥❤ ♠ét ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ✐➤➟❛♥ tè✐ ➤➵✐ mR ❱➭♥❤ R ✈õ❛ ①➞② ❞ù♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ✈➭♥❤ ➤➬② ➤ñ t❤❡♦ t➠♣➠ m✲❛❞✐❝ ❝ñ❛ R✳ ▼ét ❞➲② k ∈N (zn ) ⊆ M ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② t❤❡♦ t➠♣➠ ❝❤♦ tr➢í❝✱ tå♥ t➵✐ sè tù n0 s zn m ế ỗ zm ∈ mk M ✈í✐ ♠ä✐ n, m ≥ n0 ❚õ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❈❛✉❝❤② ♥❤➢ tr➟♥✱ t➢➡♥❣ tù t ị ĩ ợ ệ ủ t t➠♣➠ ❤✐Ư✉ ❧➭ m✲❛❞✐❝ tr➟♥ ✈➭♥❤ R✳ ▼➠➤✉♥ ♥➭② ➤➢ỵ❝ ❦Ý M Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✻ ▼Ư♥❤ ➤Ị ✶✳✷✳✷✳ ✭✐✮ P❤➬♥ tư ❬▼❛t✱ ➜Þ♥❤ ❧ý ✶✹✳✶✱ ➜Þ♥❤ ❧ý ✶✹✳✷❪✳ x∈m p ∈ Ass(M ) ❧➭ ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ s❛♦ ❝❤♦ dim R/p = d t❤❛♠ sè ❝ñ❛ M ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ xi+1 t❤♦➯ ♠➲♥ ✭✐✐✮ ❈❤♦ M ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❍Ư ❝➳❝ ♣❤➬♥ tư x∈ / p, ✈í✐ ♠ä✐ x1 , , xd ∈ m ❧➭ ❤Ö ∈ / p, ✈í✐ ♠ä✐ p ∈ AssR (M/(x1 , , xi )M ) dim R/p = d − i, ✈í✐ ♠ä✐ i = 1, , d − x1 , , xt ∈ m ❧➭ ❝➳❝ ❞➲② ❝➳❝ ♣❤➬♥ tö✱ ✈í✐ t d✳ ❑❤✐ ➤ã dim(M/(x1 , , xt )M ) ≥ dim M − t ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✭✐✐✐✮ ◆Õ✉ x1 , , x d x1 , , xt ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ M t❤× ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ a1 , , ad , t❛ ❝ã xa11 , , xadd ❝ị♥❣ ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ x1 , , xd M ✭✐✈✮ ◆Õ✉ M , tr♦♥❣ ➤ã M ✶✳✸ ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ ❧➭ ➤➬② ➤đ M M t❤× ♥ã ❝ị♥❣ ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ m✲❛❞✐❝ ❝đ❛ M ❍➭♠ tư ♠ë ré♥❣ ❚r♦♥❣ ♣❤➬♥ ♥➭② t❛ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ♠➠➤✉♥ Ext t❤➢ê♥❣ ➤➢ỵ❝ ❞ï♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥ ✭①❡♠ ❬▼❛t❪✮✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✶✳ ❈❤♦ M, N ❧➭ ❝➳❝ R✲♠➠➤✉♥ ✈➭ n ❧➭ ♠ét sè tù ♥❤✐➟♥✳ ▼➠➤✉♥ ❞➱♥ ①✉✃t ♣❤➯✐ t❤ø n ❝đ❛ ❤➭♠ tư Hom(−, N ) ø♥❣ ✈í✐ M ♠➠➤✉♥ ♠ë ré♥❣ t❤ø ♥ ❝đ❛ M ❈ơ t❤Ĩ✱ ➤Ĩ ①➞② ❞ù♥❣ ExtnR ✈➭ N, ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ ExtnR (M, N ) t❛ ❧✃② ♠ét ❣✐➯✐ ①➵ ➯♥❤ ❝đ❛ u ➤➢ỵ❝ ❣ä✐ ❧➭ u M ε → P2 →2 P1 →1 P0 → M → ❚➳❝ ➤é♥❣ ❤➭♠ tö Hom(−, N ) ✈➭♦ ❞➲② ❦❤í♣ tr➟♥ t❛ ❝ã ➤è✐ ♣❤ø❝ u∗ u∗ → Hom(P0 , N ) →1 Hom(P1 , N ) →2 Hom(P2 , N ) → Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✷✻ ❚❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ t❛ ❝ã Supp Mp p = q ●✐➯ sö p = m ì Mp tr t ị ❧ý ✶✹✳✶❪✳ ❱× t❤Õ✱ htM (p) = dim Mp = dim(Rp /qRp ) + htMp (qRp ) = dim(Rp /qRp ) + htM (q) = dim(Rp /qRp ) + dim Mq ❈❤♦ p = m ▲✃② p ∈ (Min(Supp(M )))>s s❛♦ ❝❤♦ q ⊆ q ❱× Mq ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ q Rq ∈ Ass Mq ♥➟♥ t❛ ❝ã htM (q) + ht(m/q) = htM (q ) + htM (q/q ) + dim R/q = dim(Rq /q Rp ) + dim R/q = dim Mq + dim R/q = d = htM (m) = dim M ✭✈✮ ⇒ ✭✐✐✐✮ ❱× ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ Mp ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥➟♥ depth Mp = dim Mp p ∈ (Supp(M ))>s t❤× dim Mp = d − dim R/p ♥➟♥ s✉② r❛ depth Mp = d − dim R/p ❑Õt q✉➯ s❛✉ ❝❤♦ t❤✃② r➺♥❣ tÝ♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ t➵✐ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ✈í✐ ❝❤✐Ị✉ ◆❤➽❝ ❧➵✐ r➺♥❣ ➤å♥❣ ❝✃✉ ✈➭♥❤ S ①Ðt ♥❤➢ ❦❤í♣ R✲♠➠➤✉♥ > s ➤➢ỵ❝ ❜➯♦ t♦➭♥ q✉❛ ❝➳❝ ➤å♥❣ ❝✃✉ ♣❤➻♥❣✳ f : R S ị f ợ ọ ❧➭ ➤å♥❣ ❝✃✉ ♣❤➻♥❣ ♥Õ✉ R✲♠➠➤✉♥ ♣❤➻♥❣✱ tø❝ ❧➭ ✈í✐ ỗ N N N ❝➳❝ R✲♠➠➤✉♥, ❞➲② ❝➯♠ s✐♥❤ → N ⊗R S → N ⊗R S → N ⊗R S → ❧➭ ❦❤í♣✳ ◆Õ✉ ✈➭♥❤ A ❧➭ ♠ét ✈➭♥❤ ❝♦♥ ủ B tì t ó A r trờ ợ ♥➭②✱ ♠ét ♣❤➬♥ tö b ∈ B B ❧➭ ✈➭♥❤ ♠ë ré♥❣ ❝đ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥❣✉②➟♥ tr➟♥ b ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ➤❛ t❤ø❝ ✈í✐ ❤Ư sè tr♦♥❣ A✳ ◆Õ✉ ỗ tử ủ B tr A tì t ó B ❧➭ ♥❣✉②➟♥ tr➟♥ ▼Ư♥❤ ➤Ị ✷✳✷✳✹✳ ❈❤♦ A✱ ❤♦➷❝ B A ♥Õ✉ ❧➭ ♥❣✉②➟♥ ❧➭ ♠ë ré♥❣ ♥❣✉②➟♥ ❝ñ❛ A✳ f : (R, m) −→ (S, n) ❧➭ ➤å♥❣ ữ ị ó t ❝ã✿ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ Sq ế ỗ ỗ tì ứ Sq tì Rp ❧➭ p ∈ (Spec(R))>s ; S ✭✐✐✮ ◆Õ✉ ❝é♥❣ t❤➟♠ ❣✐➯ t❤✐Õt p ∈ (Spec(R))>s q ∈ (Spec(S))>s ✱ Rp ột rộ ủ ỗ (Rp /pRp ) ⊗R S ✈➭ t✃t ❝➯ ❝➳❝ ✈➭♥❤ tớ ỗ R, q (Spec(S))>s p ∈ Spec(R) ✈í✐ dim(R/p) > s✳ ❱× S ❧➭ ♠ét R✲ ♠➠➤✉♥ ❤♦➭♥ t♦➭♥ ♣❤➻♥❣✱ ♥➟♥ tå♥ t➵✐ q ∈ Spec(S) s❛♦ ❝❤♦ p = q ∩ R ✈➭ dim(S/q) > s✳ Rp −→ Sq , ❳Ðt ➤å♥❣ ❝✃✉ ♣❤➻♥❣ ✈í✐ r/u −→ f (r)/f (u) ❚❤❡♦ ❬❇❍✱ ▼Ư♥❤ ➤Ị ✶✳✷✳✶✻❪✱ t❛ ❝ã depthSq Sq = depthRp Rp + depthSq (Sq /(pRp )Sq ) ✈➭ dimSq Sq = dimRp Rp + dimSq (Sq /(pRp )Sq ) ❱× depthSq (Sq /(pRp )Sq ) t❤❡♦ ❣✐➯ t❤✐Õt Sq ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ỗ tứ tr t ó ỗ dimSq (Sq /(pRp )Sq ) t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✺✳✹ ✭✐✮ ✈➭ q ∈ (Spec(S))>s ✱ ♥➟♥ tõ ❝➳❝ depthRp Rp = dimRp Rp , s✉② r❛ Rp p ∈ (Spec(R))>s ✭✐✐✮ ❈❤♦ q ∈ (Spec(S))>s ✈➭ p = q ∩ R ị ó ủ ì tứ (Rp /pRp ) ⊗R S, ❚❛ ❝ã Sq /(pRp )Sq R/p −→ S/q ỗ rộ từ dim R/p > s t❛ ❝ã dim S/q > s ❉♦ ➤ã✱ tõ ❣✐➯ t❤✐Õt Rp ✈➭ ❦Õt ❤ỵ♣ ✈í✐ ❝➳❝ ➤➻♥❣ t❤ø❝ ë tr➟♥ t❛ ❝ã ❧➭ ❝ò♥❣ ❧➭ depthSq (Sq /(pRp )Sq ) = dimSq (Sq /(pRp )Sq ) ữ ì s ú ý Sq /(pRp )Sq ♠➭ t❤❡♦ ❣✐➯ t❤✐Õt t✃t ❝➯ ❝➳❝ t❤í ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥➟♥ (Rp /pRp ) ⊗R S ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❤❛② ❦❤✐ ➤ã ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② depthSq Sq = dimSq Sq , ❤❛② Sq ❧➭ q ∈ (Spec(S))>s ❈❤ó ý r➺♥❣ ♥Õ✉ R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× Spec(R) ✈➭ ❞♦ ➤ã (Supp(M ))>s ❧➭ ❝❛t❡♥❛r②✳ ì từ ị ý Soỏ hoựa bụỷi Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✷✽ ✭✐✮⇔ ✭✈✮✱ t❛ ❝ã M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ♠ä✐ >s ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❧➭ Mp p ∈ (Supp(M ))>s ✈➭ dim R/p = dim M ✈í✐ ♠ä✐ p ∈ (min(Supp(M )))>s ❚❤❡♦ ❈❤ó ý ✷✳✷✳✺ ë tr➟♥✱ tÝ♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ❝ã t❤Ĩ ❝❤✉②Ĩ♥ q✉❛ ➤➬② ➤đ ♥❤➢ s❛✉✳ M ▼Ư♥❤ ➤Ị ✷✳✷✳✻✳ ◆Õ✉ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ R > s M t❤× ❝ị♥❣ ❧➭ ❧➭ ♠ét ✈➭♥❤ t❤➢➡♥❣ ủ tì ề ợ ũ ú ứ ♠✐♥❤✳ x1 , , x d ➜Ó ❝❤ø♥❣ ♠✐♥❤ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ M ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✷✳✷✱ ✭✐✈✮ t❛ ❝ã ❞➲② t❤❡♦ ❣✐➯ t❤✐Õt ✈í✐ ❝❤✐Ị✉ M ❈❤♦ xˆi ❧➭ ➯♥❤ ❝ñ❛ xˆ1 , , xˆd xi > s✱ tr♦♥❣ R✳ t❛ ❧✃② ❑❤✐ ➤ã ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M > s ♥➟♥ xˆ1 , , xˆd M ✲❞➲② ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❧➭ ❱× > s ❉♦ ➤ã t❤❡♦ ❇ỉ ➤Ò ✷✳✶✳✸ t❛ ❝ã dim(((x1 , , xi−1 )M :M xi )/(x1 , , xi−1 )M ) = dim(((ˆ x1 , , xˆi−1 )M :M xˆi )/(ˆ x1 , , xˆi−1 )M ) ✈í✐ ♠ä✐ i = 1, , d✳ ❱× t❤Õ t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✸✱ ✭✐✮ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s✳ ❱× ✈❐② M ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ M ⇔ ✭✐✐✮ t❛ ❝ã ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ s, x1 , , x d ❧➭ > s✳ R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s t❤× M ❝ị♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s t từ ú ý t t ỗ q ∈ (Supp(M ))>s , t❛ ❝ã ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ỗ Mq dim R/q = dim M ì t❤Õ✱ ❧✃② q ∈ (Min(Supp(M )))>s , t❛ ❝ã q ∈ (Supp(M ))>s ✈➭ p = q ∩ R ❱× R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥➟♥ t❛ ❝ã ✈➭♥❤ t❤í Rq /pRq = (Rp /pRp ) ⊗Rp Rq dim R/p > s ✈➭ t✃t ❝➯ ❝➳❝ ❝ñ❛ ➤å♥❣ ❝✃✉ ❝❤Ý♥❤ t➽❝ Rp −→ Rq ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❚❤❡♦ ❇r✉♥s✲❍❡r③♦❣ ❬❇❍✱ ▼Ư♥❤ ➤Ị ✶✳✷✳✶✻❪ t❛ ❝ã dim Mq = dim Mp + dim Rq /pRq , Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✷✾ ✈➭ depth Mq = depth Mp + depth Rq /pRq ❱× M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ♥➟♥ s✉② r❛ Mp ❝ị♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ ✭✐✮ ⇔ tr➟♥ t❛ ❝ã ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❍➡♥ ♥÷❛✱ ❧✃② dim Mq = depth Mq , ❤❛② Mq ✭✈✮✳ ❱× t❤Õ✱ tõ ❤❛✐ ➤➻♥❣ t❤ø❝ q ∈ (Min(Supp(M )))>s , ❦❤✐ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✷✱ t❛ ❝ã q ∈ AssR (M ) ✈➭ dim R/q > s ▲➵✐ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✷ t❛ ❝ã AssR (M ) = (AssR R/(pR)) p∈AssR (M ) ❱× t❤Õ tå♥ t➵✐ p ∈ (Ass(M ))>s s❛♦ ❝❤♦ q ∩ R = p✳ ❉♦ ➤ã✱ ✈× R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ tõ ❣✐➯ t❤✐Õt ❝❤✐Ị✉ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ > s✱ t❤❡♦ ❈❤ó ý ✷✳✷✳✺ t❛ ❝ã dim R/q = dim R/p = dim M ❱× ✈❐② M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ Số hóa Trung tâm Học liệu ĐHTN > s✳ http://www.lrc.tnu.edu.vn/ ✸✵ ❈❤➢➡♥❣ ✸ ▼ét sè ❦Õt q✉➯ tr➟♥ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② t❛ ✈➱♥ ❦Ý ❤✐Ư✉ ♣❤➢➡♥❣ ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❤➵♥ s✐♥❤ ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ ❧➭ ♠ë ré♥❣ ❝ñ❛ (R, m) ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✱ ➤Þ❛ m✱ a ❧➭ ♠ét ✐➤➟❛♥ ❝đ❛ R ✈➭ M ❧➭ R✲♠➠➤✉♥ ❤÷✉ dim M = d ❈❤➢➡♥❣ ♥➭② ♥❣❤✐➟♥ ❝ø✉ ♠ét ❧♦➵✐ ❞➲② f ✲❞➲② ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❈➢ê♥❣✲❙❝❤❡♥③❡❧✲❚r✉♥❣ ❬❈❙❚❪✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ❧➭ tr×♥❤ ❜➭② ❧➵✐ ❝❤ø♥❣ ♠✐♥❤ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✸✳✶ > s a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❚❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② tr➟♥ ✈➭♥❤ R ❣✐❛♦ tr ợ tệ tr ị ĩ ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ ▼ét ❞➲② M x1 , , x n ❝➳❝ ♣❤➬♥ tư ❝đ❛ a ➤➢ỵ❝ ❣ä✐ ❧➭ a✲❞➲② ♥Õ✉ Supp(((x1 , , xi−1 )M :M xi )/(x1 , , xi−1 )M )) ⊆ V (a), ✈í✐ ♠ä✐ i = 1, , n Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✸✶ ❈❤ó ý ✸✳✶✳✷✳ ✭✐✮ ▼ét ❞➲② ❝➳❝ ♣❤➬♥ tư ❝❤Ý♥❤ q✉② ❝đ❛ M ế ỗ tr x1 , , xn i = 1, , n a ➤➢ỵ❝ ❣ä✐ ❧➭ a✲❧ä❝ t❛ ❧✉➠♥ ❝ã p ∈ Ass M/(x1 , , xi−1 )M t❤♦➯ ♠➲♥ tÝ♥❤ ❝❤✃t a xi ∈ / p, ✈í✐ ♠ä✐ p ✭✐✐✮ ◆Õ✉ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❞✉② ♥❤✃t m t❤× m✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝❤Ý♥❤ ❧➭ ❦❤➳✐ ♥✐Ư♠ f ✲❞➲② ➤➢ỵ❝ ➤➢❛ r❛ ❜ë✐ ❈➢ê♥❣✲❙❝❤❡♥③❡❧✲❚r✉♥❣ ❬❈❙❚❪✳ ❉♦ ➤ã a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝❤Ý♥❤ ❧➭ sù ♠ë ré♥❣ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ✭✐✐✐✮ ❈❤♦ x1 , , xn ❧➭ ♠ét ❞➲② ❝➳❝ ♣❤➬♥ tư ❝đ❛ f ✲❞➲②✳ a ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭❛✮ x1 , , xn ❧➭ a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ ✭❜✮ x1 /1, , xn /1 tr♦♥❣ Rp ✭❝✮ xa11 , , xann ❧➭ M; Mp ✲❞➲② ♥❣❤❒♦✱ ∀p ∈ Supp M \ V (a); ❧➭ a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝đ❛ M, ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ a1 , , a n ệ ề s t ứ ỗ số ❞➢➡♥❣ n ❧✉➠♥ tå♥ t➵✐ ♠ét a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝đ❛ M ❝ã ➤é ❞➭✐ n ➜✐Ị✉ ➤ã ❝❤ø♥❣ tá r➺♥❣ ➤é ❞➭✐ ❝ñ❛ ♠ét a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ã t❤Ĩ ❧➭ ✈➠ ❤➵♥✳ ▼Ư♥❤ ➤Ị ✸✳✶✳✸✳ ●✐➯ sư ❧✉➠♥ tå♥ t➵✐ ♣❤➬♥ tư ❝đ❛ x1 , , xn ❧➭ ♠ét a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ M ❑❤✐ ➤ã y ∈ a s❛♦ ❝❤♦ x1 , , xn , y a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② M✳ ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ Ha0 (M ) = M, t❤× Ha0 (M ) = M ✱ t❤× t❛ ❝❤ä♥ t✉ú ý ♣❤➬♥ tö Ha0 (M/Ha0 (M )) = ❉♦ ➤ã tå♥ t➵✐ ♣❤➬♥ tö y∈I ❧➭ ♠ét a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ ✸✳✷ ❧➭ ♠ét ❙✉② r❛ y ∈ a ◆Õ✉ depth(M/Ha0 (M )) > M/Ha0 (M )✲❝❤Ý♥❤ q✉②✳ ❙✉② r❛ x1 , , xn , y ❧➭ M ▼ét sè ❦Õt q✉➯ tr➟♥ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❚r➢í❝ ❤Õt t❛ ♥❤➽❝ ❧➵✐ ❦Õt q✉➯ s❛✉✱ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❬◆❙✱ ✸✳✹❪ ♥❤➢♥❣ ë ➤➞② m ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ a ✭①❡♠ ❬❑❙✱ ▼Ư♥❤ ➤Ị ✶✳✷❪✮✳ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✸✷ ▼Ư♥❤ ➤Ị ✸✳✷✳✶✳ ✳ ❈❤♦ n > ✈➭ x1 , , xn ❧➭ a✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉②✳ ❑❤✐ ➤ã t❛ ❝ã ➤➻♥❣ ❝✃✉ tù ♥❤✐➟♥ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ Hai (M ) ∼ = ❇ỉ ➤Ị ✸✳✷✳✷✳ ❈❤♦ i H(x (M ) , ,xn ) ♥Õ✉ i < n, n Hai−n (H(x (M )) , ,xn ) ♥Õ✉ i n x1 , , x n ❧➭ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s tr♦♥❣ a✳ ❑❤✐ ➤ã tå♥ t➵✐ xn+1 ∈ a s❛♦ ❝❤♦ (Supp(Han+1 (M ))) s n+1 ⊆ (Supp(H(x (M ))) , ,xn ,xn+1 ) (Ass(Han+1 (M ))) s n+1 ⊆ (Ass(H(x (M ))) s , ,xn ,xn+1 ) s ✈➭ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ x1 , , x n ∈ q ❣✐➯ t❤✐Õt t❛ ❝ã t❤Õ ✈➭ p ∈ Supp(Han+1 (M )) s❛♦ ❝❤♦ dim R/p s✳ ❈❤♦ qRp ∈ Supp(Mp ) \ {pRp }✳ ❑❤✐ ➤ã dim R/q > s✳ ❚❤❡♦ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s, ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✸✱ ✭✐✮⇔✭✐✐✐✮ x1 /1, , xn /1 ❧➭ ♠ét Mq ✲❞➲② ♥❣❤❒♦✳ ▲➢✉ ý r➺♥❣ Mq ∼ = (Mp )qRp , ✈× x1 , , x n ❧➭ x1 /1, , xn /1 ❧➭ f ✲❞➲② ❝đ❛ Mp ♣❤➬♥ tư a✲❧ä❝ ❝❤Ý♥❤ q✉② ❝đ❛ t❤❡♦ ❈❤ó ý ✶✳✺✳✾✱ ✭✐✐✮✳ ❈❤♦ xn+1 ∈ a ❧➭ M/(x1 , , xn )M ❑❤✐ ➤ã xn+1 ∈ ap ❧➭ ♣❤➬♥ tư ap ✲❧ä❝ ❝❤Ý♥❤ q✉② ❝đ❛ Mp /(x1 , , xn )Mp ✈➭ ✈× t❤Õ x1 , , xn , xn+1 ∈ ap ❧➭ ap ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ Mp t❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✶✳✸✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✷✳✶ t❛ ❝ã n+1 Han+1 (Mp ) ∼ (Mp )) = Ha0p (H(x p , ,xn ,xn+1 )Rp ❱× p ∈ Supp(Han+1 (M )) ♥➟♥ pRp ∈ Supp(Han+1 (Mp )) ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ p n+1 n+1 pRp ∈ Supp(Ha0p (H(x (Mp ))) ⊆ Supp(H(x (Mp )) , ,xn ,xn+1 )Rp , ,xn ,xn+1 )Rp ❉♦ ➤ã n+1 p ∈ Supp(H(x (M )) ❜❛♦ ❤➭♠ t❤ø ♥❤✃t ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ , ,xn ,xn+1 ) ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ➤è✐ ✈í✐ ❜❛♦ ❤➭♠ t❤ø ❤❛✐✳ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✸✸ ❇ỉ ➤Ị ✸✳✷✳✸✳ ◆Õ✉ x1 , , xn ∈ a ❧➭ ♠ét M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s t❤× (Supp(Hai (M ))) s i (Supp(H(x (M ))) , ,xn ) = s n (Supp(Hai−n (H(x (M )))) , ,xn ) ♥Õ✉ i < n, ♥Õ✉ i n s ✈➭ (Ass(Hai (M ))) s ❈❤ø♥❣ ♠✐♥❤✳ s❛♦ ❝❤♦ i (Ass(H(x (M ))) , ,xn ) = n (Ass(Hai−n (H(x (M )))) , ,xn ) ❚❛ ❝❤ø♥❣ ♠✐♥❤ ➤➻♥❣ t❤ø❝ ➤➬✉ t✐➟♥✳ ❈❤♦ s✳ dim R/p x1 /1, , xn /1 ∈ aRp HaiRp (M ) ❈❤♦ i s ❧➭ ❑❤✐ ➤ã i < n, ♥Õ✉ i n p ∈ (Supp(Hai (M )) a ⊆ p✳ ❚❤❡♦ ❝❤ø♥❣ ♠✐♥❤ ë ❇ỉ ➤Ị ✸✳✷✳✷✱ aRp ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝đ❛ Mp ✳ ❱× t❤Õ t❛ ❝ã i H(x (M ) , ,xn )Rp ∼ = s ♥Õ✉ n Hai−n Rp (H(x1 , ,xn )Rp (M )) ♥Õ✉ ♥Õ✉ i i < n, n n✳ ❚❛ ❝ã p ∈ Supp(Hai (M )) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ pRp ∈ Supp(HaiRp (Mp )) ❉♦ ➤ã✱ t❤❡♦ ➤➻♥❣ ❝✃✉ tr➟♥ t❛ ❝ã p ∈ Supp(Hai (M )) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ n p ∈ Supp(Hai−n (H(x (M ))) ❚r➢ê♥❣ ❤ỵ♣ i < n ❝❤ø♥❣ ♠✐♥❤ ❤♦➭♥ t♦➭♥ , ,xn ) t➢➡♥❣ tù✳ ➜Þ♥❤ ❧ý ✸✳✷✳✹✳ ❈❤♦ ✐➤➟❛♥ ❝ñ❛ ✐➤➟❛♥ R M s❛♦ ❝❤♦ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ Haj (M ) = ✈í✐ j > d − dim M/aM ✳ > s ❈❤♦ a ❧➭ ❑❤✐ ➤ã tå♥ t➵✐ b ⊇ a s❛♦ ❝❤♦ d − dim M/bM = j − ✈➭ (Supp(Haj (M ))) s = (Supp(Hbj (M ))) s ✈➭ (Ass(Haj (M )))>s = (Ass(Hbj (M )))>s ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ➤➻♥❣ t❤ø❝ ➤➬✉ t✐➟♥✳ ❈❤♦ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ x1 , , xd−t ❝ñ❛ ❝ñ❛ d − t < j − 1✳ M dim M/aM = t✳ ❚❛ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ♣❤➬♥ ❝ñ❛ ❤Ö t❤❛♠ sè tr♦♥❣ a✳ ❈❤♦ p1 , , pn ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt M/(x1 , , xd−t )M s❛♦ ❝❤♦ a ⊆ p1 ∩ ∩ pr , ✈➭ a Số hóa Trung tâm Học liệu ĐHTN pr+1 ∪ ∪ pn http://www.lrc.tnu.edu.vn/ ✸✹ ❑❤✐ ➤ã ✈➭ ✈× r < n✱ ✈× ♥Õ✉ ❦❤➠♥❣ t❤× ♥➟♥ d−t < j −1 √ a + :R M = d−t < j (x1 , , xd−t ) + :R M t❤❡♦ ▼Ö♥❤ ➤Ò ✶✳✹✳✷✱ ✭✐✐✮ s✉② r❛ j Haj (M ) = H(x (M ) = ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt✳ , ,xd−t ) ❈❤♦ A = {pi : tè✱ t❛ ❝❤ä♥ y∈ ❝ñ❛ ♠➠➤✉♥ E= r, dim M/pi M = t}✳ ❙ư ❞ơ♥❣ ➜Þ♥❤ ❧ý tr➳♥❤ ♥❣✉②➟♥ i n i=r+1 pi \ pi ∈A pi ✳ ❚✐Õ♣ t❤❡♦ ①Ðt ❞➲② ▼❛②❡r✲❱✐❡t♦r✐s ❬❇❙❪ d−t H(x (M ) ➤è✐ ✈í✐ ✐➤➟❛♥ a ✈➭ (y) t❤✉ ➤➢ỵ❝ ❞➲② ❦❤í♣ , ,xd−t ) j−d+t j−d+t Haj−d+t−1 (E) −→ H(j−d+t (E)⊕H(y) (E) −→ Haj−d+t ∩(y) a,y) (E) −→ Ha ∩(y) (E) ❱× j −1 > d−t ✈➭ (x1 , , xd−t ) + :R M ✱ a ∩ (y) ⊆ t❛ ❧➵✐ ❝ã j−d+t Haj−d+t−1 (E) = Haj−d+t (E) = 0, ✈× t❤Õ t❛ ❝ã ➤➻♥❣ ❝✃✉ ∩(y) ∩(y) (E) = H(y) j−d+t Haj−d+t (E) ∼ = H(a,y) (E) ❚✐Õ♣ t❤❡♦✱ ✈× tr♦♥❣ x1 , , xd−t ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ M tr♦♥❣ a ✭❝ò♥❣ ♥❤➢ b = (a, y)✮✱ ✈➭ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✱ t❤❡♦ ❇ỉ ➤Ị ✸✳✷✳✸ t❛ ❝ã (Supp(Haj (M ))) ❚❤❡♦ ❝➳❝❤ ❝❤ä♥ ♣❤➬♥ tö y✱ t❛ ❝ã s y = (Supp(Hbj (M ))) s ❧➭ ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ M/aM ✳ ❉♦ ➤ã dim M/bM = dim M/aM − 1✱ ♥❣❤Ü❛ ❧➭ d − dim M/bM = d − dim M/aM + = d − t + ❚❤❡♦ ❣✐➯ sö ✈× d−t < j−1 d − dim M/bM = j − s✉② r❛ d − dim M/bM < j ♥➟♥ t❛ ❝ã ❱× ✈❐②✱ ❜➺♥❣ q✉② ♥➵♣ t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❇ỉ ➤Ị ✸✳✷✳✺✳ ❝❤✐Ị✉ > s ●✐➯ sö dim M > s✳ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❑❤✐ ➤ã M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ depth(p, M, > s) = d dim M/pM ỗ p (Supp(M ))>s ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ p ∈ Supp(M ) s❛♦ ❝❤♦ dim M/pM = t > s✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè Số hóa Trung tâm Học liệu x1 , , xd−t ĐHTN ❝đ❛ M ❝❤ø❛ tr♦♥❣ p✳ ❱× http://www.lrc.tnu.edu.vn/ M ❧➭ ✸✺ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❉♦ ➤ã✱ > s ♥➟♥ d − t depth(p, M, > s) x1 , , xd−t ❧➭ M ✲❞➲② ▼➷t ❦❤➳❝✱ ✈× ♠ä✐ ✈í✐ ❝❤✐Ị✉ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ ❝ã ➤é ❞➭✐ ❧í♥ ♥❤✃t ❧➭ d − t ➤Ị✉ ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❝đ❛ M depth(p, M, > s) d − t✳ ❙✉② r❛ depth(p, M, > s) = d t ợ ỗ = d dim M/pM ❚❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ ✭✐✐✐✮ ⇒ ✭✐✮✱ M depth Mp ♥➟♥ t❛ ❝ã d − dim M/pM ♥➟♥ t❛ ❝ã depth Mp + dim M/pM ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✹✱ t❛ ❝ã ➤é ❞➭✐ ❝đ❛ t✃t ❝➯ tr♦♥❣ >s p ∈ (Supp(M ))>s ✱ tõ ❣✐➯ t❤✐Õt t❛ ❝ã d − dim M/pM = depth(p, M, > s) ❙✉② r❛ depth Mp > s = d > s M ✲❞➲② ❝ù❝ ➤➵✐ ✈í✐ ❝❤✐Ị✉ > s a ❧➭ sè ♥❣✉②➟♥ i ♥❤á ♥❤✃t s❛♦ ❝❤♦ dim(Supp(Hai (M ))) > s ì từ ị ❧ý ✶✳✶❪ t❛ ❝ã ❤Ö q✉➯ s❛✉✳ ❍Ö q✉➯ ✸✳✷✳✻✳ ❈❤♦ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > ❑❤✐ ➤ã ✈í✐ ỗ t s p (Supp(M ))>s i < d − dim M/pM ✱ t❐♣ (Supp(Hpi (M )))>s (Ass(Hpi (M ))) s ữ ỗ i t rỗ d dim M/pM ệ ề s❛✉ ❧➭ ♠ë ré♥❣ ♠ét ❦Õt q✉➯ ❝ñ❛ ❍❡❧❧✉s ❬❍✱ ❍Ư q✉➯ ✷❪✳ ▼Ư♥❤ ➤Ị ✸✳✷✳✼✳ ❈❤♦ ✈➭ j M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s, a ❧➭ ✐➤➟❛♥ ❝đ❛ R ❑❤✐ ➤ã (Supp(Haj (M ))) s ∩ {q ∈ Spec(R) : d − dim M/qM = j}, ❧➭ t ữ ứ ì ỗ q Supp M/aM s❛♦ ❝❤♦ dim M/qM = dim M/aM ➤Ò✉ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ❝đ❛ tè a + :R M ✈➭ ✈× ❦❤➠♥❣ ❝ã ✐➤➟❛♥ ♥❣✉②➟♥ q ∈ Supp M/aM ♥➭♦ t❤á❛ ♠➲♥ dim M/qM > dim M/aM ♥➟♥ t❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ d − dim M/aM ❣✐➯ sư t✐Õ♣ r➺♥❣ j − ❍➡♥ ♥÷❛✱ t❤❡♦ ➜Þ♥❤ ❧ý ✸✳✷✳✹ t❛ ❝ã t❤Ĩ d − dim M/aM = j − 1✳ ❈❤♦ x1 , , xj−1 Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ✸✻ ❤Ư t❤❛♠ sè ❝ñ❛ ❝ñ❛ M tr♦♥❣ a✳ ❈❤♦ p1 , , pn ❧➭ ♥❤÷♥❣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt M/(x1 , , xj−1 )M ➤➢ỵ❝ ➤➳♥❤ sè ❧➵✐ s❛♦ ❝❤♦ a ⊆ p1 ∩ ∩ pr ✈➭ a ❚❛ ❧✉➠♥ ❝ã t❤Ó ❣✐➯ sö r➺♥❣ r s✳ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ s ❧➭ ❤÷✉ ❤➵♥ ỗ > s ó t i ỗ a ủ R ế ỉ ế ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥✿ ✭✐✮ (Ass(H(x,y) (M ))) s ữ ỗ tử t số x ủ M ỗ y R; (Ass(H(x,y,z) (M ))) ỗ s ữ ỗ ệ t số x, y ủ M z ∈ R ❈❤ø♥❣ ♠✐♥❤✳ ➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ❝đ❛ ➤Þ♥❤ ❧ý ➤➢ỵ❝ s✉② r❛ ♥❣❛② tõ ▼Ư♥❤ ➤Ị ✸✳✷✳✽✳ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư ✭✐✮ ✈➭ ✭✐✐✮ t❤á❛ ♠➲♥✳ ❚❤❡♦ ▼Ư♥❤ ➤Ò ✸✳✷✳✽✱ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ t❐♣ t❐♣ (Ass(H(x,y) (M ))) (Ass(H(x,y,z) (M ))) x, y, z ∈ m✳ ❚❛ ➤➷t s ❧➭ ❤÷✉ ❤➵♥ ✈í✐ ỗ s ữ ỗ x, y, z ∈ m✳ a = (x, y, z)R✳ ❑❤✐ ➤ã dim M/aM dim M/aM = d − 3, t❤× x, y, z tå♥ t➵✐ ♠ét ✐➤➟❛♥ b ⊇ a s❛♦ ❝❤♦ (Ass(Hb3 (M ))) dim M/bM = d − 2✳ M tr♦♥❣ ❧➭ M ✲❞➲② s❛♦ ❝❤♦ dim M/aM > d − 3✳ ❱× ✈❐②✱ tå♥ t➵✐ x ,y ✈➭ ❚❤❐t ✈❐②✱ ❧✃② d − ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❝đ❛ ▼Ư♥❤ ➤Ị ✶✳✷✳✷ ✭✐✮✳ ❉♦ ➤ã✱ t❤❡♦ tết t ó t ữ ì ❝❤♦ x, y ∈ m, ◆Õ✉ M t❤❡♦ (Ass(H(x,y,z) (M ))) s ❧➭ ❑❤✐ ➤ã✱ t❤❡♦ ➜Þ♥❤ ❧ý ✸✳✷✳✹ s = (Ass(Ha3 (M ))) s ✈➭ ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❝đ❛ b✳ ❚❤❡♦ ❣✐➯ t❤✐Õt M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ♥➟♥ x , y ✈í✐ ❝❤✐Ị✉ tr♦♥❣ > s (Ass(Hb3 (M ))) ⊆ (Ass(H(x ,y ,z ) (M ))) s ❚❤❡♦ ❣✐➯ t❤✐Õt ✭✐✐✮✱ t❛ (Ass(Hb3 (M ))) s ❧➭ ❤÷✉ ❤➵♥✳ ❱í✐ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù✱ t❛ ❝ị♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r ỗ x, y m ó t (Ass(H(x ,y ,z ) (M ))) s b✳ ❱× t❤Õ t❤❡♦ ❇ỉ ➤Ị ✸✳✷✳✷✱ tå♥ t➵✐ z ∈ b (Ass(H(x,y) (M ))) s t ữ ì s ữ ị ý ợ ứ ♠✐♥❤✳ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✸✽ ❑Õt ❧✉❐♥ ❚ã♠ ❧➵✐✱ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❜➭✐ ❜➳♦✿ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼♦❞✉❧❡s ✐♥ ❉✐♠❡♥s✐♦♥ ❘❡s✉❧ts ♦♥ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣② > s ❛♥❞ ❝ñ❛ ◆❛s❡r ❩❛♠❛♥✐ ✭✷✵✵✾✮✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❧➵✐ ❝➳❝ ♥é✐ ❞✉♥❣ s❛✉✿ ✶✳ ◆❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥✿ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ❤Ö t❤❛♠ sè✱ ❤➭♠ tư ♠ë ré♥❣✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳✳✳ ✷✳ ◆❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❝❤Ý♥❤ q✉②✱ ❞➲② ❝❤Ý♥❤ q✉② ❧ä❝✱ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ✈➭ t➢➡♥❣ ø♥❣ ❧➭ ❝➳❝ ❧í♣ ♠➠➤✉♥ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ f ✲♠➠➤✉♥✱ f ✲♠➠➤✉♥ s✉② ré♥❣ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ✸✳ ❚r×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ❞➲② ❝❤Ý♥❤ q✉② ✈í✐ ❝❤✐Ị✉ >s ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ ❞➲② ♥➭② tr♦♥❣ ❜➭✐ ❜➳♦ ❝đ❛ ❇r♦❞♠❛♥♥✲◆❤➭♥✳ ✹✳ ❚r×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ò✉ >s ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❧➵✐ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ✈Ị ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s q✉❛ ➤➬② ➤đ m✲❛❞✐❝✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ✈➭ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ➤➻♥❣ ❝❤✐Ị✉ tí✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝ã ❝❤✐Ị✉ >s ❝đ❛ t❐♣ s✉♣♣♦rt ❝đ❛ M✳ ❈❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✳ ❈➳❝ ❦Õt q✉➯ ♥➭② t❤✉é❝ ✈Ò ◆✳ ❩❛♠❛♥✐ ❬◆❩❪✳ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✸✾ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬❆❙❪ ❆s❛❞♦❧❧❛❤✐✱ ❏✳ ❙❝❤❡♥③❡❧✱ P✳ ✭✷✵✵✸✮✳ ❙♦♠❡ r❡s✉❧t ♦♥ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❏❛♣❛♥❡s❡ ❏✳ ▼❛t❤ ✷✾✿✷✽✺✲✷✾✻✳ ❬❇◆❪ ❇r♦❞♠❛♥♥✱ ▼✳✱ ◆❤❛♥✱ ▲✳ ❚✳ ✭✷✵✵✻✮✳ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❝❡rt❛✐♥ ❊①t✲♠♦❞✉❧s✳❈♦♠♠✳ ❆❧❣✳ ✸✻✿✶✺✷✼✲✶✺✸✻✳ ❬❇❙❪ ❇r♦❞♠❛♥♥✱ ▼✳✱ ❙❤❛r♣✱ ❘✳ ❨✳ ✭✶✾✾✽✮✳ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡❜r❛✐❝ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ●❡♦♠❡tr✐❝ ❆♣♣❧✐❝❛t✐♦♥s✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬❇❍❪ ❇r✉♥s ❲✳ ✱ ❏✳ ❍❡r③♦❣✳ ✭✶✾✾✽✮✳ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❘✐♥❣s✱ r❡✈✐s❡❞ ❡❞✳✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬❈❙❚❪ ❈✉♦♥❣✱ ◆✳ ❚✳✱ ❙❝❤❡♥③❡❧✱ P✳✱ ❚r✉♥❣✱ ◆✳ ❱✳ ✭✶✾✼✽✮✳ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧♥✳ ▼❛t❤✳ ◆❛❝❤r ✽✺✿ ✺✼✲✼✸✳ ❬❍❪ ❍❡❧❧✉s✱ ▼✳ ✭✷✵✵✶✮✳ ❖♥ t❤❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❛ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❏ ❆❧❣❡❜r❛ ✷✸✼❀✹✵✻✲✹✶✾✳ ❬❑❪ ❑✳ ❑❤❛s❤②❛r♠❛♥rs❤✳ ✭✷✵✵✻✮✳ ❖♥ t❤❡ ❢✐♥✐t❡♥❡ss ♣r♦♣❡rt✐❡s ♦❢ n n i AssR ExtR (R/a , M )✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✹✿✼✼✾✲✼✽✹ ❬❑❙❪ ❑❤❛s❤②❛r♠❛♥rs❤✱ ❑✳✱ ❙❛❧❛r✐❛♥✱ ❙❤✳ ✭✶✾✾✽✮✳ ❋✐❧t❡r r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✷✻✿✷✹✽✸✲ ✷✹✾✵✳ ❬▲❚❪ ▲✉✱ ❘✳✱ ❚❛♥❣✱ ❩✳ ✭✷✵✵✶✮✳ ❚❤❡ ❢✲❞❡♣t❤ ♦❢ ❛♥ ✐❞❡❛❧ ♦♥ ❛ ♠♦❞✉❧❡✳ Pr♦❝ ❆▼❙ ✶✸✵✭✼✮✿✶✾✵✺✲✶✾✶✶✳ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ✹✵ ❬▼❛t❪ ▼❛ts✉♠✉r❛✱ ❍✳ ✭✶✾✽✻✮✳ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣ ❚❤❡♦r②✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡r✲ s✐t② Pr❡ss✳ ❬◆❙❪ ◆❛❣❡❧✱ ❯✳✱ ❙❝❤❡♥③❡❧✱ P✳ ✭✶✾✾✹✮✳ ❈♦❤♦♠♦❧♦❣✐❝❛❧ ❛♥♥✐❤✐❧❛t♦rs ❛♥❞ ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ r❡❣✉❧❛r✐t②✳ ■♥✿ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✿ ❙②③②❣✐❡s✱ ▼✉❧t✐♣❧✐❝✐t✐❡s✱ ❛♥❞ ❇✐r❛t✐♦♥❛❧ ❆❧❣❡❜r❛✳ ❈♦♥t❡♠♣✳ ▼❛t❤✳ ✶✺✾✿✸✵✼✲✸✷✽✳ ❬◆❪ ◆❤❛♥✱ ▲✳ ❚✳ ✭✷✵✵✺✮✳ ❖♥ ❣❡♥❡r❛❧✐③❡❞ r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❈♦♠♠✉♥✐❝❛t✐♦♥ ✐♥ ❆❧❣❡❜r❛✳ ✼✾✸✿✽✶✲✾✹✳ ❬◆▼❪ ◆❤❛♥ ▲✳ ❚✳ ❛♥❞ ▼❛r❝❡❧ ▼♦r❛❧❡s✳ ✭✷✵✵✻✮✳ ●❡♥❡r❛❧✐③❡❞ ❢✲♠♦❞✉❧❡s ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✸✹✱ ✽✻✸✲✽✼✽✳ ❬❙❝❤❪ ❙❝❤❡♥③❡❧✱ P✳ ✭✶✾✽✷✮✳ ❉✉❛❧✐s✐❡r❡♥❞❡ ❑♦♠♣❧❡①❡ ✐♥ ❞❡r ❧♦❦❛❧❡♥ ❆❧❣❡❜r❛ ✉♥❞ ❇✉❝❤s❜❛✉♠ ❘✐♥❣❡✳ ▲❡❝t✉r❛ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝❛s✳ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✱ ♣✳ ✾✵✼✳ ❬❙❱❪ ❙t✉❝❦r❛❞✱ ❏✳✱ ❱♦❣❡❧✱ ❲✳ ✭✶✾✽✻✮✳ ❇✉❝❤s❜❛✉♠ ❘✐♥❣s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳ ❇❡r❧✐♥✿ ❲❊❇ ❉❡✉ts❡❝❤❡r ❱❡r❧❛❣ ❞❡r ❲✐ss❡♥s❝❤❛❢t❡♥✳ ❬❩❛❪ ❩❛♠❛♥✐✱ ◆✳ ✭✷✵✵✸✮✳ ❆ ♥♦t❡ ♦♥ t❤❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✶✿✶✷✵✸✲✶✷✵✻✳ ❬◆❩❪ ❩❛♠❛♥✐ ◆✳ ✭✷✵✵✾✮✳ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼♦❞✉❧❡s ✐♥ ❉✐♠❡♥s✐♦♥ ❃s ❛♥❞ ❘❡s✉❧ts ♦♥ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✸✼✱ ✶✷✾✼✲ ✶✸✵✼ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ ... M ✲❞➲②✳ ❑❤✐ ➤ã ✭❛✮ ▼ä✐ ❤♦➳♥ ✈Þ ❝➳❝ ♣❤➬♥ tư ✭❜✮ tì ọ số t ị ý ❈❤♦ ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R M IM = M ✳ s❛♦ ❝❤♦ ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ ❱í✐ ỗ số n trớ ệ ề s ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ ExtiR (N, M )... ▼Ư♥❤ ➤Ị ✶✳✺✳✶✸✳ ❈➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ Số hóa Trung tâm Học liệu ĐHTN http://www.lrc.tnu.edu.vn/ M ❧➭ ✶✸ f ✲♠➠➤✉♥✳ ✭✐✮ ▼ ❧➭ ỗ ệ t số x1 , , xt ủ M ỗ tè p ∈ Ass(M/(x1 , ,... tứ ệ số tr A ế ỗ ♣❤➬♥ tư ❝đ❛ B tr➟♥ A t❤× t❛ ♥ã✐ B ❧➭ ♥❣✉②➟♥ tr➟♥ ▼Ư♥❤ ➤Ị ✷✳✷✳✹✳ ❈❤♦ A✱ ❤♦➷❝ B A ♥Õ✉ ❧➭ ♥❣✉②➟♥ ❧➭ ♠ë ré♥❣ ♥❣✉②➟♥ ❝ñ❛ A✳ f : (R, m) −→ (S, n) ❧➭ ➤å♥❣ ❝✃✉ ữ ị ó t ó Số hóa