Một số đặc trưng của mô đun cohen macaulay với chiều

ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM PHẠM THỊ LÝ MỘT SỐ ĐẶC TRƯNG CỦA MÔ ĐUN COHEN-MACAULAY VỚI CHIỀU >S Chuyên ngành: ĐẠI SỐ VÀ LÝ THUYẾT SỐ Mã số: 60.46.01.04 LUẬN VĂN THẠC SỸ TOÁN HỌC Người hướng dẫn khoa học: TS.NGUYỄN THỊ DUNG Thái Nguyên, năm 2014 ▲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❝ ✈✐➟♥ P❤➵♠ ❚❤Þ ▲ý ❳➳❝ ♥❤❐♥ ❳➳❝ ♥❤❐♥ ❝ñ❛ tr➢ë♥❣ ❦❤♦❛ ❝❤✉②➟♥ ♠➠♥ ❝đ❛ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ ❉✉♥❣ ✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝➷♥ ❦Ï ❝đ❛ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ ❉✉♥❣✱ ❈➠ ➤➲ ❞➭♥❤ ♥❤✐Ị✉ t❤ê✐ ❣✐❛♥ ✈➭ ❝➠♥❣ sø❝ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ ❝❤➞♥ t❤➭♥❤ tí✐ ❝➠ ❤➢í♥❣ ❞➱♥✳ ❚➠✐ ❝ị♥❣ ①✐♥ ❜➭② tá ❧ß♥❣ ❝➯♠ ➡♥ s➞✉ s➽❝ tí✐ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ ❝đ❛ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠✱ ➜➵✐ ❤ä❝ ❑❤♦❛ ❤ä❝ t❤✉é❝ ➜➵✐ ❤ä❝ ❚❤➳✐ ệ ọ ữ t tì ❞➵② ✈➭ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ t➵✐ tr➢ê♥❣✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❝➯♠ ➡♥ ❜➵♥ ❜❒ ♥❣➢ê✐ t❤➞♥ ✈➭ ➤å♥❣ ♥❣❤✐Ö♣ ➤➲ ➤é♥❣ ✈✐➟♥ ✈➭ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ t➠✐ ➤Ĩ t➠✐ ❝ã t❤Ĩ ❤♦➭♥ t❤➭♥❤ ❦❤ã❛ ❤ä❝ ❝đ❛ ♠×♥❤✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✻ ♥➝♠ ✷✵✶✹ ❍ä❝ ✈✐➟♥ P❤➵♠ ❚❤Þ ▲ý ✐✐ ▼ơ❝ ❧ơ❝ ❚r❛♥❣ ▼ơ❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳ ❍Ö t❤❛♠ sè✱ sè ❜é✐ ✈➭ ❦✐Ó✉ ➤❛ t❤ø❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ▼➠ ➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳ ❇✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣✱ ❝❤✐Ò✉ ◆♦❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❈❤➢➡♥❣ ✷✳ ▼ét sè ➤➷❝ tr➢♥❣ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ♠ét sè ♠ë ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷✳ ▼ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✐✐✐ ▼ë ➤➬✉ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈➭ M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ dim M = d✳ ❚❛ ➤➲ ❜✐Õt r➺♥❣ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ❧ý t❤✉②Õt ❝➳❝ ✈➭♥❤ ◆♦❡t❤❡r ✈➭ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ♠➠➤✉♥ M ➤➢ỵ❝ ❣ä✐ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥Õ✉ ♠ä✐ ❤Ư t❤❛♠ sè ❧➭ M ✲❞➲② ❝❤Ý♥❤ q✉②✳ ❈✃✉ tró❝ ❝đ❛ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤➲ ➤➢ỵ❝ ❜✐Õt râ t❤➠♥❣ q✉❛ ❧ý t❤✉②Õt ❜é✐✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ➤➬② ➤đ m✲❛❞✐❝✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛✱ ✳✳✳✭①❡♠ ❬❇❍❪✱ ❬▼❛t❪✮✳ ➜➲ ❝ã ♠ét sè ♠ë ré♥❣ ❝đ❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ tr♦♥❣ sè ➤ã ❧➭ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ M ✲❞➲② M ✲❞➲② ❝❤Ý♥❤ q✉② ✈í✐ ❝❤✐Ị✉ > s ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❇r♦❞♠❛♥♥✲◆❤❛♥ ❬❇◆❪ ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ợ ị ĩ ị ĩ (x1 , , xn ) ♠ä✐ ❈❤♦ tr♦♥❣ −1 s ❧➭ ♠ét sè ♥❣✉②➟♥✳ m ➤➢ỵ❝ ❣ä✐ ❧➭ M ✲❞➲② ▼ét ❞➲② ❝➳❝ ♣❤➬♥ tư ✈í✐ ❝❤✐Ị✉ > s ♥Õ✉ xi ∈ / p, ✈í✐ p ∈ AssR (M/(x1 , , xi−1 )M ) t❤á❛ ♠➲♥ dim(R/p) > s✱ ✈í✐ ♠ä✐ i = 1, , n✳ ❚❛ ♥ã✐ r➺♥❣ ♠ä✐ ❤Ö t❤❛♠ sè ❝đ❛ M ❧➭ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ♥Õ✉ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s✳ ❘â r➭♥❣ r➺♥❣ ♠ét M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > M ✲❞➲②✱ ❢✲❞➲② ø♥❣ ✈í✐ M >s s ✈í✐ s = −1, 0, t➢➡♥❣ ø♥❣ ❧➭ ♠ét t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❈➢ê♥❣✲❙❝❤❡♥③❡❧✲❚r✉♥❣ ❬❈❙❚❪✱ ✈➭ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ø♥❣ ✈í✐ M t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ▲✳ ❚✳ ◆❤➭♥ ❬◆❤❪✳ ❱× t❤Õ ❝➳❝ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > t➢➡♥❣ ø♥❣ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ s tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s = −1, 0, f ✲♠➠➤✉♥ ➤Þ♥❤ ♥❣❤Ü❛ ❜ë✐ ❬❈❙❚❪ ✈➭ ❢✲ ♠➠➤✉♥ s✉② ré♥❣ ợ tệ rs ữ ❝ø✉ ❣➬♥ ➤➞② ✈➱♥ ❝❤♦ t❤✃② r➺♥❣ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✈í✐ s>1 > s, ❧➭ ♠ét sè ♥❣✉②➟♥ tï② ý ✈➱♥ ❝ß♥ ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t t➢➡♥❣ tù ♥❤➢ ❝➳❝ ❧í♣ ♠➠➤✉♥ q✉❡♥ ❜✐Õt tr➟♥✳ ❚❤❐t ✈❐②✱ ♥➝♠ ✷✵✵✾✱ ◆✳ ❩❛♠❛♥✐ ❬❩❪ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✶ >s t❤➠♥❣ q✉❛ ➤➬② ➤đ m✲❛❞✐❝✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛✱ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ➤➻♥❣ ❝❤✐Ị✉ ➤Õ♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ tè ✈í✐ ❝❤✐Ị✉ >s ❝đ❛ t❐♣ ❣✐➳ ❝ñ❛ M✳ ◆❣♦➭✐ r❛✱ ♠ét sè ❦Õt q✉➯ ❧✐➟♥ q✉❛♥ tí✐ tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ♥❤➢ ❧➭ sù ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ tr➢í❝ ➤➞② ❝đ❛ ❍❡❧❧✉s ❬❍❪ ✈➭ ◆❤➭♥✲▼♦r❛❧❡s ❬◆▼❪ ❝ị♥❣ ➤➲ ➤➢ỵ❝ ➤➢❛ r❛ tr♦♥❣ ❬❩❪✳ ❚✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ◆✳ ❩❛♠❛♥✐✱ ♠ét ✈✃♥ ➤Ị ➤➢ỵ❝ ➤➷t r❛ ❧➭✿ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ▲✐Ư✉ r➺♥❣ > s ❝ã ❝➳❝ ➤➷❝ tr➢♥❣ q✉❛ sè ❜é✐✱ ❦✐Ĩ✉ ➤❛ t❤ø❝ ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❤❛② ❦❤➠♥❣❄ ❈➞✉ ❤á✐ ♥➭② ➤➲ ➤➢ỵ❝ tr➯ ❧ê✐ tr♦♥❣ ♠ét ♥❣❤✐➟♥ ❝ø✉ ❣➬♥ ➤➞② ❝đ❛ ◆✳ ❚✳ ❉✉♥❣ ❬❉❪✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ➤ä❝ ✈➭ tr×♥❤ ❜➭② ❧➵✐ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❜➭✐ ❜➳♦ ✧❙♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ✐♥ ❞✐♠❡♥s✐♦♥ > s✧ ❝đ❛ ❬❉❪ ➤➢ỵ❝ ➤➝♥❣ tr➟♥ t➵♣ ❝❤Ý ❇✉❧❧❡t✐♥ ♦❢ t❤❡ ❑♦r❡❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ♥➝♠ ✷✵✶✹✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❤❛✐ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ❜❛♦ ❣å♠ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ✿ ❤Ư t❤❛♠ sè ✈➭ sè ❜é✐✱ ❦✐Ĩ✉ ➤❛ t❤ø❝✱ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ▼ơ❝ ❝đ❛ ❈❤➢➡♥❣ ❞➭♥❤ ➤Ĩ ♥❤➽❝ ❧➵✐ ❝➳❝ ❦Õt q✉➯ ✈Ị ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ♠ét sè ♠ë ré♥❣✱ tr♦♥❣ ➤ã ❣✐í✐ t❤✐Ư✉ ✈Ị ❧í♣ ♠➠➤✉♥ ✈í✐ ❝❤✐Ị✉ > s ✈➭ ♠ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ❧í♣ ♠➠➤✉♥ ♥➭② t❤➠♥❣ q✉❛ ➤➬② ➤đ m✲❛❞✐❝✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛✱ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ➤➻♥❣ ❝❤✐Ị✉ ➤Õ♥ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ tè ✈í✐ ❝❤✐Ị✉ > s ❝đ❛ t❐♣ ❣✐➳ ❝đ❛ M ✳ ❈➳❝ ➤➷❝ tr➢♥❣ ♥➭② ➤➲ ➤➢ỵ❝ ứ tr ợ trì tr♦♥❣ ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ❝đ❛ ❉➢➡♥❣ ❚❤Þ ●✐❛♥❣ ❬●❪✳ ▼ơ❝ ❝đ❛ ❈❤➢➡♥❣ ❧➭ ♥é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥✱ ❞➭♥❤ ➤Ó ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s t❤➠♥❣ q✉❛ sè ❜é✐ e(x; M ) ❝đ❛ M ✱ ❝❤✐Ị✉ ◆♦❡t❤❡r N-dimR Hmi (M ) ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ M Hmi (M )✱ ✈➭ ❦✐Ó✉ ➤❛ t❤ø❝ p(M ) ❝đ❛ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❬❈❪✳ P❤➬♥ ❦Õt ❧✉❐♥ ❝đ❛ ❧✉❐♥ ✈➝♥ tỉ♥❣ ❦Õt ❝➳❝ ❦Õt q✉➯ ➤➲ ➤➵t ợ ế tứ ị r t ❜é ❝❤➢➡♥❣ ♥➭②✱ t❛ ❧✉➠♥ ❦ý ❤✐Ö✉ ◆♦❡t❤❡r✱ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥ ✈➭ M ❧➭ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ R✲♠➠➤✉♥✳ ❈❤➢➡♥❣ ♥➭② ❞➭♥❤ ➤Ó ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ➤➢ỵ❝ ❞ï♥❣ tr♦♥❣ ❝❤➢➡♥❣ t✐Õ♣ t❤❡♦✿ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣✱ ❝❤✐Ị✉ ◆♦❡t❤❡r✱ sè ❜é✐✱ ❦✐Ĩ✉ ➤❛ t❤ø❝✱ ✶✳✶ ❍Ư t❤❛♠ sè ✈➭ sè ❜é✐ ▼ơ❝ ♥➭② ❞➭♥❤ ➤Ĩ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ✈Ị ❤➭♠ ❍✐❧❜❡rt✱ ❤Ö t❤❛♠ sè ✈➭ sè ❜é✐✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ➤➢ỵ❝ ❞ï♥❣ tr♦♥❣ ❝❤➢➡♥❣ s❛✉ ✈➭ ❝ã t❤Ĩ ➤➢ỵ❝ ①❡♠ tr♦♥❣ ❬▼❛t❪✱ ❬❇❍❪✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ♠ét ✐➤➟❛♥ I ủ (R, m) ợ ọ ị ĩ √ √ ❝ñ❛ R ♥Õ✉ tå♥ t➵✐ n > s❛♦ ❝❤♦ mn ⊆ I ⊆ m ❑❤✐ ➤ã mn ⊆ I ⊆ m ❤❛② √ I = m ♥➟♥ t❛ ❝ã I ❧➭ m✲♥❣✉②➟♥ s➡✳ ❱❐② I ❧➭ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ I ❧➭ ❝đ❛ R✲♠➠➤✉♥ M ❈❤♦ sö m✲♥❣✉②➟♥ s➡✳ ❚➢➡♥❣ tù ✐➤➟❛♥ I ⊂ m ➤➢ỵ❝ ❣ä✐ ❧➭ I ♥Õ✉ tå♥ t➵✐ n > s❛♦ ❝❤♦ mn M ⊆ IM ❧➭ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ {a1 , , ar } ❧➭ ❤Ư s✐♥❤ ❝đ❛ ❆rt✐♥✱ tø❝ ❧➭ R (R/I) ➤❛ t❤ø❝ ❍✐❧❜❡rt✱ ✈í✐ < ∞ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ I R ✈➭ M ❚❛ ❝ã ❧➭ R✲ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ●✐➯ dim(R/I) = ❉♦ ✈❐② R (I k M/I k+1 M ) ♥➟♥ < ∞ R/I ❧➭ ✈➭♥❤ ❚❤❡♦ ➜Þ♥❤ ❧ý k ➤đ ❧í♥✱ tå♥ t➵✐ ➤❛ t❤ø❝ ✈í✐ ❤Ư sè ❤÷✉ tØ PM,I (k) s❛♦ ❝❤♦ ✸ PM,I (k) = R (I k M/I k+1 M ) ➜➷t n n ❙❛♠✉❡❧ M/I k+1 M ) = R (M/I n+1 M ) n ➤đ ❧í♥✱ PM,I (n) ❧➭ ♠ét ➤❛ t❤ø❝ ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤❛ t❤ø❝ ❍✐❧❜❡rt✲ ❝đ❛ PM,I (n) k k=0 k=0 ❑❤✐ ➤ã ✈í✐ R (I PM,I (k) = PM,I (n) = ➤è✐ ✈í✐ M I✳ ◆❣➢ê✐ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❜❐❝ ❝ñ❛ ➤❛ t❤ø❝ ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ❝➳❝❤ ❝❤ä♥ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝❤✐Ị✉ r I ữ ế dim M = d tì ❧✉➠♥ tå♥ t➵✐ ❝➳❝ ♣❤➬♥ tö x1 , , xd ∈ m s❛♦ ❝❤♦ R (M/(x1 , , xd )M ) < ∞ ❑❤✐ ➤ã t❛ ❝ã ❦Õt q✉➯ s❛✉✳ ➜Þ♥❤ ❧ý ✶✳✶✳✶✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ♥❤➢ tr➟♥✱ t❛ ❝ã dim M = deg(PM,I (n)) = min{t | ∃x1 , , xt ∈ m : R (M/(x1 , , xt )M ) < ∞} ❚õ ➤Þ♥❤ ❧ý tr➟♥ t❛ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷✳ sè ❝đ❛ M ✭✐✐✮ ◆Õ✉ ❧➭ ♥Õ✉ ✭✐✮ ▼ét ❤Ö x := x1 , , xd ∈ m ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❤Ư t❤❛♠ (M/(x))M ) < ∞ x ∈ m ❧➭ ♠ét ❤Ö t❤❛♠ sè ❝đ❛ M ✱ ✈í✐ ♠ä✐ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❈❤ó ý ✶✳✶✳✸✳ ✭✐✐✮ ◆Õ✉ ❝đ❛ sè R (q x x1 , , xi ➤➢ỵ❝ ❣ä✐ i = 1, , d ✭✐✮ ❍Ö t❤❛♠ sè ❧✉➠♥ tå♥ t➵✐✳ ❧➭ ♠ét ❤Ö t❤❛♠ số ủ M tì tử ữ M t❤× q = (x1 , , xd ) ❣ä✐ ❧➭ ✐➤➟❛♥ t❤❛♠ q + AnnR M ❧➭ ♠ét ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ R, tø❝ ❧➭ + AnnR M ) < ∞ ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ❝❤♦ t❛ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❤Ư t❤❛♠ sè✱ ✭①❡♠ ❬▼❛t✱ ➜Þ♥❤ ❧ý ✶✹✳✶✱ ➜Þ♥❤ ❧ý ✶✹✳✷❪✮✳ ▼Ư♥❤ ➤Ị ✶✳✶✳✹✳ ♠ét ❜é ❣å♠ ❝ñ❛ d ✭✐✮ ◆Õ✉ x ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ sè ♥❣✉②➟♥ ❞➢➡♥❣ t❤× M ✈➭ x(n) = xn1 , , xnd d M✳ ✹ n = n1 , , n d ❧➭ ❝ị♥❣ ❧➭ ❤Ư t❤❛♠ sè ✭✐✐✮ ❈❤♦ x1 , , x t ❧➭ ♠ét ❞➲② ❝➳❝ ♣❤➬♥ tư ❝đ❛ x ∈ m ✭✐✐✐✮ ❍Ö x1 , , xt ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ p ∈ Ass M/(x1 , , xi−1 )M t❤á❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ xi ∈ / p, dim R/p = d − i + 1, ✈í✐ ♠ä✐ ✈í✐ i = 1, , d ➜➷❝ ❜✐Ưt✱ ♠ét ♣❤➬♥ tư x ∈ m ❧➭ ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ M ❝❤Ø ❦❤✐ x∈ / p, ✈í✐ ♠ä✐ p ∈ Ass M ✭✐✈✮ ◆Õ✉ ➤ã M ❧➭ t➠♣➠ ➤➬② ➤đ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳ dim M = d✳ t❤ø❝ ✈í✐ s❛♦ ❝❤♦ x ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M n t❤× M ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ M ♠➲♥ d✳ ❑❤✐ ➤ã✱ dim M − t dim(M/(x1 , , xt )M ) ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ m, ✈í✐ t ♠ä✐ ❦❤✐ ✈➭ dim R/p = d x ❝ị♥❣ ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ M , tr♦♥❣ m✲❛❞✐❝ ❝ñ❛ M ❈❤♦ I ❧➭ ✐➤➟❛♥ m✲♥❣✉②➟♥ s➡ ❝đ❛ R, ✈➭ ❝❤✐Ị✉ ❑r✉❧❧ ❚❤❡♦ ➜Þ♥❤ ❧ý ✶✳✶✳✶✱ t❛ ❝ã R (M/I n+1 M ) ➤đ ❧í♥✱ tr♦♥❣ ➤ã deg PM,I (n) = d = PM,I (n) ❧➭ ➤❛ ❑❤✐ ➤ã tå♥ t➵✐ ❝➳❝ sè ♥❣✉②➟♥ e0 , e1 , , ed , e0 > s❛♦ ❝❤♦ n+d n+d−1 − e1 + + (−1)d ed d d−1 PM,I (n) = e0 ❈➳❝ sè e0 , , ed ❣ä✐ ❧➭ ❤Ư sè ❍✐❧❜❡rt ❝đ❛ M ➤è✐ ✈í✐ I✱ ❦Ý ❤✐Ö✉ ❧➭ ei (I, M ) ➜➷❝ ❜✐Öt✱ sè ♥❣✉②➟♥ ❞➢➡♥❣ e0 tr♦♥❣ ❜✐Ĩ✉ ❞✐Ơ♥ tr➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤è✐ ✈í✐ x = x1 , xd ❝đ❛ I ❧➭ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ M s❛♦ ❝❤♦ (x)I n M = I n+1 M ✮✳ sè ❜é✐ ❝đ❛ M I, ❦Ý ❤✐Ư✉ ❧➭ e(I, M ) ❈❤ó ý r➺♥❣ ♥Õ✉ ❧➭ sè ❜é✐ e(I; M ) ❝đ❛ M M t❤× ❧✉➠♥ tå♥ t➵✐ ❤Ư t❤❛♠ sè (x) ❧➭ ✐➤➟❛♥ rót ❣ä♥ ❝đ❛ I ❍➡♥ ♥÷❛ e(I; M ) = e(x; M )✳ ø♥❣ ✈í✐ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ I ø♥❣ ✈í✐ M ✭♥❣❤Ü❛ ❱× t❤Õ✱ ✈✐Ư❝ tÝ♥❤ t♦➳♥ ❝ã t❤Ĩ ➤➢ỵ❝ q✉② ✈Ị tr➢ê♥❣ ❤ỵ♣ I ❧➭ ✐➤➟❛♥ s✐♥❤ ❜ë✐ ❤Ư t❤❛♠ sè ❝đ❛ M ❜ë✐ ✈× ❦❤✐ ➤ã ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ sè ❜é✐ t❤➠♥❣ q✉❛ ➤å♥❣ ➤✐Ị✉ ❑♦s③✉❧ H• ✭①❀ ▼✮ ✭①❡♠ ❬❇❍❪✮✳ ❉♦ ➤ã✱ ➤Ĩ t✐Ư♥ ❤➡♥ tr♦♥❣ ✈✐Ư❝ tÝ♥❤ t♦➳♥ tr➟♥ sè ❜é✐✱ t ệ số ộ ì tứ ợ ❣✐í✐ t❤✐Ư✉ ✺ ❜ë✐ ◆♦rt❤❝♦tt✳ ❱➱♥ ❣✐➯ t❤✐Õt ❤÷✉ ❤➵♥ s✐♥❤✳ ▼ét ❤Ư ❝➳❝ ♣❤➬♥ tư M ♥Õ✉ ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ (R, m) x1 , , xt tr♦♥❣ M ❧➭ R✲♠➠➤✉♥ m ➤➢ỵ❝ ❣ä✐ ❧➭ ❤Ư ❜é✐ ❝đ❛ (M/(x1 , , xt )M ) < ∞, ❤❛② ♠ét ❝➳❝❤ t➢➡♥❣ ➤➢➡♥❣✱ (x1 , , xt ) ❧➭ ♠ét ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❧➭ ♠ét ❞➲② ❦❤í♣ ❝➳❝ ✈➭ ❝❤Ø ♥Õ✉ r➺♥❣ ♥Õ✉ R✲♠➠➤✉♥✳ x1 , , xt x1 , , x t M ❈❤♦ −→ M −→ M −→ M −→ ❑❤✐ ➤ã ❧➭ ❤Ư ❜é✐ ❝đ❛ ❧➭ ❤Ư ❜é✐ ❝đ❛ M M t❤× x1 , , xt ✈➭ M ❧➭ ♠ét ❤Ư ❜é✐ ❝đ❛ ♥Õ✉ M ✳ ❚õ ➤ã ❞Ơ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ x2 , , x t ❧➭ ❤Ư ❜é✐ ❝đ❛ M/x1 M :M x1 ì tế t ó ị ♥❣❤Ü❛ s❛✉✳ ❈❤♦ x1 , , xt ❧➭ ❤Ư ❜é✐ ❝đ❛ M ✳ ◆Õ✉ t ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✻✳ = 0✱ tø❝ ❧➭ (M ) < ∞ t❤× t❛ ➤➷t e(∅; M ) = (M ) ◆Õ✉ t > 0, tø❝ ❧➭ (M/(x1 , , xt )M ) < ∞ t❤× t❛ ❝ã ((0 :M x1 )/(x2 , , xt )(0 :M x1 )) < ∞, tø❝ ❧➭ (x2 , , xt ) ❧➭ ❤Ư ❜é✐ ❝đ❛ :M x1 ❚❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ t❤× e(x2 , , xt ; M/x1 M ) ✈➭ e(x2 , , xt ; :M x1 ) ❧➭ tå♥ t➵✐✳ ❑❤✐ ➤ã e(x1 , , xt ; M ) = e(x2 , , xt ; M/x1 M ) − e(x2 , , xt ; :M x1 ) ➤➢ỵ❝ ❣ä✐ ❧➭ sè ❜é✐ ❝đ❛ M ø♥❣ ✈í✐ ❤Ư ❜é✐ (x1 , , xt ) ❙❛✉ ➤➞② ❧➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ sè ❜é✐✳ ▼Ư♥❤ ➤Ị ✶✳✶✳✼✳ ❜✐Ưt✱ t❤× ♥Õ✉ ✭✐✮ tå♥ t➵✐ i e(x1 , , xt ; M ) s❛♦ ❝❤♦ xni M = 0, ✈í✐ ♠ä✐ (M/(x1 , , xt )M ) n ❧➭ sè tù ♥❤✐➟♥ e(x1 , , xt ; M ) = ✭✐✐✮ ●✐➯ sö −→ Mn −→ −→ M1 −→ M0 −→ ✻ ➜➷❝ ♥➭♦ ➤ã ❈❤ø♥❣ ♠✐♥❤✳ (a) ⇒ (c)✳ ✭✐✮✳ ➜✐Ị✉ ❦✐Ư♥ t➢➡♥❣ ➤➢➡♥❣ ❈❤♦ d = 1✳ ❑❤✐ ➤ã (a) ⇔ (b) ❞♦ ❇ỉ ➤Ị ✷✳✷✳✸✱ ✭✐✮✳ ✈➭ s=0 M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❚❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✹✭✐✈✮✱ tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝ x1 ❝ñ❛ M✱ ♥❣❤Ü❛ ❧➭ s✱ tr♦♥❣ ➤ã I(x21 ; M ) = I(x1 ; M )✳ ❱× t❤Õ ✭❝✮ ❧➭ ➤ó♥❣✳ ❈❤♦ d > 1✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❦Õt q✉➯ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ s < d✳ ❈❤♦ s = ❑❤✐ ➤ã N-dimR Hmi (M ) i (M )) ➤Ị ✶✳✸✳✺✱ ✭✐✮✱ R (Hm ✈í✐ ♠ä✐ i < d ❚❤❡♦ ❇ỉ < ∞ ✈í✐ ♠ä✐ i < d, ♥❣❤Ü❛ ❧➭ M ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❱× t❤Õ✱ t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✹✱ ✭✐✈✮✱ tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝ x = x1 , , xd ❝ñ❛ M s❛♦ ❝❤♦ I(x21 , , x2d ; M ) = I(x1 , , xd ; M )✳ ♥➭② ♥❣❤Ü❛ ❧➭ ➤✐Ị✉ ❦✐Ư♥ ✭❝✮ ➤ó♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❣✐➯ sư r➺♥❣ ❦Õt q✉➯ ➤ó♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ s = s − ◆Õ✉ ❈❤♦ p(M ) ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❙✉② r❛ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè ❝❤✉➮♥ t➽❝ ➜✐Ị✉ s ✈➭ ❝❤♦ x1 ∈ m s❛♦ ❝❤♦ x1 ∈ / p ✈í✐ ♠ä✐ Att(Hmi (M )))\{m} ❈❤ó ý r➺♥❣ t❤❡♦ ❣✐➯ t❤✐Õt ❝đ❛ (a) ✈➭ t❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✸✱ ✭✐✮ t❛ ❝ã p(M ) ➤Ị ✷✳✷✳✸✱ ✭✐✐✮✳ ❱× ✈❐②✱ ❇ỉ ➤Ị ✷✳✷✳✸✱ ✭✐✮✳ sè ❝❤✉➮♥ t➽❝ s✳ ❉♦ ➤ã✱ p(M/x1 M ) = p(M ) − N-dim(Hmi (M/x1 M ) s − 1✱ t❤❡♦ ❇ỉ s − ✈í✐ ♠ä✐ i < d − t❤❡♦ ➳♣ ❞ô♥❣ ❣✐➯ t❤✐Õt q✉② ♥➵♣ ❝❤♦ M/x1M ✱ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ x2 , , xd ❝ñ❛ M ✈➭ ❝➳❝ sè ♥❣✉②➟♥ k2 , , ks ∈ {2, , d} s❛♦ ❝❤♦ I(y2 , , yd ; M ) = I(x2 , , xd ; M ), tr♦♥❣ ➤ã yj = x2j ✈í✐ ♠ä✐ j = 2, , d ✐❢ j ∈ / {k2 , , ks } ✈➭ yj = x j ♥Õ✉ j ∈ {k2 , , ks }, ❑❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ sö r➺♥❣ ✷✻ k2 = 2, , ks = s✱ ♥❣❤Ü❛ ❧➭ I(x2 , , xs , x2s+1 , , x2d ; M/x1 M ) = I(x2 , , xd ; M/x1 M ) ❚❤❡♦ ❝➳❝❤ ❝❤ä♥ x1 ✱ ➳♣ ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✷✳✶✱ t❛ ❝ã dim(0 :M x1 ) (1) 0✳ ❱× d > 1, ♥➟♥ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✼✱✭✸✐✮ t❛ ❝ã e(x2 , , xs , x2s+1 , , x2d ; :M x1 ) = = e(x2 , , xs , xs+1 , , xd ; :M x1 ) ❱× t❤Õ t❤❡♦ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ ➤é ❞➭✐ ✈➭ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ sè ❜é✐ ❤×♥❤ t❤ø❝ I(x2 , , xs ,x2s+1 , , x2d ; M/x1 M ) = (M/x1 M/(x2 , , xs , x2s+1 , , x2d )M/x1 M ) − e(x2 , , xs , x2s+1 , , x2d ; M/x1 M ) 2 R (M/(x1 , , xs , xs+1 , , xd )M ) = − e(x1 , , xs , x2s+1 , , x2d ; M ) + e(x2 , , xs , x2s+1 , , x2d ; :M x1 ) = I(x1 , , xs , x2s+1 , , x2d ; M ) ❚➢➡♥❣ tù✱ t❛ ❝ò♥❣ ❝ã I(x2 , , xd ;M/x1 M ) = (M/x1 M/(x2 , , xd )M/x1 M ) − e(x2 , , xd ; M/x1 M ) = R (M/(x1 , x2 , , xd )M ) − e(x1 , x2 , , xd ; M ) + e(x2 , , xd ; :M x1 ) = I(x1 , , xd ; M ) ❱× ✈❐②✱ tõ (1) ❦Ð♦ t❤❡♦ I(x1 , , xs , x2s+1 , , x2d ; M ) = I(x1 , , xd ; M ), ✈➭ ✭❝✮ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ (c) ⇒ (d) ❈❤♦ d = 1✳ ❑❤✐ ➤ã s=0 ➤ã tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè ❝❤✉➮♥ t➽❝ ✈➭ x1 ✷✼ M ❝ñ❛ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❉♦ M ✈➭ t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✹✱ ✭✐✈✮✱ t❛ ❝ã I(x1 ; M ) = I(x21 ; M ) = I(xn1 ; M ) ✈í✐ ♠ä✐ n ∈ N✳ ❈❤♦ Cx = I(x1 ; M ) I(xn1 ; M ) = Cx = n0 Cx ❑❤✐ ➤ã ✈í✐ ♠ä✐ n ì (d) ợ ứ d > 1✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ s✱ tr♦♥❣ ➤ã s = 0✳ ❚õ ❣✐➯ t❤✐Õt ✭❝✮✱ tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè x ❝đ❛ M s < d✳ ❈❤♦ s❛♦ ❝❤♦ I(x21 , , x2d ; M ) = I(x1 , , xd ; M ) ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ r➺♥❣ ✭✐✈✮ t❛ ❝ã M ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ✈➭ t❤❡♦ ❇æ ➤Ị ✷✳✶✳✹✱ x ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝ ❝ñ❛ M ✳ ➜➷t Cx = I(x1 , , xd ; M )✳ ❑❤✐ ➤ã I(xn1 , , xnd ; M ) = n0 Cx ✈í✐ ♠ä✐ n ✈➭ (d) ➤ó♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ r➺♥❣ ❦Õt q✉➯ ➤ó♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ t❤❛♠ sè ❝ñ❛ M t❤á❛ ♠➲♥ s − s > ✈➭ ❣✐➯ sö x = (x1 , , xd ) ❧➭ ♠ét ❤Ö s = ❈❤♦ ❈❤♦ (c)✳ ❑❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t t❛ ❝ã t❤Ĩ ❣✐➯ sö r➺♥❣ k1 = d − s + 1, , ks = d ✱ ♥❣❤Ü❛ ❧➭ I(x21 , , , x2d−s , xd−s+1 , , xd ; M ) = I(x1 , , xd ; M ) (2) ❚❤❡♦ tÝ♥❤ ❝❤✃t ❝ñ❛ sè ❜é✐ tr♦♥❣ ❇ỉ ➤Ị ✶✳✶✳✼✱ t❛ ❝ã I(x21 , , x2d−s ,xd−s+1 , , xd ; M ) = (M/(x21 , , x2d−s , xd−s+1 , , xd )M ) − e(x21 , , x2d−s , xd−s+1 , , xd ; M ) = (M/xd M/(x21 , , x2d−s , xd−s+1 , , xd−1 )M/xd M ) − e(x21 , , x2d−s , xd−s+1 , , xd−1 ; M/xd M + e((x21 , , x2d−s , xd−s+1 , , xd−1 ); :M xd ) = I(x21 , , x2d−s , xd−s+1 , , xd−1 ; M/xd M ) + 2d−s e(x1 , , xd−1 ; :M xd ) ❚➢➡♥❣ tù t❛ ❝ò♥❣ ❝ã I(x1 , , xd ; M ) = I(x1 , , xd−1 ; M/xd M ) + e(x1 , , xd−1 ; :M xd ) ✷✽ ❈❤ó ý r➺♥❣ I(x21 , , x2d−s , xd−s+1 , , xd−1 ; M/xd M ) t❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✹✳ ❱× I(x1 , , xd−1 ; M/xd M ) s < d✱ ♥➟♥ t❛ ❝ã 2d−s e(x1 , , xd−1 ; :M xd ) > e(x1 , , xd−1 ; :M xd ) ❱× ✈❐②✱ t❤❡♦ (2) t❛ ❝ã e(x1 , , xd−1 ; :M xd ) = ✈➭ I(x1 , , xd−s , xd−s+1 , , xd−1 ; M/xd M ) = I(x21 , , x2d−s , xd−s+1 , , xd−1 ; M/xd M ) ❉♦ ➤ã✱ dim(0 :M xd ) d − t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✼ ✈➭ ✈× t❤Õ✱ t❛ ❝ã ➤➻♥❣ t❤ø❝ e(xn1 , , xnd−1 ; :M xd ) = 0✱ ♠➠➤✉♥ ✈í✐ ♠ä✐ n > 0✳ M/xd M ✱ tå♥ t➵✐ ♠ét ❤➺♥❣ sè Cx I(xn1 , ,xnd ; M ) ❉ï♥❣ ❣✐➯ t❤✐Õt q✉② ♥➵♣ ❝❤♦ s❛♦ ❝❤♦ nI(xn1 , , xnd−1 , xd ; M ) = n (M/(xn1 , , xnd−1 , xd )M ) − e(xn1 , , xnd−1 , xd ; M ) = n (M/xd M/(xn1 , , xnd−1 )M/xd M ) − e(xn1 , , xnd−1 ; M/xd M ) + e(xn1 xnd−1 ; :M xd ) = n I(xn1 , , xnd−1 ; M/xd M ) + e(xn1 , , xnd−1 ; :M xd ) nns−1 Cx = ns Cx ✈í✐ ♠ä✐ sè n > ì (d) ợ ứ (d) (b)✳ ❱× I(xn1 , , xnd ; M ) ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❦✐Ĩ✉ ➤❛ t❤ø❝ ✭✐✐✮ ●✐➯ sư (a) ➤ó♥❣✳ ➜➷t m ns I(x; M ) p(M ) t❛ ❝ã p(M ) ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ n✱ t❤❡♦ s = mR ❱× t❛ ❝ã ➤➻♥❣ ❝✃✉ Hmi (M ) ∼ = Hmi (M ) ❝➳❝ R✲♠➠➤✉♥ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✶✳✸✳✺ ✈➭ ❣✐➯ t❤✐Õt ❝đ❛ ✭❛✮ t❛ ❝ã N-dimR (Hmi (M )) s ✈í✐ ♠ä✐ i < d ❚r➢í❝ ❤Õt t❛ ❦❤➻♥❣ ➤Þ♥❤ r➺♥❣ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ d✳ ❈❤♦ d ✈➭ M = ❑❤✐ ➤ã s = ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❚❤❡♦ ❬❈❙❚❪✱ ỗ ệ t số ủ M M ❝❤Ý♥❤ q✉② ✈í✐ ❝❤✐Ị✉ > s✉② r❛ M > ❈❤♦ d>1 ✈➭ ❣✐➯ sư ♠Ư♥❤ ➤Ị ➤ó♥❣ ✈í✐ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ k=d r➺♥❣ t❤× ❧➭ ♠➠➤✉♥ ❈♦❤❡♥ ▼❛❝❛✉❧❛② ❝❤✐Ò✉ x1 ∈ / p M ✳ ❈❤♦ p ∈ AssR M ✈× x1 d − 1✳ s❛♦ ❝❤♦ ❧➭ ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ ❈❤♦ x = x1 , , x d ❧➭ dim(R/p) := k > s ◆Õ✉ M✳ ❉♦ ➤ã✱ t❛ ❝ã t❤Ĩ ❣✐➯ sư k < d✳ ❈❤ó ý r➺♥❣ p ∈ AttR (Hmk (M )) t❤❡♦ ❬❇❙✱ ❍Ö q✉➯ ✶✶✳✸✳✸❪✳ ❱× ✈❐② p ⊇ AnnR (Hmk (M )) t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✸✳✷✳ ❱× t❤Õ t❤❡♦ ❇ỉ ➤Ị ✶✳✸✳✺✱ ✭✐✐✐✮ t❛ ❝ã N-dimR (Hmk (M )) = dim(R/ AnnR (Hmk (M )) ▼➷t ❦❤➳❝✱ N-dimR (Hmk (M )) r❛✳ ❱× t❤Õ x1 ❧➭ s t❤❡♦ ❣✐➯ t❤✐Õt ✭❛✮✳ ➜✐Ò✉ ♥➭② ❦❤➠♥❣ t❤Ĩ ①➯② M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s✳ ✷✳✶✳✶✵✱ ✭✐✮✳ ❱❐② s✉② r❛ dim(R/p) = k > s ❉♦ ➤ã s t❤❡♦ ❇ỉ ➤Ị dim(0 :M x1 ) Hmi (0 :M x1 ) = ✈í✐ ♠ä✐ i > s✳ ❚õ ❞➲② ❦❤í♣ −→ :M x1 −→ M −→ M /(0 :M x1 ) −→ ❜➺♥❣ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❇ỉ ➤Ị ✷✳✷✳✸✱ t❛ ❝ã ➤➻♥❣ ❝✃✉ s❛✉ Hmi (M ) ∼ = Hmi (M /(0 :M x1 ))) ✈í✐ ♠ä✐ i > s✳ ❉♦ ➤ã tõ ❞➲② ❦❤í♣ x M −→ M /x1 M −→ 0, −→ M /(0 :M x1 ) −→ Hmi (M ) −→ Hmi (M /x1 M ) −→ Hmi+1 (M ) t❛ ❝ã ❞➲② ❦❤í♣ ❱× N-dimR (Hmi (M )) ✈í✐ ♠ä✐ s ✈í✐ ♠ä✐ i s✳ ❱× ✈❐②✱ x1 , , xd ❧➭ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s✱ ♥❣❤Ü❛ ❧➭ M ✈í✐ ♠ä✐ i t❤❡♦ ❇ỉ ➤Ị ✶✳✸✳✺✳ ❉♦ ➤ã✱ ➳♣ ❞ô♥❣ ❣✐➯ t❤✐Õt q✉② ♥➵♣ ❝❤♦ i < d−1 > s✳ ❱× t❤Õ✱ M ✈í✐ ♠ä✐ i s t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺✳ dimR (R/ AnnR (Hmi (M ))) s p ∈ AttR (Hmi (M )) ❑❤✐ ➤ã t❛ ❝ã ❝❤✐Ò✉ ❑r✉❧❧ i < d t❤❡♦ ❬❇❙✱ ✶✶✳✸✳✺❪ ✈➭ p ∈ AssR M t❤❡♦ ❬❇❙✱ ✶✶✳✸✳✸❪✳ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥➟♥ t❤❡♦ ❬❑❲✱ ❍Ư q✉➯ ✶✳✷❪ t❛ ❝ã R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❉♦ ➤ã ✸✵ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✺ t❛ ❝ã k > s✳ ❱× k < d✱ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ M tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè x ❦❤➠♥❣ ❧➭ M ✲❞➲② ✈í✐ ❝❤✐Ị✉ > s✳ x ❝đ❛ M > s✳ ●✐➯ sö r➺♥❣ s❛♦ ❝❤♦ x1 ∈ p✳ ➜✐Ị✉ ♥➭② ❦❤➠♥❣ t❤Ĩ ①➯② r❛✳ ❱× t❤Õ ❱× t❤Õ k s✳ ❱❐② dimR (R/ AnnR (Hmi (M ))) = max dim(R/p) s i (M ) p∈AttR Hm ◆❤➢ ❧➭ ❤Ư q✉➯ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ t❛ ❝ã ❝➳❝ ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s t❤➠♥❣ q✉❛ ❝❤✐Ị✉ ❝đ❛ q✉ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥❤➢ s❛✉✳ ◆Õ✉ ◆❈(M ) ❧➭ t❐♣ q✉ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝ñ❛ ♥Õ✉ R M ✈➭ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ❝➳❝ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× ◆❈(M ) ❧➭ t❐♣ ➤ã♥❣ tr♦♥❣ t❐♣ dim(◆❈(M )) ➤ã (M ) Spec R t t rs ì tế ợ ➤Þ♥❤✳ ➜➷t a(M ) = a0 (M ) ad−1 (M )✱ tr♦♥❣ = AnnR (Hmi (M )) ✈í✐ ♠ä✐i ❍Ư q✉➯ ✷✳✷✳✻✳ ◆Õ✉ R d − ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② t❤× ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ dim(R/a(M )) ✭✐✐✮ ✭✐✐✐✮ dim NC(M ) ❈❤ø♥❣ ♠✐♥❤✳ > s✳ s s ✈➭ dim(R/p) = d ✈í✐ ♠ä✐ p ∈ (min(SuppR M ))>s ✳ ✭✐✮⇔ ✭✐✐✮✳ ❚❤❡♦ ❬❈✶✱ ➜Þ♥❤ ❧ý ✶✳✷❪ t❛ ❝ã p(M ) = dim(R/a(M )) ❉♦ ➤ã ❦Õt q✉➯ ➤➢ỵ❝ s r từ ị ý í ợ s✉② r❛ tõ ➜Þ♥❤ ❧ý ✷✳✶✳✶✸✱ ✭✐✮⇒ ✭✐✈✮✳ ✭✐✐✐✮⇒ ✭✐✮ ❈❤♦ Mp p ∈ (SuppR M )>s ✳ ❱× dim NC(M ) ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈❤♦ dim(R/q) = d t❤❡♦ ✭✐✐✐✮✳ ❱× s t❤❡♦ ❣✐➯ t❤✐Õt ✭✐✐✐✮✱ q ∈ min(SuppR M )>s s❛♦ ❝❤♦ q ⊆ p✳ ❑❤✐ ➤ã R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ♥➟♥ R ❧➭ ❝❛t❡♥❛r②✳ ❉♦ ➤ã d ≥ dim(R/p) + dim Mp ≥ dim(R/p) + ❤t(p/q) = dim(R/q) = d ✸✶ ❱× ✈❐② M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > ❍Ư q✉➯ ✷✳✷✳✼✳ R ●✐➯ sư r➺♥❣ s t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✶✳✶✸✱ ✭✐✈✮⇒✭✐✮✳ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ò s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M = ⊕ni=1 Mi i✱ Mi ❝ã ❝❤✐Ị✉ ❧í♥ ♥❤✃t ❧➭ ❝❤✐Ị✉ > s✳ ✭✐✐✮ ❈❤♦ x1 , , xd−s ❧➭ s ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✈í✐ ❝❤✐Ị✉ ➤ã M ❤♦➷❝ ❝ã ❝❤✐Ị✉ ❧➭ ♠ét ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❈❤ø♥❣ ♠✐♥❤✳ ♣❤➬♥ > s ❤Ư d t❤❛♠ ✈➭ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ sè ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ N-dim(Hmj (M )) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ N-dim(Hmj (Mi )) ❑❤✐ ➤ã (x1 , , xd−s )M ✭✐✮ ❚❤❡♦ ❣✐➯ t❤✐Õt ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ t❛ ❝ã > s M✳ ❝đ❛ M ❧➭ ❝ị♥❣ ❧➭ > s✳ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✭✐✐✮ ➜➷t > s ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✈í✐ ♠ä✐ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s s ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ♠ä✐ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ s ✈í✐ ♠ä✐ j < d ❤♦➷❝ dim Mi N = (x1 , , xd−s )M ✳ M ❚õ ❞➲② ❦❤í♣ j < d✳ dim Mi = d ❉♦ ✈➭ s ✈í✐ ♠ä✐ i = 1, , n✳ → N → M → M/N → ✈➭ ❣✐➯ t❤✐Õt x1 , , xd−s ❧➭ ♠ét ♣❤➬♥ ❤Ö t❤❛♠ sè ❝đ❛ M ✱ t❤❡♦ tÝ♥❤ ❝❤✃t ❝đ❛ ❤Ư t❤❛♠ sè t❛ ❝ã dim M/N = s✳ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ t❛ ❝ã ❱× ✈❐②✱ ➳♣ ❞ơ♥❣ tÝ♥❤ tr✐Ưt t✐➟✉ ❝đ❛ ♠➠➤✉♥ ➤è✐ Hmi (M/N ) = ✈í✐ ♠ä✐ i > s ❉♦ ➤ã✱ ➳♣ ❞ô♥❣ tÝ♥❤ ❝❤✃t δ ✲❤➭♠ tư ➤å♥❣ ➤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈➭♦ ❞➲② ❦❤í♣ ♥❣➽♥ ë tr➟♥ t❛ ❝ã ❞➲② ❦❤í♣ ❞➭✐ → Hmi (M/N ) → Hmi+1 (N ) → Hmi+1 (M ) → Hmi+1 (M/N ) → ✈➭ ✈× Hmi (M/N ) = ✈í✐ ♠ä✐ i>s ♥➟♥ t❛ ❝ã Hmi (N ) ∼ = Hmi (M ) ✈í✐ ♠ä✐ i > s t ợ ết q từ ị ý ❚✐Õ♣ t❤❡♦✱ t❛ q✉❛♥ t➞♠ ➤Õ♥ tÝ♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ủ tứ ỗ ❧ị② t❤õ❛ ❤×♥❤ t❤ø❝✳ ▼Ư♥❤ ➤Ị ✷✳✷✳✽✳ t❤ø❝ ❈❤♦ t ❜✐Õ♥ x1 , , xt S = R[[x1 , , xt ]] ✈í✐ ❝➳❝ ệ số tr ỗ ũ từ ❤×♥❤ R ❑❤✐ ➤ã p(S) = p(R) + t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ q✉② ♥➵♣✱ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❘â r➭♥❣ r➺♥❣ t = 1✳ n = (m, x1 , , xt ) ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❞✉② ♥❤✃t ❝ñ❛ S ✈➭ dim S = dim R+t✳ ➜➷t x1 = x ✈➭ ❝❤♦ (a1 , , ad ) ❧➭ ❤Ö t❤❛♠ sè ❝ñ❛ R✳ ❑❤✐ ➤ã t❛ ❝ã ❝➳❝ t♦➭♥ ❝✃✉ í t ữ ị ci xi ) = c0 ❉♦ ➤ã✱ t❛ ❝ã t❤Ó ❝♦✐ ỗ R S t ( ĩ ❝ñ❛ ❝➳❝ ϕ : S −→ R ϕ✳ ❘â r➭♥❣ r➺♥❣ Ker ϕ = xS ✳ ❱× ✈❐② tå♥ t➵✐ ♠ét ➤➻♥❣ ❝✃✉ ❣✐÷❛ S ✲♠➠➤✉♥ S/(a1 , , ad , x)S ∼ = R/(a1 , , ad )R ➜✐Ò✉ ♥➭② s✉② r❛ S/(a1 , , ad , x)S ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥✱ ♥❣❤Ü❛ ❧➭ (a1 , , ad , x) ❧➭ ♠ét ♣❤➬♥ ❤Ö t❤❛♠ sè ❝đ❛ ♥❣✉②➟♥ ❞➢➡♥❣✳ ❱× x ❧➭ S✳ ❈❤♦ S ✲❝❤Ý♥❤ n1 , , nd , n xn q✉②✱ ❧➭ ♠ét ❜é ❝ò♥❣ ❧➭ S✲ (d + 1) ❝➳❝ sè ❝❤Ý♥❤ q✉② ✈➭ ✈× t❤Õ (0 :S xn ) = ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ tÝ♥❤ ❝❤✃t ❝đ❛ sè ❜é✐ tr♦♥❣ ▼Ư♥❤ ➤Ị ✶✳✶✳✼✱ ✭✐✐✐✮✱ t❛ ❝ã ❦Õt q✉➯ s❛✉ e(an1 , , and d , xn ; S) = e(an1 , , and d ; S/xn S) − e(an1 , , and d ; :S xn ) = e(an1 , , and d ; S/xn S) ❍✐Ó♥ ♥❤✐➟♥ r➺♥❣ ➳♥❤ ✈í✐ ψ : S −→ Rn ❝❤♦ ❜ë✐ ψ( ci xi ) = (c0 , , cn−1 ) ❧➭ t♦➭♥ Ker ψ = xn S ✳ ❱× t❤Õ S/xn S ∼ = Rn ✳ ❱❐② s✉② r❛ e(an1 , , and d ; S/xn S) = e(an1 , , and d ; Rn ) = ne(an1 , , and d ; R) ▼➷t ❦❤➳❝✱ tõ ➤➻♥❣ ❝✃✉ S S/xn S ∼ = Rn ✱ t❛ ❝ã S/(an1 , , and d , xn )S = S/xn S/(an1 , , and d )S/xn S S =n R R/(an1 , , and d )R ❱× t❤Õ t❛ ❝ã I(an1 , , and d , xn ; S) = S S/(an1 , , and d , xn )S −e(an1 , , and d , xn ; S) = S S/xn S/(an1 , , and d )S/xn S −e(an1 , , and d ; S/xn S) =n R R/(an1 , , and d )R −ne(an1 , , and d ; R) = nI(an1 , , and d ; R) ✸✸ ❉♦ ➤ã✱ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❦✐Ĩ✉ ➤❛ t❤ø❝ t❛ t❤✉ ➤➢ỵ❝ ❈❤♦ ❧➭ ✈➭♥❤ ỗ ũ từ ì tứ S = R[[x1 , , xt ]] R[x1 , , xt ] p(S) = p(R) + ❧➭ ✈➭♥❤ ➤❛ t❤ø❝ t ❜✐Õ♥ tr➟♥ R✳ ❚❛ ➤➲ ❜✐Õt r➺♥❣ ▼❛❝❛✉❧❛② ✭♥❣❤Ü❛ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > ❝ị♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❤ỵ♣ −1✮ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ S ✈➭ S > −1✳ ❉➢í✐ ➤➞②✱ t❛ q✉❛♥ t➞♠ ➤Õ♥ tr➢ê♥❣ ❈❤♦ s ≥ ❧➭ sè ♥❣✉②➟♥✳ ●✐➯ sö r➺♥❣ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈❤♦ ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ t❤✉➬♥ ♥❤✃t ❝ñ❛ R ❧➭ ❝❛t❡♥❛r② ♣❤æ n = (m, x1 , , xt )S S ✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ R ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s✳ S ✭✐✐✮ ✭✐✐✐✮ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ Sn > s + t✳ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮⇒✭✐✐✮✳ ❱× > s + t✳ R ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > s ✈➭ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ọ tớ ì tứ t ó t ị ❧ý ✷✳✷✳✺✳ ❱× t❤Õ ➤ã ❧➭ ❈♦❤❡♥✲ s ≥ 0✳ ❍Ö q✉➯ ✷✳✷✳✾✳ ✭✐✮ R S = S p(S) = p(R) + t ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ ✭✐✐✮⇒✭✐✮✳ ❱× R I[[x]] S = R[[x]] s + t t❤❡♦ ▼Ö♥❤ ➤Ị ✷✳✷✳✽✳ ❉♦ ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲ R ❧➭ ✈➭♥❤ t❤➢➡♥❣ A/I ❝đ❛ ✈➭♥❤ A✳ ❚õ ➤➻♥❣ ❝✃✉ R[[x]] ∼ = tr♦♥❣ ➤ã s > s + t t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ▼❛❝❛✉❧❛②✱ t❤❡♦ ❬❑❲✱ ❍Ö q✉➯ ✶✳✷❪ t❛ ❝ã ❈♦❤❡♥✲▼❛❝❛✉❧❛② p(R) ❧➭ ✐➤➟❛♥ ❝ñ❛ A [[x]] ∼ = A[[x]]/I[[x]], I A[[x]] ❝ã ❤Ư sè tr♦♥❣ I✱ ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤✐Ị✉ ♥➭② ❦Ð♦ t❤❡♦ A[[x]]✳ ❱× ✈❐② S ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❉♦ ➤ã✱ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ ✭✐✐✐✮ t❛ ❝ã p(S) < s + t ✈➭ ✈× ✈❐② p(R) < s t❤❡♦ ệ ề ì tế ết q ợ s r tõ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ ✭✐✐✮✳ ❚➢➡♥❣ tù ❝❤♦ ❝❤ø♥❣ ♠✐♥❤ tr➢ê♥❣ ❤ỵ♣ ✭✐✮ ⇔ ✭✐✐✐✮✳ ✸✹ ❚➢➡♥❣ tù ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❢✲♠➠➤✉♥ ✈➭ ❢✲♠➠➤✉♥ s✉② ré♥❣✱ ❣✐➯ t❤✐Õt R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ♠ä✐ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② tr ị ý tể ỏ ợ ❚❛ ❝ã t❤Ĩ ♠✐♥❤ ❤ä❛ ❜➺♥❣ ✈Ý ❞ơ s❛✉✳ ❱Ý ❞ơ ✷✳✷✳✶✵✳ ✭✐✮ ❚å♥ t➵✐ ♠ét ♠✐Ị♥ ♥❣✉②➟♥ ◆♦❡t❤❡r dim S = 4, depth S = ✈➭ S (S, n) s❛♦ ❝❤♦✿ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ > 2✳ N-dim(Hn3 (S)) = 3✱ dim(S/ AnnS (Hn3 (S)) = ✈➭ dim S/a(S) = ✭✐✐✮ ✭✐✐✐✮ p(S) = 3, dim(S/a(S)) = ✈➭ S > 2✱ tr♦♥❣ ➤ã S ❈❤ø♥❣ ♠✐♥❤✳ ❧➭ ➤➬② ➤đ ❦❤➠♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ n✲❛❞✐❝ ❝đ❛ S ✳ ❈❤♦ (R, m) ❧➭ ♠✐Ị♥ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ ❝❤✐Ị✉ ➤➢ỵ❝ ①➞② ❞ù♥❣ ❜ë✐ ❉✳ ❋❡rr❛♥❞ ✈➭ ▼✳ ❘❛②♥❛✉❞ ❬❋❘❪ tr♦♥❣ ➤ã ➤➬② ➤ñ ✐➤➟❛♥ ♥❣✉②➟♥ tè ♥❤ó♥❣ m✲❛❞✐❝ ❝đ❛ R ❝ã ♠ét q ❝❤✐Ị✉ 1✳ ❚❤❡♦ ❬❈◆✱ ❱Ý ❞ô ✹✳✶❪ t❛ ❝ã dimR (Hm1 (R)) = N-dim(Hm1 (R)) = < dimR (Hm1 (R)) = ỗ ũ từ ì tứ ❤❛✐ ❜✐Õ♥ S = R[[x, y]] ✈í✐ ❤Ư sè tr♦♥❣ R ❑❤✐ ➤ã dim S = ✈➭ depth S = 3✳ ❱× S ❝❤✐Ị✉ 4✱ ♥➟♥ t❛ t❤✃② r➺♥❣ ✭✐✐✮✳ ❘â r➭♥❣ r➺♥❣ ❧➭ ♠✐Ò♥ ◆♦❡t❤❡r ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ S x, y > 2✳ n = (m, x, y)S ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❞✉② ♥❤✃t ❝ñ❛ S ✈➭ S = R[[x, y]] ➤➬② ➤ñ n✲❛❞✐❝ ❝ñ❛ S ❱× p ∈ Ass R✱ ♥➟♥ tå♥ t➵✐ ♣❤➬♥ tư a ∈ R s❛♦ ❝❤♦ p = AnnR a ➜➷t ∞ xi + bi y i ∈ S | , bi ∈ p, ∀i p[[x, y]] = i=0 ❑❤✐ ➤ã p[[x, y]] ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ S ✈➭ ∞ ∞ i (aai )xi + (abi )y i = = p[[x, y]] x + bi y ∈ S | AnnS a = i=0 ❉♦ ➤ã✱ i i=0 p[[x, y]] ∈ Ass S ✈➭ dim( R S ) = dim( )[[x, y]] = dim(R/p) + = p[[x, y]] p ✸✺ ❚❤❡♦ ❬❇❙✱ ❍Ö q✉➯ ✶✶✳✸✳✸❪✱ t❛ ❝ã ❱× ✈❐② p[[x, y]] ∈ AttS (Hn3 (S)) ∼ = AttS (Hn3 (S)) p[[x, y]] ⊇ AnnS (Hn3 (S)) ❱× p[[x, y]] ∈ Ass S ∩ AttS (Hn3 (S)) ♥➟♥ t❛ ❝ã p[[x, y]] ∩ S ∈ Ass(S) ∩ AttS (Hn3 (S)) = ❞♦ S ❧➭ ♠✐Ị♥ ♥❣✉②➟♥✳ ❉♦ ➤ã t❛ t❤✉ ➤➢ỵ❝ AnnS (Hn3 (S)) = AnnS (Hn3 (S)) ∩ S ⊆ p[[x, y]] ∩ S = ❱❐②✱ dimS S/ AnnS (Hn3 (S)) = dim S/a(S) = dim S = 4✳ ✭✐✐✐✮✳ ❚❤❡♦ ▼Ö♥❤ ➤Ị ✷✳✷✳✽ t❛ ❝ã p(S) = t❤❡♦ ❬❈❪✳ ❱× ✈❐② S p(S) = 3✳ ❙✉② r❛ dim S/a(S) = p(S) = ❦❤➠♥❣ ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣ ✈í✐ ❝❤✐Ị✉ > t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ✸✻ ❑Õt ❧✉❐♥ ❚ã♠ ❧➵✐✱ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ t❤ù❝ ❤✐Ư♥ ➤➢ỵ❝ ❝➳❝ ✈✃♥ ➤Ị s❛✉✳ ✶✳ ❚r×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ✈Ị ❤Ư t❤❛♠ sè ✈➭ sè ❜é✐✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r✱ ❦✐Ĩ✉ ➤❛ t❤ø❝✱✳✳✳ ✷✳ ❚r×♥❤ ❜➭② ❧➵✐ ♠ét sè ❦Õt q✉➯ ✈Ò ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ♠ét sè ♠ë ré♥❣ ♥❤➢ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❢✲♠➠➤✉♥✱ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ f ✲♠➠➤✉♥ s✉② ré♥❣✱ > s✳ ✸✳ ❚r×♥❤ ❜➭② ❧➵✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ >s t❤➠♥❣ q✉❛ sè ❜é✐✱ ❦✐Ĩ✉ ➤❛ t❤ø❝✱ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ✸✼ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬❇❍❪ ❇r✉♥s ❲✳ ✱ ❏✳ ❍❡r③♦❣✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❘✐♥❣s ✱ r❡✈✐s❡❞ ❡❞✳✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✽✳ ❬❇◆❪ ❇r♦❞♠❛♥♥ ▼✳ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥✱ ♦❢ ❝❡rt❛✐♥ ❆ ❢✐♥✐t❡♥❡ss r❡s✉❧t ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s Ext✲♠♦❞✉❧❡s✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✸✻ ✭✷✵✵✽✮✱ ✶✺✷✼✲✶✺✸✻✳ ❬❇❙❪ ❇r♦❞♠❛♥♥ ▼✳ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣✱ ❵❵▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✽✳ ❬❈❪ ❈✉♦♥❣ ◆✳ ❚✳ ✱ ❖♥ t❤❡ ❧❡❛st ❞❡❣r❡❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ❜♦✉♥❞✐♥❣ ❛❜♦✈❡ t❤❡ ❞✐❢❢❡r❡♥❝❡s ❜❡t✇❡❡♥ ❧❡♥❣t❤s ❛♥❞ ♠✉❧t✐♣❧✐❝✐t✐❡s ♦❢ ❝❡rt❛✐♥ s②st❡♠ ♦❢ ♣❛r❛♠❡t❡rs ✐♥ ❧♦❝❛❧ r✐♥❣s✱ ❬❈✶❪ ❈✉♦♥❣ ◆✳ ❚✳ ✱ ◆❛❣♦②❛ ▼❛t❤ ❏✳✱ ✶✷✺ ✭✶✾✾✷✮✱ ✶✵✺✲✶✶✹✳ ❖♥ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ♥♦♥✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧♦❝✉s ♦❢ ❧♦❝❛❧ r✐♥❣s ❛❞♠✐tt✐♥❣ ❞✉❛❧✐③✐♥❣ ❈♦♠♣❧❡①❡s✱ ✶✵✾ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜✳ P❤✐❧✳ ❙♦❝✳ ✭✶✾✾✶✮✱ ✹✼✾✲✹✽✽✳ ❬❈▼◆❪ ❈✉♦♥❣ ◆✳ ❚✳ ✱ ▼❛r❝❡❧ ▼♦r❛❧❡s ❛♥❞ ▲✳ ❚✳ ◆❤❛♥✱ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✱ t❤❡ ❧❡♥❣t❤ ♦❢ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛✱ ✷✻✺ ✭✷✵✵✸✮ ✶✵✵➊✶✶✸✳ ❬❈◆❪ ❈✉♦♥❣ ◆✳ ❚✳ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥✱ ♠♦❞✉❧❡s✱ ❖♥ ❖♥ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥s✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✸✵ ✭✷✵✵✷✮✱ ✶✷✶✲✶✸✵✳ ✸✽ ❬❈◆◆❪ ❈✉♦♥❣ ◆✳ ❚✳✱ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ◆✳ ❚✳ ❑✳ ◆❣❛✱ ❖♥ ♣s❡✉❞♦ s✉♣♣♦rts ❛♥❞ ♥♦♥✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧♦❝✉s ♦❢ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠♦❞✉❧❡s✱ ✸✷✸ ❏✳ ❆❧❣❡❜r❛✱ ✭✷✵✶✵✮✱ ✸✵✷✾✲✸✵✸✽✳ ❬❈❙❚❪ ❈✉♦♥❣ ◆✳ ❚✳ ✱ P✳ ❙❝❤❡♥③❡❧✱ ◆✳ ❱✳ ❚r✉♥❣✱ ▼❛❝❛✉❧❛② ♠♦❞✉❧♥ ❬❉❪ ◆✳ ❚✳ ❉✉♥❣✱ ❞✐♠❡♥s✐♦♥ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ❈♦❤❡♥✲ ▼❛t❤✳ ◆❛❝❤r✱ ✽✺ ✭✶✾✼✽✮ ✺✺✲✼✸✳ ❙♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ✐♥ > s✱❇✉❧❧✳ ❑♦r❡❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✭✺✶✮ ✭✷✵✶✹✮✱ ◆♦ ✷✱ ♣♣ ✺✶✾✲✺✸✵✳ ❬●❪ ❉➢➡♥❣ ❚❤Þ ●✐❛♥❣✱ ▼➠ ➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝❤✐Ị✉ > s ✱▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❑✶✾✱ ✭✹✵✮ ✭✷✵✶✸✮✳ ❬❋❘❪ ❋❡rr❛♥❞ ❉✳ ❛♥❞ ▼✳ ❘❛②♥❛✉❞✱ ◆♦❡t❤❡r✐❛♥✱ ❬❍❪ ❍❡❧❧✉s ▼✳✱ ♠♦❞✉❧❡s✱ ❋✐❜r❡s ❢♦r♠❡❧❧❡s ❞✬✉♥ ❛♥♥❡❛✉ ❧♦❝❛❧ ❆♥♥✳ ❙❝✐✳ ❊✬❝♦❧❡ ◆♦r♠✳ ❙✉♣✳✱ ✭✹✮ ✸ ✭✶✾✼✵✮✱ ✷✾✺✲✸✶✶✳ ❖♥ t❤❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❛ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ❏✳ ❆❧❣❡❜r❛ ✷✸✼ ✭✷✵✵✶✮✱ ✹✵✻➊✹✶✾ ❬❑❲❪ ❑❛✇❛s❛❦✐ ❚✳ ✱ ✱ ❖♥ ❛r✐t❤♠❡t✐❝ ▼❛❝❛✉❧❛②❢✐❝❛t✐♦♥ ♦❢ ◆♦❡t❤❡r✐❛♥ r✐♥❣s ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✉♠❡ ✸✺✹✱ ◆✉♠✲ ❜❡r ✶ ✭✷✵✵✶✮✱ ✶✷✸✲✶✹✾✳ ❬❑❪ ❑✐r❜② ❉✳✱ ✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❉✐♠❡♥s✐♦♥ ❛♥❞ ❧❡♥❣t❤ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ❖①❢♦r❞✱ ✭✷✮✱ ✹✶ ✭✶✾✾✵✮✱ ✹✶✾✲✹✷✾✳ ❬▼❛❝❪ ▼❛❝❞♦♥❛❧❞ ■✳ ●✳✱ ♠✉t❛t✐✈❡ r✐♥❣✱ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠✲ ❙②♠♣♦s✳ ▼❛t❤✳✱ ✶✶ ✭✶✾✼✸✮✱ ✷✸✲✹✸✳ ❬▼❛t❪ ▼❛ts✉♠✉r❛ ❍✳✱ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣ ✱ ❈❛♠❜r✐❞❣❡✿ ❈❛♠❜r✐❞❣❡ ❚❤❡♦r② ❯♥✐✈❡rs✐t② Pr❡ss✱ ✭✶✾✽✻✮✳ ❬◆❉❪ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ◆✳ ❚✳ ❉✉♥❣✱ ✧❆ ❋✐♥✐t❡♥❡ss ❘❡s✉❧t ❢♦r ❆tt❛❝❤❡❞ Pr✐♠❡s ♦❢ ❈❡rt❛✐♥ ❚♦r✲♠♦❞✉❧❡s✧✱ ❆❧❣❡❜r❛ ❈♦❧❧♦q✉✐✉♠ ✱ ✼✽✼✲✼✾✻✳ ✸✾ ✶✾✱ ✭❙♣❡❝ ✶✮✱ ✭✷✵✶✷✮ ❬◆▼❪ ◆❤❛♥ ▲✳ ❚✳ ❛♥❞ ▼✳ ▼♦r❛❧❡s✱ ●❡♥❡r❛❧✐③❡❞ ❢✲♠♦❞✉❧❡s ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡ ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✸✹ ✭✷✵✵✻✮✱ ✽✻✸✲✽✼✽✳ ❬◆❤❪ ◆❤❛♥ ▲✳ ❚✳✱ ❛ss♦❝✐❛t❡❞ ❖♥ ❣❡♥❡r❛❧✐③❡❞ r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ❢♦r ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ♠♦❞✉❧❡s ❆❧❣❡❜r❛✱ ✸✸ ✭✷✵✵✺✮✱ ✼✾✸✲✽✵✻✳ ❬❘❪ ❘♦❜❡rts ❘✳ ◆✳✱ ❑r✉❧❧ ❞✐♠❡♥s✐♦♥ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦✈❡r q✉❛s✐✲❧♦❝❛❧ ✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞✱ ✷✻ ✭✶✾✼✺✮✱ ✷✻✾✲✷✼✸✳ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s ❬❙❤❪ ❙❤❛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❞ ❛ t❤❡♦r② ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ ◆❛❣♦②❛ ▼❛t❤ ❏✳✱ ✶✵✷ ✭✶✾✽✻✮✱ ✶✲✹✾✳ ❬❩❪ ❩❛♠❛♥✐ ◆✳✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼♦❞✉❧❡s ✐♥ ❉✐♠❡♥s✐♦♥ > s ❛♥❞ ❘❡s✉❧ts ♦♥ ✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✸✼✱ ✭✷✵✵✾✮✱ ✶✷✾✼✲✶✸✵✼ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣② ✹✵ ... ❑♦s③✉❧ H• ✭①❀ ▼✮ ✭①❡♠ ❬❇❍❪✮✳ ❉♦ ➤ã✱ ➤Ĩ t✐Ư♥ ❤➡♥ tr♦♥❣ ✈✐Ö❝ tÝ♥❤ t♦➳♥ tr➟♥ sè ❜é✐✱ t❛ ♥❤➽❝ ❧➵✐ ệ số ộ ì tứ ợ tệ ❜ë✐ ◆♦rt❤❝♦tt✳ ❱➱♥ ❣✐➯ t❤✐Õt ❤÷✉ ❤➵♥ s✐♥❤✳ ▼ét ❤Ư ❝➳❝ ♣❤➬♥ tư M ♥Õ✉ ❧➭ ✈➭♥❤ ➤Þ❛... ➤➷t N-dimR A = −1 ❱í✐ A = 0, N-dimR A < d ❝♦♥ ❝đ❛ A, ❝❤♦ ♠ét sè ♥❣✉②➟♥ d ≥ 0, ❧➭ s❛✐ ỗ t tồ t số n0 s❛♦ ❝❤♦ t❛ ➤➷t N-dimR A = d A0 ⊆ A1 ⊆ ♥Õ✉ ❝➳❝ ♠➠➤✉♥ N-dimR (An+1 /An ) < d, ✈í✐ ♠ä✐ n... ✭✐✮ M ●✐➯ sö dim M = d > s✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭✐✐✮ ỗ ệ t số ề x1 , , xr ❝ñ❛ > s M ✈➭ ỗ tố p (Ass(M/(x1 , , xr )M ))>s , t❛ ❝ã dim R/p = d r ỗ