F môđun suy rộng và tập iđêan nguyên tố liên kết của môđun đối đồng điều địa phương

ĐẠI HỌC THÁI NGUYÊN ĐẠI HỌC SƯ PHẠM ĐOÀN THỊ THU THẢO F-MÔĐUN SUY RỘNG VÀ TẬP IĐÊAN NGUYÊN TỐ LIÊN KẾT CỦA MÔĐUN ĐỐI ĐỒNG ĐIỀU ĐỊA PHƯƠNG CHUYÊN NGÀNH: ĐẠI SỐ VÀ LÝ THUYẾT SỐ ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ❝➳❝ ❦Õt q✉➯ ♥❣❤✐➟♥ ứ ợ trì tr t♦➭♥ tr✉♥❣ t❤ù❝✱ ❝❤➢❛ ➤➢ỵ❝ sư ❞ơ♥❣ ❝❤♦ ❜➯♦ ✈Ư ♠ét ❤ä❝ ✈Þ ♥➭♦✳ ◆❣✉å♥ t➭✐ ❧✐Ư✉ sư ❞ơ♥❣ ❝❤♦ ✈✐Ư❝ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ➤➲ ➤➢ỵ❝ sù ➤å♥❣ ý ❝đ❛ ❝➳❝ ❝➳ ♥❤➞♥ ✈➭ tỉ ❝❤ø❝✳ ❈➳❝ t❤➠♥❣ t✐♥✱ t ệ trì tr ợ ❣❤✐ râ ♥❣✉å♥ ❣è❝✳ ❚❤➳✐ ◆❣✉②➟♥✱ t❤➳♥❣ ✹ ♥➝♠ ✷✵✶✸ ❍ä❝ ✈✐➟♥ ➜♦➭♥ ❚❤Þ ❚❤✉ ❚❤➯♦ ❳➳❝ ♥❤❐♥ ❝đ❛ tr➢ë♥❣ ❦❤♦❛ ❝❤✉②➟♥ ♠➠♥ ❳➳❝ ♥❤❐♥ ❝đ❛ ♥❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ◆❣✉②Ơ♥ ❚❤Þ ❉✉♥❣ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ ❚✳❙ ễ ị trự tế ỉ ì ❞➽t✱ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥ ✈➭ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ t➠✐ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❚➠✐ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ ❇❛♥ ❣✐➳♠ ❤✐Ö✉ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❚❤➳✐ ◆❣✉②➟♥✱ P❤ß♥❣ ➜➭♦ t➵♦ s❛✉ ➜➵✐ ❤ä❝✱ ❝➳❝ t❤➬② ❣✐➳♦ ❱✐Ö♥ t♦➳♥ ❤ä❝ ❍➭ ◆é✐ ✈➭ ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ ❑❤♦❛ ❚♦➳♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❚❤➳✐ ◆❣✉②➟♥ ➤➲ ❣✐➯♥❣ ❞➵②✱ ❣✐ó♣ ➤ì ❝❤♦ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ♥➭②✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ ➤Õ♥ ❣✐❛ ì t tt ữ ❣✐ó♣ ➤ì✱ ➤é♥❣ ✈✐➟♥ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ✐✐✐ ▼ô❝ ❧ô❝ ❚r❛♥❣ ▲ê✐ ❝❛♠ ➤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ▲ê✐ ❝➯♠ ➡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▼ë ➤➬✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ❍Ö t❤❛♠ sè ✈➭ sè ❜é✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸✳ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹✳ ❱Ò ♠ét sè ❞➲② ❝❤Ý♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❈❤➢➡♥❣ ✷✳ F ✲♠➠➤✉♥ s✉② ré♥❣ ✷✳✶✳❚Ý♥❤ ❝❤✃t ❝ñ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ f ✲♠➠➤✉♥ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷✳ ➜➷❝ tr➢♥❣ ❝ñ❛ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤➠♥❣ q✉❛ sè ❜é✐ ✈➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✸✻ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✶ ▼ë ➤➬✉ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈➭ M ✈í✐ ❝❤✐Ị✉ ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ dim M = d✳ ◆✳ ❚✳ ❈➢ê♥❣✱ ◆✳ ❱✳ ❚r✉♥❣ ✈➭ P✳ ❙❝❤❡♥③❡❧ ❬❈❙❚❪ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ✭f ✲❞➲②✮ ♥❤➢ ❧➭ ♠ë ré♥❣ ❝ñ❛ ❞➲② ❝❤Ý♥❤ q✉② q✉❡♥ ❜✐Õt✱ ✈➭ ➤å♥❣ t❤ê✐ ❤ä ❝ò♥❣ ➤➢❛ r❛ ❧í♣ ♠➠➤✉♥ t❤á❛ ♠➲♥ ♠ä✐ ❤Ư t❤❛♠ sè ➤Ị✉ ❧➭ ❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ t❤✐Ư✉ ♠ét ❧í♣ ♠➠➤✉♥ t❤á❛ ♠➲♥ f ✲♠➠➤✉♥✳ ❈ị♥❣ tr♦♥❣ ❜➭✐ ❜➳♦ ➤ã✱ ❤ä l(HIi (M )) < ∞, ✈í✐ ♠ä✐ i < d ợ ọ s rộ ì ❝❤✉♥❣✱ ♠ä✐ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② s✉② ré♥❣ ➤Ò✉ ❧➭ f ✲♠➠➤✉♥ ✈➭ ➤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ❝ị♥❣ ➤ó♥❣ ❦❤✐ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈✃✉ tró❝ ❝đ❛ f ✲♠➠➤✉♥ ❧➭ R ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ➤➲ ➤➢ỵ❝ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ♥❣➭② ♥❛② ❝➳❝ ❧í♣ ♠➠➤✉♥ ♥➭② ➤➲ trë ♥➟♥ q✉❡♥ t❤✉é❝ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❍×♥❤ ❤ä❝ ➤➵✐ sè✳ ❚✐Õ♣ t❤❡♦✱ ♥➝♠ ✷✵✵✺✱ ý t➢ë♥❣ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ f ✲❞➲② t❤✉é❝ ✈Ò ▲✳ ❚✳ ◆❤➭♥ ❬◆❪✿ ▼ét ❞➲② ❝➳❝ ♣❤➬♥ tö x1 , , xr tr♦♥❣ m ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ❝ñ❛ t❤á❛ ♠➲♥ ❝❤♦ M ♥Õ✉ dim R/p > 1✱ dim M/IM > ❦ý ❤✐Ö✉ ❧➭ xi ∈ / p, ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ p ∈ AssR M/(x1 , , xi−1 )M i = 1, , r ❈❤♦ I ❧➭ ✐➤➟❛♥ ❝ñ❛ ❑❤✐ ➤ã✱ ❦❤➳✐ ♥✐Ư♠ ➤é s➞✉ s✉② ré♥❣ ❝đ❛ M R s tr I gdepth(I; M ) ũ ợ ị ♥❣❤Ü❛ ♠ét ❝➳❝❤ tù ♥❤✐➟♥ ❧➭ ➤é ❞➭✐ ❝ù❝ ➤➵✐ ❝ñ❛ ♠ét ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ❝ñ❛ M tr♦♥❣ I✳ ❉➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ✈➭ ➤é s➞✉ s✉② ré♥❣ ✈➱♥ ❝ß♥ ❝ã ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t ➤Đ♣ ✈➭ ❝✉♥❣ ❝✃♣ ♠ét sè t❤➠♥❣ t✐♥ ❤÷✉ Ý❝❤ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝ñ❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✳ ❈❤➻♥❣ ❤➵♥✱ t❐♣ t1 , ,tn ∈N Ass(M/(xt11 , , xtnn )M ) ữ ỗ x1 , , xn ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ❝đ❛ tr♦♥❣ I ❧➭ M✳ ❍➡♥ t❤Õ ♥÷❛✱ ♥Õ✉ ➤é s➞✉ s✉② ré♥❣ ❝ñ❛ ❧➭ M gdepth(I, M ) = r t❤× r ❝❤Ý♥❤ ❧➭ sè ♥❣✉②➟♥ i ♥❤á ♥❤✃t s❛♦ ❝❤♦ t❐♣ Supp(HIi (M )) ❧➭ ✈➠ ❤➵♥✱ ✈➭ t❐♣ Ass(HIr (M )) ❧➭ ❤÷✉ ❤➵♥ ✭①❡♠ ❬◆❪✮✳ ▼ét ❝➳❝❤ tù ♥❤✐➟♥✱ tõ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣✱ ▲✳ ❚✳ ◆❤➭♥ ✈➭ ✷ ▼✳ ▼♦r❛❧❡s ❬◆▼❪ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ❧í♣ ♠➠➤✉♥ ❣ä✐ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ♠ä✐ ❤Ư t❤❛♠ sè ❧➭ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣✳ ❍ä ➤➲ ❝❤ø♥❣ tá r➺♥❣ f ✲♠➠➤✉♥ ❝❤✃t ❝đ❛ s✉② ré♥❣ ✈➱♥ ❝ã ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t tèt t➢➡♥❣ tù ✈í✐ ♠ét sè tÝ♥❤ f ✲♠➠➤✉♥ ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr×♥❤ ❜➭② ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❧➵✐ ❝❤✐ t✐Õt ❜➭✐ ❜➳♦ ✧●❡♥❡r❛❧✐③❡❞ ❋✲♠♦❞✉❧❡s ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧ ❝ñ❛ ▲✳ ❚✳ ◆❤➭♥ ✈➭ ▼✳ ▼♦r❛❧❡s ➤➝♥❣ tr➟♥ t➵♣ ❝❤Ý ❈♦♠♠✉♥✐❝❛t✐♦♥ ✐♥ ❆❧❣❡❜r❛✱ ♥➝♠ ✷✵✵✻✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❤❛✐ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ❞➭♥❤ ➤Ĩ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥❤➢ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ❤Ö t❤❛♠ sè✱ sè ❜é✐✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱✳✳✳ ➜Ĩ t❤❡♦ ❞â✐ ♠ét ❝➳❝❤ t➢➡♥❣ ➤è✐ ❤Ư t❤è♥❣✱ ▼ơ❝ 1.4 ❝đ❛ ❈❤➢➡♥❣ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❝❤Ý♥❤ q✉②✱ ❞➲② ❝❤Ý♥❤ q✉② ❧ä❝✱ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ✈➭ t➢➡♥❣ ø♥❣ ❧➭ ❝➳❝ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ f ✲♠➠➤✉♥ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➺♠ ë ❈❤➢➡♥❣ ré♥❣❀ ➤➷❝ tr➢♥❣ ❝đ❛ f ✲♠➠➤✉♥ 2✿ ❑❤➳✐ ♥✐Ư♠ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤➠♥❣ q✉❛ ❤Ư t❤❛♠ sè ❝đ❛ M, s✉② ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ✈➭ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ➤➻♥❣ ❝❤✐Ị✉ tí✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝ã ❝❤✐Ị✉ > ❝ñ❛ t❐♣ s✉♣♣♦rt ❝ñ❛ ♣❤➢➡♥❣❀ ◆Õ✉ ✈➭♥❤ R M; sè ❜é✐ ✈➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉ t❤× ❧í♣ f ✲♠➠➤✉♥ s✉② ré♥❣ ❝❤Ý♥❤ ❧➭ ❧í♣ ♠➠➤✉♥ ❝ã q✉ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝ã ❝❤✐Ị✉ ❧í♥ ♥❤✃t ❧➭ ❝➯ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ➤Ị✉ ❝ã ❤♦➷❝ ❝❤✐Ò✉ ✈➭ t✃t d ❤♦➷❝ ❝❤✐Ò✉ 1❀ ❚Ý♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❑Õt q✉➯ ♥➭② ❧➭ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❍❡❧❧✉s ❬❍✱ ➜Þ♥❤ ❧ý ✹❪ ✈➭ ❆s❛❞♦❧❧❛❤✐✲❙❝❤❡♥③❡❧ ❬❆❙✱ ➜Þ♥❤ ❧ý ✶✳✶❪✳ P❤➬♥ ❦Õt ❧✉❐♥ ❝đ❛ ❧✉❐♥ ✈➝♥ tỉ♥❣ ❦Õt ❧➵✐ ❝➳❝ ❦Õt q✉➯ ➤➲ tr×♥❤ ❜➭②✳ ✸ ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ t❛ ❧✉➠♥ ❦Ý ❤✐Ö✉ ❧➭ R✲♠➠➤✉♥✳ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r ✈➭ M ❈❤➢➡♥❣ ♥➭② ❞➭♥❤ ➤Ó ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝➳❝ ❦Õt q✉➯ ❝ñ❛ ❧✉❐♥ ✈➝♥ ë ❝➳❝ ❝❤➢➡♥❣ s❛✉ ♥❤➢ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ❤Ö t❤❛♠ sè✱ sè ❜é✐✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱✳✳✳ ✶✳✶ ❚❐♣ ✐➤➟❛♥ ♥❣✉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 M ♥Õ✉ s❛♦ ❝❤♦ an (M/Q) = AnnR (M/Q) ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ R, t❛ ♥ã✐ Q ❧➭ ♠ét ♠➠➤✉♥ ❝♦♥ p✲♥❣✉②➟♥ s➡ ❝ñ❛ M ❧➭ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ R ♠➠➤✉♥ M s➡ ♥Õ✉ tå♥ t➵✐ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ♥❣✉②➟♥ s➡ N = Q1 ∩ ∩ Qn s➡✳ ◆Õ✉ =x∈M t❛ ♥ã✐ Qi ✈í✐ N ❝ã ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ i = 1, , n, t❤➭♥❤ ❣✐❛♦ ❝đ❛ ❤÷✉ ❤➵♥ ❝➳❝ ♠➠➤✉♥ ❝♦♥ s❛♦ ❝❤♦ pi ✲♥❣✉②➟♥ N = ❤♦➷❝ N = ❝ã ♠ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤× t❛ ♥ã✐ N ❧➭ ♣❤➞♥ tÝ❝❤ ➤➢ỵ❝✳ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ tè✐ t❤✐Ó✉ ✭t❤✉ ❣ä♥✮ ♥Õ✉ ❝➳❝ ✹ ✐➤➟❛♥ ♥❣✉②➟♥ tè pi ❧➭ ➤➠✐ ♠ét ❦❤➳❝ ♥❤❛✉ ✈➭ ❦❤➠♥❣ ❝ã ❤➵♥❣ tư ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ Qi ♥➭♦ ❧➭ t❤õ❛✱ i = 1, , n n Qj ⊆ Qi i=1;i=j ✭✐✈✮ ❉Ô t❤✃② r➺♥❣ ♠ä✐ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝ñ❛ ❞➵♥❣ 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➡ ❝ñ❛ Qi N ❈➳❝ ❤➵♥❣ tö N ✳ ◆Õ✉ pi Qi , i = 1, , n✱ ❧➭ tè✐ t❤✐Ĩ✉ tr♦♥❣ ➤➢ỵ❝ ọ t ợ tì Qi ợ ọ tì AssR M/N ợ ọ t ú ột ỗ (xn ) R ợ ọ ❧➭ ♠ét ❞➲② ❈❛✉❝❤② t❤❡♦ t➠♣➠ m✲❛❞✐❝ ♥Õ✉ ✈í✐ k ∈ N ❝❤♦ tr➢í❝✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ n0 n, m ≥ n0 ❉➲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➠♣➠ m✲❛❞✐❝ tr➟♥ ✈➭♥❤ R✳ ▼➠➤✉♥ ♥➭② ➤➢ỵ❝ ❦Ý ✺ ❤✐Ư✉ ❧➭ M ▼Ư♥❤ ➤Ị s❛✉ ❝❤♦ t❛ ♠ét sè tÝ♥❤ ❝❤✃t ❝ñ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ✭①❡♠ ❬▼❆❚✱ ➜Þ♥❤ ❧ý ✻✳✶✱ ➜Þ♥❤ ❧ý ✻✳✸✱ ➜Þ♥❤ ❧ý ✻✳✺❪✮✳ ❚❛ ❝ã ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉✿ ▼Ư♥❤ ➤Ị ✶✳✶✳✷✳ ✭✐✮ ■➤➟❛♥ p ❧➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠ét ♠➠➤✉♥ ❝♦♥ ➤➻♥❣ ❝✃✉ ✈í✐ M ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ M ❝❤ø❛ R/p ✭✐✐✮ ❈❤♦ p ❧➭ ♣❤➬♥ tö tè✐ ➤➵✐ ❝ñ❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ❝ã ❞➵♥❣ ➤ã Ann(x), tr♦♥❣ = x ∈ M ❑❤✐ ➤ã p ∈ AssR (M ) ❱× t❤Õ✱ M = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ AssR (M ) = ∅ ❍➡♥ ♥÷❛✱ t❐♣ ZD(M ) ❝➳❝ ➢í❝ ❝đ❛ ❦❤➠♥❣ ❝đ❛ M ❝❤Ý♥❤ ❧➭ ❤ỵ♣ ❝đ❛ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ✭✐✐✐✮ ❈❤♦ ❞➲② ❦❤í♣ ♥❣➽♥ ❝➳❝ M R✲♠➠➤✉♥ −→ M −→ M −→ M −→ ❑❤✐ ➤ã Ass M ⊆ Ass M ⊆ AssR M ∪ Ass M ế M R ữ s tì ➤ã t❛ ❝ã Ass M ❧➭ t❐♣ ❤÷✉ ❤➵♥✱ Ass M ⊆ Supp M ✈➭ V (Ann M ) = SuppR M ❍➡♥ ♥÷❛✱ ❝➳❝ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ ✭✈✮ Ass M ✈➭ Supp M ❧➭ ♥❤➢ ♥❤❛✉✳ AssRp (Mp ) = {qRp : q ∈ AssR (M ), q ⊆ p} ✭✈✐✮ ✭✈✐✐✮ ✶✳✷ AssR M = {p ∩ R : p ∈ AssR M } AssR M = p∈AssR M AssR M /pM ❍Ö t❤❛♠ sè ✈➭ sè ❜é✐ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ◆♦❡t❤❡r ✈➭ M ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ dim M = d, R ữ s ị ♥❣❤Ü❛ ✶✳✷✳✶✳ sè ❝ñ❛ M ✭✐✐✮ ◆Õ✉ ♥Õ✉ x∈m ✭✐✮ ▼ét ❤Ö x := (x1 , , xd ) ∈ m ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❤Ư t❤❛♠ (M/(x)M ) < ∞ ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ ❣ä✐ ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè✱ ✈í✐ ♠ä✐ M t❤× ❝➳❝ ♣❤➬♥ tư (x1 , , xi ) ➤➢ỵ❝ i = 1, , d ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ❝❤♦ t❛ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❤Ư t❤❛♠ sè✱ ✭①❡♠ ❬▼❆❚✱ ➜Þ♥❤ ❧ý ✶✹✳✶✱ ➜Þ♥❤ ❧ý ✶✹✳✷❪✮✳ ▼Ư♥❤ ➤Ị ✶✳✷✳✷✳ ♠ét ❜é ❣å♠ sè ❝đ❛ ✭✐✮ ◆Õ✉ x ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝ñ❛ M ✈➭ n = (n1 , , nd ) ❧➭ d sè ♥❣✉②➟♥ ❞➢➡♥❣ t❤× x(n) = (xn1 , , xnd d ) ❝ị♥❣ ❧➭ ❤Ư t❤❛♠ M✳ ✭✐✐✮ ❈❤♦ x1 , , xt ❧➭ ♠ét ❞➲② ❝➳❝ ♣❤➬♥ tư ❝đ❛ m, ✈í✐ t dim M − t dim(M/(x1 , , xt )M ) ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✭✐✐✐✮ ❍Ö ∈ Ass M/(x1 , , xi−1 )M t❤á❛ ♠➲♥ dim R/p = d − i + ➜➷❝ ❜✐Ưt✱ ♠ét ♣❤➬♥ tư x ∈ m ❧➭ ♣❤➬♥ tư t❤❛♠ sè ❝ñ❛ M ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x ∈ / p, ✈í✐ ∈ Ass M s❛♦ ❝❤♦ dim R/p = d ✭✐✈✮ ◆Õ✉ ➤ã x1 , , xt ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ M (x1 , , xd ) ∈ m ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ xi ∈ / p, ✈í✐ ♠ä✐ p ♠ä✐ p d✳ ❑❤✐ ➤ã✱ x ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ M t❤× x ❝ị♥❣ ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ M , tr♦♥❣ M ❧➭ t➠♣➠ ➤➬② ➤ñ m✲❛❞✐❝ ❝ñ❛ M ị ĩ ữ s ó I ❧➭ ✐➤➟❛♥ R (M/I n+1 m✲♥❣✉②➟♥ s➡ ❝ñ❛ R M ❧➭ R✲♠➠➤✉♥ M ) = PM,I (n) ✈í✐ n ➤đ ❧í♥✱ tr♦♥❣ ➤ã deg PM,I (n) = d ❑❤✐ ➤ã tå♥ t➵✐ ❝➳❝ sè ♥❣✉②➟♥ e0 , e1 , , ed , e0 > s❛♦ ❝❤♦ PM,I (n) = e0 n+d n+d−1 + + (−1)d ed − e1 d d−1 ❈➳❝ sè e0 , , ed ❣ä✐ ❧➭ ❤Ö sè ❍✐❧❜❡rt ❝đ❛ M ➤è✐ ✈í✐ I ❦Ý ❤✐Ư✉ ❧➭ ei (I, M ) ➜➷❝ ❜✐Öt✱ sè ♥❣✉②➟♥ ❞➢➡♥❣ e0 tr♦♥❣ ❜✐Ĩ✉ ❞✐Ơ♥ tr➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ sè ❜é✐ ❝đ❛ ➤è✐ ✈í✐ I ❑Ý ❤✐Ư✉ ❧➭ e(I, M ) M ✸✵ ✭✐✐✐✮✳ ●✐➯ sö r➺♥❣ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❚❛ ➤✐ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ✭❛✮✳ ❚❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✷ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ✈í✐ ♠ä✐ ❚❤❐t ✈❐②✱ ❝❤♦ i < d ❝❤♦ k > 1✳ x ∈ p✳ ♥❤➢♥❣ ✈× R ✈➭ ❱× 1, p ∈ AttR (Hmi (M )) ❑❤✐ ➤ã i < d t❤❡♦ ❬❇❙✱ ✶✶✳✸✳✺❪ ✈➭ p ∈ AssR M dim(R/p) := k ●✐➯ sö i < d dim(R/ AnnR (Hmiˆ (M ))) t❤❡♦ ❬❇❙✱ ✶✶✳✸✳✸❪✳ k < d ♥➟♥ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè (x1 , , xd ) ❝đ❛ M ❱× t❤Õ (x1 , , xd ) ❦❤➠♥❣ ❧➭ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ❝ñ❛ ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥➟♥ M ❧➭ s❛♦ M✱ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✻✱ ✭✐✐✮✱ ➤✐Ò✉ ♥➭② s✉② r❛ ♠➞✉ t❤✉➱♥✳ ❱❐②✱ k 1✳ ❉♦ ➤ã dimR (R/ AnnR (Hmi (M ))) = ❙✉② r❛ dim(R/ AnnR (Hmiˆ (M ))) max ▼❛❝❛✉❧❛② ❝ñ❛ M ◆❈✭▼✮ Spec R M✳ f ✲♠➠➤✉♥ s✉② ré♥❣ q✉❛ t❐♣ q✉ü ◆❤➽❝ ❧➵✐ r➺♥❣ q✉ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲ ❧➭ = {p ∈ Spec R : Mp ❈❤ó ý r➺♥❣ ♥Õ✉ tr♦♥❣ 1, ✈í✐ ♠ä✐ i < d ❚õ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ t❛ ❝ã ➤➷❝ tr➢♥❣ ❝đ❛ tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝ñ❛ dim(R/p) i (M ) p∈AttR Hm R ❧➭ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛②} ❝❤ø❛ ♣❤ø❝ ➤è✐ ♥❣➱✉✱ t❤× ◆❈✭▼✮ ❧➭ t❐♣ ❝♦♥ ➤ã♥❣ ✈í✐ t➠♣➠ ❩❛r✐s❦✐✳ ❱× ✈❐② dim(N C(M )) ợ ị t a(M ) = a0 ad−1 (M ), tr♦♥❣ ➤ã (M ) = AnnR (Hmi (M )), ✈í✐ ♠ä✐ i d − ❍Ư q✉➯ ✷✳✷✳✻✳ ●✐➯ sư r➺♥❣ R ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉✳ ❑❤✐ ➤ã ❝➳❝ ♣❤➳t ❜✐Ó✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ ✭✐✐✮ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ dim R/a(M ) ✭✐✐✐✮ ❚å♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè ✈➭ ❤➺♥❣ sè ♥❣✉②➟♥ x = (x1 , , xd ) ❝ñ❛ M, sè ♥❣✉②➟♥ k d Cx , Dx s❛♦ ❝❤♦ I(xn1 , , xnd d ; M ) = nk Cx + Dx , ✈í✐ ♠ä✐ sè n1 , , nd ✸✶ ✭✐✈✮ ✈➭ dim R/p = d, ✈í✐ ♠ä✐ p ∈ min(Supp M ) t❤á❛ dim N C(M ) ♠➲♥ dim R/p = ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ⇒ ✭✐✐✮✳ ❚❛ ❝ã p(M ) t❤❡♦ ị í ữ t ứ tr♦♥❣ ◆✳ ❚✳ ❈➢ê♥❣ ❬❈✷❪ t❛ ❝ã dim R/a(M ) = p(M ) ❉♦ ➤ã t❛ ❝ã ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✭✐✐✮ ✭✐✐✐✮✳ ❚å♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè ⇒ (x1 , , xd ) ❝ñ❛ M t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t xi ∈ a(M/(xi+1 , , xd )M ), ✈í✐ ♠ä✐ i = 1, , d ❚❤❡♦ ❬❈✷❪✱ ♠ét ❤Ö t❤❛♠ sè ♥❤➢ ✈❐② ➤➢ỵ❝ ❣ä✐ ❧➭ sè p✲❝❤✉➮♥ t➽❝ ❝đ❛ ✭✐✐✐✮ ⇒ Cx , Dx M p✲❝❤✉➮♥ t➽❝ ❝ñ❛ M, ✈➭ ❦❤✐ p(M ) ♠ä✐ ❤Ư t❤❛♠ ➤Ị✉ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮✳ ✭✐✮ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ ✈× I(xn1 , , xnd d ; M ) = nk Cx + Dx ✱ ❧➭ ❝➳❝ ❤➺♥❣ sè ♥➟♥ t❛ ❝ã p(M ) tr♦♥❣ ➤ã ❑❤✐ ➤ã ❦Õt q✉➯ s✉② r❛ tõ ➜Þ♥❤ ❧Ý ✷✳✷✳✺✱ ✭✐✐✮✳ ✭✐✮ ⇔ ✭✐✈✮ ❱× ❍Ư q✉➯ ✷✳✶✳✺✱ R ❧➭ ✈➭♥❤ ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉ ♥➟♥ M ❧➭ f ✲♠➠➤✉♥ Supp M ❧➭ ❝❛t❡♥❛r②✳ ❚❤❡♦ s✉② ré♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ t✃t ❝➯ ❝➳❝ ✐➤➟❛♥ p ∈ Supp M s❛♦ ❝❤♦ Mp ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤Ò✉ ❝ã ❝❤✐Ò✉ dim R/p > ❉♦ ➤ã dim(N C(M )) ❈❤ó ý r➺♥❣ tå♥ t➵✐ ❧➭ R ♠➠➤✉♥ M ✈í✐ dim(N C(M )) 1, ♥❤➢♥❣ M ❦❤➠♥❣ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❱Ý ❞ô ✷✳✷✳✼✳ tr➟♥ tr➢ê♥❣ ❈❤♦ R = K[[x, y, z, t, s]] ỗ ũ từ ì tứ K ➜➷t M = R/(x, y, z) ⊕ R/(t, s) ❑❤✐ ➤ã✱ Ass M = {p = (x, y, z)R, q = (t, s)R}, min(Supp M ) = {q, p} ✈➭ dim M = dim R/q = 3, dim R/p = < dim M ➜Þ♥❤ ❧Ý ✷✳✶✳✹✱ M dim N C(M ) = ❦❤➠♥❣ ❧➭ f ✲♠➠➤✉♥ ❱× t❤Õ✱ t❤❡♦ s✉② ré♥❣✳ ❚✉② ♥❤✐➟♥✱ râ r➭♥❣ r➺♥❣ ✸✷ ❱Ý ❞ơ s❛✉ ➤➞② ✈Ị t❤✐Õt R f ✲✈➭♥❤ s✉② ré♥❣ ❦❤➠♥❣ ❝❛t❡♥❛r② ❝❤♦ t❛ t❤✃② r➺♥❣ ❣✐➯ ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② tr♦♥❣ ▼Ư♥❤ ➤Ị ✷✳✶✳✻ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✺ ❧➭ t❤ù❝ sù ❝➬♥ t❤✐Õt✳ ❱Ý ❞ô ✷✳✷✳✽✳ ✭✐✮ ❚å♥ t➵✐ ♠✐Ị♥ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ (R, m) s❛♦ ❝❤♦ dim R = ✈➭ R ❧➭ ❦❤➠♥❣ ❝❛t❡♥❛r②✳ ✭✐✐✮ R ❧➭ f ✲✈➭♥❤ s✉② ré♥❣✱ ♥❤➢♥❣ R ❦❤➠♥❣ ❧➭ f ✲✈➭♥❤ s✉② ré♥❣✳ ✭✐✐✐✮ N-dimR (Hm2 (R)) = 3, dim R/ AnnR (Hm2 (R)) = ✈➭ p(R) = ✭✐✈✮ dim R/a(R) = ✈➭ dim R/a(R) = 2✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ S ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ◆♦❡t❤❡r ❝❤Ý♥❤ q✉② ❤♦➭♥ ❤➯♦ ❝❤✐Ị✉ ➤➢ỵ❝ ①➞② ❞ù♥❣ ❜ë✐ ❇r♦❞♠❛♥♥ ❬❇❪ s❛♦ ❝❤♦ ❝ù❝ ➤➵✐ ✭✐✮ ht m1 = ✈➭ ht m2 = ✭✐✐✐✮ Q −→ S/m1 ✈➭ Q −→ S/m2 ❧➭ ❝➳❝ ➤➻♥❣ ❝✃✉✳ m1 ∩ m2 ❦❤➠♥❣ ❝➳❝ ❝❤ø❛ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❦❤➳❝ ❦❤➠♥❣ ♥➭♦ ❝ñ❛ S ➜➷t R = Q + (m1 ∩ m2 ) dim R = 3, ✈➭ m = m1 ∩ m2 ❑❤✐ ➤ã ❘ ❧➭ ♠✐Ị♥ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐ ❞✉② ♥❤✃t ❝đ❛ t❤Õ✱ t❤❡♦ ❱Ý ❞ơ ✷✳✶✳✷✱ ✭✐✐✐✮ t❛ ❝ã R tr➟♥✱ tå♥ t➵✐ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ❱× ✈❐② R ❧➭ f ✲✈➭♥❤ ✭①❡♠ ❬❇❪✮✳ ❱× p ∈ Supp R s❛♦ ❝❤♦ dim R/p + ht p = dim(R/ AnnR (Hm2 (M ))) = R s✉② ré♥❣✳ ❚❤❡♦ ❝➳❝❤ ①➞② ❞ù♥❣ ❧➭ ❦❤➠♥❣ ❝❛t❡♥❛r②✳ ❚❤❡♦ ❬❈❉◆❪ t❛ ❝ã N-dimR (Hm1 (R)) ➤ã S ❧➭ Q✲➤➵✐ sè ✈➭ ❝❤ø❛ ➤ó♥❣ ✐➤➟❛♥ m1 , m2 t❤á❛ ♠➲♥ ❜❛ ➤✐Ị✉ ❦✐Ư♥ s❛✉✳ ✭✐✐✮ ❈➳❝ ➳♥❤ ①➵ tù ♥❤✐➟♥ ✈➭ ❙✉② r❛ N-dimR (Hm2 (R)) = dim R/a(R) = t❤❡♦ ❬❈◆❪ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✸ t❛ ❝ã dim R/a(R) = p(R) = p(R) = ❱× t❤Õ R ❦❤➠♥❣ ❧➭ ❍➡♥ ữ ì p(M ) = f s rộ t ị í sử r ã DR ❝➳❝ R✲♠➠➤✉♥ R ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ♣❤ø❝ ❜Þ ❝❤➷♥ ♥é✐ ①➵ i DR s❛♦ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✸✸ • H i (DR ), i ∈ Z s✐♥❤ M ó ề R ữ s ỗ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ dim M = d✱ ♠➠➤✉♥ ➤å♥❣ ➤✐Ị✉ • )) K i (M ) ∼ = H −i (Hom(M, DR ❝ị♥❣ ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ ❣ä✐ ❧➭ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ✈➭ ✈í✐ ♠ä✐ K i (M ) i = 0, , d✳ ❑❤✐ ➤ã K d (M ) ➤➢ỵ❝ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➠➤✉♥ ❦❤✉②Õt t❤ø ✐ ❝đ❛ M✱ ✭①❡♠ ❙✮✳ ❍➡♥ ♥÷❛✱ t❤❡♦ ➤è✐ ♥❣➱✉ ➤Þ❛ ♣❤➢➡♥❣ ❬❙✱ ✶✳✶❪✱ tå♥ t➵✐ ❝➳❝ ➤➻♥❣ ❝✃✉ Hmi (M ) ∼ = Hom(K i (M ), E), ✈í✐ ♠ä✐ i✱ tr♦♥❣ ➤ã ❍Ư q✉➯ ✷✳✷✳✾✳ E ❧➭ ❜❛♦ ♥é✐ ①➵ ❝đ❛ tr➢ê♥❣ t❤➷♥❣ ❞➢ ●✐➯ sư r➺♥❣ R/m✳ R ❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉✳ ❈❤♦ K(M ) ❧➭ ♠➠➤✉♥ ❝❤Ý♥❤ t➽❝ ❝đ❛ M ❈➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ➤➞② ❧➭ ➤ó♥❣✱ ✭✐✮ ◆Õ✉ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✱ t❤× K(M ) ❝ị♥❣ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ✭✐✐✮ ❈❤♦ ❑❤✐ ➤ã K(M ) ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ M dim M ❧➭ ❦❤➠♥❣ tré♥ tì M ó tể ợ ú tr ột f ✲♠➠➤✉♥ s✉② ré♥❣✳ ✭✐✐✐✮ ◆Õ✉ Mi ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ ❝ã ❝❤✐Ò✉ d ❤♦➷❝ ❝❤✐Ò✉ ❦❤➠♥❣ q✉➳ ✶ ✈í✐ ♠ä✐ n i=1 Mi ❧➭ i = 1, , n t❤× M = f ✲♠➠➤✉♥ s✉② ré♥❣✳ ✭✐✈✮ ❈❤♦ x1 , , xd−1 ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ s✉② ré♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ M ❑❤✐ ➤ã M ❧➭ f ✲♠➠➤✉♥ (x1 , , xd−1 )M ❝ò♥❣ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ▲✃② ✐➤➟❛♥ ♥❣✉②➟♥ tè p ∈ Supp K(M ) s❛♦ ❝❤♦ dim R/p ❑❤✐ ➤ã dim Mp = d−dim R/p ✈➭ Mp ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✶✳✹✳ ❉♦ ➤ã K(Mp ) ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❚❤❡♦ ❬❙❪✱ ✈× (K(M ))p ∼ = K(Mp ) ♥➟♥ (K(M ))p ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❱× t❤Õ K(M ) ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤❡♦ ❍Ö q✉➯ ✷✳✶✳✺✳ ✭✐✐✮ ❚r➢ê♥❣ ❤ỵ♣ d ➤✐Ị✉ ❦✐Ư♥ ❙❡rr❡ S2 , ❧➭ t➬♠ t❤➢ê♥❣✳ ❈❤♦ ♥❣❤Ü❛ ❧➭ d = 4, depth((K(M ))p ) ✈× K(M ) t❤á❛ ♠➲♥ min(2, dim((K(M ))p ) ✸✹ ♥➟♥ dim(R/ Ann(Hmi (M ))) ❧➭ K(M ) ❧➭ f ✲♠➠➤✉♥ i = 1, 2, ✭①❡♠ ❬❙✱ ✸✳✷✳✶❪✮✳ ❙✉② r❛ d 4, ❑❤✐ ▼ ❧➭ ❦❤➠♥❣ tré♥ ❧➱♥✱ t❤❡♦ ❬❙❪ M ♥➟♥ K(K(M )) ➤➢ỵ❝ ♥❤ó♥❣ ✈➭♦ K(K(M )) ✭✐✐✐✮ ❚❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✺ t❛ ❝ã s rộ t ị í ì f ✲♠➠➤✉♥ s✉② ré♥❣✳ tr♦♥❣ 1, N-dimR (Hmj (Mi )) j < d ❉♦ ➤ã N-dim(Hmj (M )) 1, ✈í✐ ♠ä✐ i = 1, , n 1, ✈í✐ ♠ä✐ j < d ❙✉② r❛ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✺✳ ✭✐✈✮ ➜➷t N = (x1 , , xd−1 )M ❑❤✐ ➤ã dim M/N = ❚õ ❞➲② ❦❤í♣ −→ N −→ M −→ M/N −→ 0, t❤❡♦ tÝ♥❤ ❝❤✃t δ ✲❤➭♠ tư ➤å♥❣ ➤✐Ị✉ t❛ ❝ã ❞➲② ❦❤í♣ −→ Hmi (M/N ) −→ Hmi+1 (N ) −→ Hmi+1 (M ) −→ Hmi+1 (M/N ) −→ N-dimR (Hmi (M )) ❉♦ ➤ã ♠ä✐ 1, ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ N-dimR (Hmi (N )) 1, ✈í✐ i < d ị í t t ợ ề ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❍Ö q✉➯ ✷✳✷✳✶✵✳ ❈❤♦ T = R[[X]] ứ S = R[X] ỗ ũ t❤õ❛ ❤×♥❤ t❤ø❝ ✭t➢➡♥❣ ø♥❣ ✈➭♥❤ ➤❛ t❤ø❝✮ t❤❡♦ ♠ét ❜✐Õ♥ X tr➟♥ R✳ ➜➷t n = (m, X) ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐ t❤✉➬♥ ♥❤✃t ❞✉② ♥❤✃t ❝ñ❛ S ✳ ❑❤✐ ➤ã t❛ ❝ã✱ ✭✐✮ ◆Õ✉ R ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ t❤× T, Sn ❧➭ ❝➳❝ f ✲✈➭♥❤ s✉② ré♥❣✳ ✭✐✐✮ ●✐➯ sö r➺♥❣ ❧➭ ❝➳❝ R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝ñ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ T ❤♦➷❝ Sn f ✲✈➭♥❤ s✉② ré♥❣✳ ❑❤✐ ➤ã R ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜➬✉ t✐➟♥✱ t❛ ❧✃② x ❑❤✐ ➤ã = (x1 , , xd ) ❧➭ ♠ét ❤Ö t❤❛♠ sè tï② ý ❝ñ❛ R (x1 , , xd , X) ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛ T ❱× X q✉② ❝đ❛ T, ♥➟♥ X n ❝ị♥❣ ❧➭ ♣❤➬♥ tư ❝❤Ý♥❤ q✉② ❝đ❛ ❧➭ ♠ét ♣❤➬♥ tư ❝❤Ý♥❤ T ✱ ✈× t❤Õ (0 :T X n ) = ❉♦ ✈❐②✱ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ sè ❜é✐ ✈➭ ▼Ư♥❤ ➤Ị ✶✳✷✳✺✱ ✭✐✐✐✮✱ t❛ ❝ã e(xn1 , , xnd d , X n ; T ) = e(xn1 , , xnd d ; T /X n T ) − e(xn1 , , xnd d ; :T X n ) = e(xn1 , , xnd d ; T /X n T ) ✸✺ ψ : T −→ Rn ci xi ) = (c0 , , cn−1 ) ♠ét t♦➭♥ ❝✃✉ ✈í✐ Ker ψ = X n T ✳ ❱× t❤Õ T /X n T ∼ = Rn ✳ ❱❐② ❘â r➭♥❣ r➺♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ❜ë✐ ψ( ❧➭ e(xn1 , , xnd d ; T /X n T ) = e(xn1 , , xnd d ; Rn ) = ne(xn1 , , xnd d ; R) ▼➷t ❦❤➳❝✱ tõ ➤➻♥❣ ❝✃✉ T /X n T ∼ = Rn ✱ t❛ ❝ã T /(xn1 , , xnd d , X n )T = T T /X n T /(xn1 , , xnd d )T /X n T T =n R R/(xn1 , , xnd d )R ì tế t t ợ I(xn1 , , xnd d , X n ; T ) = = T /(xn1 , , xnd d , X n )T −e(xn1 , , xnd d , X n ; T ) T /X n T /(xn1 , , xnd d )T /X n T −e(xn1 , , xnd d ; T /xn T ) T =n T R/(xn1 , , xnd d )R −ne(xn1 , , xnd d ; R) R = nI(xn1 , , xnd d ; R) ✭✐✮ ❱× R ❧➭ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② s✉② ré♥❣✱ ♥➟♥ p(R) 1✳ ❙✉② r❛✱ T p(T ) ✭✐✐✮ ●✐➯ sö r➺♥❣ ❝ã p(T ) 1✳ T ❧➭ f ✲✈➭♥❤ ❉♦ ➤ã ❤➭♠ n1 , , nd ❉♦ ➤ã p(R) r➺♥❣ R s✉② ré♥❣✳ ❑❤✐ ➤ã t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✺✱ t❛ I(xn1 , , xnd d , X n ; T ) n1 , , nd , n I(xn1 , , xnd d ; R) ❈❤ó ý ✷✳✷✳✶✶✳ ❱× t❤Õ✱ t❤❡♦ (∗) t❛ ❝ã f ✲✈➭♥❤ s✉② ré♥❣ t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✺✱ ✭✐✐✮✳ ❧➭ ➤❛ t❤ø❝ t❤❡♦ ❝➳❝ ❜✐Õ♥ ❤➭♠ (∗) ❜Þ ❝❤➷♥ tr➟♥ ❜ë✐ ❝➳❝ ❝ã ❜❐❝ ❦❤➠♥❣ q✉➳ ❧➭ ❜Þ ❝❤➷♥ tr➟♥ ❜ë✐ ♠ét ❤➺♥❣ sè ❦❤➠♥❣ ♣❤ô t❤✉é❝ 0, tø❝ ❧➭ R ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❈❤♦ T, S ✈➭ n ♥❤➢ tr♦♥❣ ❍Ö q✉➯ ✷✳✷✳✶✵✳ ◆❣➢ê✐ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ T ❤♦➷❝ Sn ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❚❤❡♦ ❍Ư q✉➯ ✷✳✷✳✶✵✱ t❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ♥Õ✉ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× Sn ❱× t❤Õ✱ t❤❡♦ ✭✯✮ R ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ T ❤♦➷❝ ✸✻ ✷✳✸ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❚r♦♥❣ ♠ơ❝ ♥➭②✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ♠ét ❦Õt q✉➯ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét f ✲ ♠➠➤✉♥ s✉② ré♥❣✳ ❈❤ó ý r➺♥❣ ❦Õt q✉➯ t➢➡♥❣ tù ➤➲ ➤➢ỵ❝ ❍❡❧❧✉s ❬❍❪ ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ✈➭♥❤ R ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❚✐Õ♣ t❤❡♦✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❞➲② ❝❤Ý♥❤ q✉② ❧ä❝ t❤❛② ❝❤♦ ❞➲② ❝❤Ý♥❤ q✉② tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝ñ❛ ❍❡❧❧✉s✱ ❆s❛❞♦❧❧❛❤✐ ✈➭ ❙❝❤❡♥③❡❧ ❬❆❙❪ ➤➲ ❝➯✐ t✐Õ♥ ❦Õt q✉➯ ♥➭② ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ M ❧➭ ë ➤➞②✱ ✈✐Ö❝ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤➻♥❣ ❝✃✉ ❣✐÷❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ♣❤➢➡♥❣ ❝đ❛ ❆s❛❞♦❧❧❛❤✐ ✈➭ ❙❝❤❡♥③❡❧ ❬❆❙❪ ➤➢ỵ❝ t❤❛② ❜➺♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤➻♥❣ t❤ø❝ ❣✐÷❛ ❝➳❝ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✳ ❈❤ó ý r➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ♥➭② ❝❤Ø ➤ó♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ●✐➯ sö r➺♥❣ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❑❤✐ ➤ã Ass(HIj (M )) t ữ ỗ I ỗ j N ế ỉ ♥Õ✉ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ ➜Þ♥❤ ❧ý ✷✳✸✳✶✳ s❛✉ ➤➞② t❤á❛ (M )) ữ ỗ tử t số x ủ M ỗ Ass(H(x,y)R y ∈ R ✭✐✐✮ Ass(H(x,y,z)R (M )) ❧➭ t❐♣ ữ ỗ ủ ệ t số (x, y) ủ M ỗ z R ể ứ ♠✐♥❤ ➤Þ♥❤ ❧Ý tr➟♥ t❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❜ỉ ➤Ị s❛✉✳ ❚r➢í❝ ❤Õt t❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ▼ét ❞➲② x1 , , x n ∈ I I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉②✳ ❈❤♦ I ❧➭ ♠ét ✐➤➟❛♥ ❝đ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ i = 1, , n, xi / p, ỗ p Ass M/(x1 , , xi−1 )M ❈❤ó ý r➺♥❣ ỗ số ó ộ M R ế ✈í✐ ♠ä✐ t❤á❛ ♠➲♥ p I n ❧✉➠♥ tå♥ t➵✐ ♠ét I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ M n ❍➡♥ ♥÷❛✱ ♥Õ✉ x1 , , xn ∈ I ❧➭ ♠ét I ✲❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ M ✸✼ t❤× ❧✉➠♥ tå♥ t➵✐ ❝➳❝ ➤➻♥❣ ❝✃✉ tù ♥❤✐➟♥ s❛✉ ✭①❡♠ ❬❆❙❪✮ HIi (M ) ❇ỉ ➤Ị ✷✳✸✳✷✳ ❈❤♦ q✉② s✉② ré♥❣ ❝ñ❛ = i H(x (M ), , ,xn )R ♥Õ✉ n HIi−n (H(x (M )), , ,xn )R ♥Õ✉ i 0✳ ❱× ✈❐② ❉♦ ➤ã y ❧➭ ♠ét ♣❤➬♥ tö d − dim(M/JM ) = d − dim(M/IM ) + d − dim(M/JM ) = j − t❤× J ❧➭ ♠ét ✐➤➟❛♥ t❤á❛ ♠➲♥ ❜ỉ ➤Ị✳ ◆Õ✉ d − dim(M/JM ) = j − 1✱ t❛ ❝ã t❤Ĩ ❧❐♣ ❧➵✐ q✉➳ tr×♥❤ tr➟♥ ❝❤♦ ➤Õ♥ ❦❤✐ ♥❤❐♥ ➤➢ỵ❝ ♠ét ✐➤➟❛♥ J ♥❤➢ ②➟✉ ❝➬✉✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜Þ♥❤ ❧Ý ✷✳✸✳✶ ●✐➯ sư r➺♥❣ ❝➳❝ ề ệ ủ ị í ợ t❤á❛ ♠➲♥✳ ❚❤❡♦ ❬❩✱ ❍Ö q✉➯ ✷✳✷❪✱ t❛ ❝❤Ø ❝➬♥ ứ ỗ x, y m t ❧✃② ◆Õ✉ ✈➭ Ass H(x,y,z)R (M ) x, y ∈ m ➜➷t Ass HI2 (M ) ❉♦ ➤ã✱ t❤× x, y ❑❤✐ ➤ã x J ⊇I = d − dim(M/JM ) ✈➭ Ass HI2 (M ) \ {m} = HJ2 (M ) \ {m} ❱× s❛♦ ❝❤♦ ▲✃② x, y, z ❝ñ❛ M tr♦♥❣ J ❑❤✐ M ✳ ❚❤❡♦ ❇ỉ ➤Ị ✷✳✸✳✷ tå♥ t➵✐ ♣❤➬♥ tư Ass HJ2 (M ) ⊆ Ass H(x ,y )R (M ) ∪ {m}✳ ❧➭ t❐♣ ❤÷✉ ❤➵♥ t❤❡♦ ❣✐➯ t❤✐Õt ✭✐✮✳ ❉♦ ➤ã ❤➵♥✳ M✳ ❳Ðt tr➢ê♥❣ ❤ỵ♣ ❑❤✐ ➤ã✱ t❤❡♦ ❇ỉ ➤Ị ✷✳✸✳✹ tå♥ t➵✐ ♠ét ✐➤➟❛♥ ❧➭ ♣❤➬♥ tư ❝❤Ý♥❤ q✉② s✉② ré♥❣ ❝ñ❛ y ∈J d − ❧➭ ♠ét ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ dim(M/JM ) = d−1 ♥➟♥ tå♥ t➵✐ ♠ét ♣❤➬♥ tö t❤❛♠ sè x ➤ã x, y, z ∈ m✳ dim(M/IM ) ❧➭ t❐♣ ❤÷✉ ❤➵♥ t❤❡♦ ❣✐➯ t❤✐Õt ✭✐✮✳ dim(M/IM ) > d s t ữ ỗ I = (x, y)R dim(M/IM ) = d − 2 (M ) t ữ Ass H(x,y)R ì t❤Õ Ass HJ2 (M ) Ass HI2 (M ) ❝ò♥❣ ❧➭ t❐♣ ❤÷✉ ❤➵♥✳ ∈ m✳ ❚➢➡♥❣ tù ♥❤➢ tr➟♥ t❛ ❝ị♥❣ ❝ã Ass H(x,y,z)R (M )❧➭ t❐♣ ❤÷✉ ✹✵ ❑Õt ❧✉❐♥ ❚ã♠ ❧➵✐✱ ❧✉❐♥ ✈➝♥ ♥➭② ➤➲ tr×♥❤ ❜➭② ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ò ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❜➭✐ ❜➳♦ ✧●❡♥❡r❛❧✐③❡❞ F ✲♠♦❞✉❧❡s ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧ ❝ñ❛ ▲✳ ❚✳ ◆❤➭♥ ✈➭ ▼❛r❝❡❧ ▼♦r❛❧❡s ➤➝♥❣ tr➟♥ t➵♣ ❝❤Ý ❈♦♠♠✉♥✐❝❛t✐♦♥ ✐♥ ❆❧❣❡❜r❛✱ ♥➝♠ ✷✵✵✺✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ❣å♠ ❝➳❝ ♥é✐ ❞✉♥❣ s❛✉✿ ✶✳ ◆❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥✿ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt✱ ❤Ö t❤❛♠ sè✱ sè ❜é✐✱ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳✳✳ ✷✳ ◆❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❝❤Ý♥❤ q✉②✱ ❞➲② ❝❤Ý♥❤ q✉② ❧ä❝✱ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣ ✈➭ t➢í♥❣ ø♥❣ ❧➭ ❝➳❝ ❧í♣ ♠➠➤✉♥ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ f ✲♠➠➤✉♥✱ f ✲♠➠➤✉♥ s✉② ré♥❣ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ✸✳ ●✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❢✲♠➠➤✉♥ s✉② ré♥❣ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ➤➷❝ tr➢♥❣ ❝đ❛ ♥ã q✉❛ ➤➬② ➤đ m✲❛❞✐❝✱ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ✈➭ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ➤➻♥❣ ❝❤✐Ị✉ tí✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ♥❣✉②➟♥ s➡ ❝ã ❝❤✐Ị✉ >1 ❝đ❛ t❐♣ s✉♣♣♦rt ❝đ❛ M✳ ✹✳ ❈❤ø♥❣ ♠✐♥❤ ➤➷❝ tr➢♥❣ ❝ñ❛ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤➠♥❣ q✉❛ sè ❜é✐✱ ❦✐Ĩ✉ ➤❛ t❤ø❝ ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ✺✳ ❈❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét ré♥❣✳ f ✲♠➠➤✉♥ s✉② ✹✶ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬❆❙❪ ❆s❛❞♦❧❧❛❤✐✱ ❏✳✱ ❙❝❤❡♥③❡❧✱ P✳ ✭✷✵✵✸✮✱ ❙♦♠❡ r❡s✉❧t ♦♥ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❏❛♣❛♥❡♥❡s ❏✳ ▼❛t❤✳ ✷✾✿✷✽✺✲✷✾✻✳ ❬❇❪ ❇r♦❞♠❛♥♥✱ ▼✳ ✭✶✾✼✽✮✳ ❆ ♣❛rt✐❝✉❧❛r ❝❧❛ss ♦❢ r❡❣✉❧❛r ❞♦♠❛✐♥s✳ ❏✳ ❆❧❣❡❜r❛ ✺✹✿✸✻✻ ✲ ✸✼✸✳ ❬❇❙❪ ❇r♦❞♠❛♥♥✱ ▼✳ ❛♥❞ ❙❤❛r♣✱ ❘✳ ❨✳ ✭✶✾✾✽✮✳ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡✲ ❜r❛ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ●❡♦♠❡tr② ❆♣♣❧✐❝❛t✐♦♥s✳ ❈❛♠❜r✐❞❣❡✿ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② ♣r❡ss✳ ❬❇❍❪ ❇r✉♥s✱ ❲✳✱ ❍❡r③♦❣✱ ❏✳ ✭✶✾✾✸✮✳ ❈♦❤❡♥ ✲ ▼❛❝❛✉❧❛② ❘✐♥❣s✳ ❈❛♠❜r✐❞❣❡✿ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② ♣r❡ss✳ ❬❈✶❪ ❈✉♦♥❣✱ ◆✳ ❚✳ ✭✶✾✾✷✮✳ ❖♥ t❤❡ ❧❡❛st ❞❡❣r❡❡ ♦❢ ♣♦❧②♥♦♠✐❛❧s ❜♦✉♥❞✐♥❣ ❛❜♦✈❡ t❤❡ ❞✐❢❢❢❡r❡♥❝❡s ❜❡t✇❡❡♥ ❧❡♥❣❤ts ❛ ♠✉❧t✐♣❧✐❝✐t✐❡s ♦❢ ❝❡rt❛✐♥ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs ✐♥ ❧♦❝❛❧ r✐♥❣s✳ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✶✷✺✿ ✶✵✺ ✲ ✶✶✹✳ ❬❈✷❪ ❈✉♦♥❣✱ ◆✳ ❚✳ ✭✶✾✾✺✮✳ ♣✲st❛♥❞❛r❞ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣✲st❛♥❞❛r❞ ✐❞❡❛❧s ✐♥ ❧♦❝❛❧ r✐♥❣s✳ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠ ✷✵✿✶✹✺✲✶✻✶✳ ❬❈◆❪ ❈✉♦♥❣✱ ◆✳ ❚✳✱ ◆❤❛♥✱ ▲✳ ❚✳ ✭✷✵✵✷✮✳ ❖♥ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥s✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✳ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ✸✵✿✶✷✶✲✶✸✵✳ ❬❈❙❚❪ ❈✉♦♥❣✱ ◆✳ ❚✳✱ ❙❝❤❡♥③❡❧✱ P✳✱ ❚r✉♥❣✱ ◆✳ ❱✳ ✭✶✾✼✽✮✳ ❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧♥✳▼❛t❤✳ ◆❛❝❤r ✽✺✿ ✺✼✲✼✸✳ ✹✷ ❬❈❉◆❪ ◆✳ ❚✳ ❈✉♦♥❣✱ ◆✳ ❚✳ ❉✉♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✷✵✵✼✮✱ ✧❚♦♣ ❧♦❝❛❧ ❝♦❤♦♠♦❧✲ ♦❣② ❛♥❞ t❤❡ ❝❛t❡♥❛r✐❝✐t② ♦❢ t❤❡ ✉♥♠✐①❡❞ s✉♣♣♦rt ♦❢ ❛ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠♦❞✉❧❡✧✱ ❈♦♠♠✉♥✐❝❛t✐♦♥ ✐♥ ❆❧❣❡❜r❛✱ ✸✺✭✺✮✱ ♣♣✳ ✶✻✾✶✲✶✼✵✶✳ ❬❈▼◆❪ ❈✉♦♥❣✱ ◆✳ ❚✳✱▼♦r❛❧❡s✱ ▼✳✱ ◆❤❛♥✱ ▲✳ ❚✳ ✭✷✵✵✸✮✳ ❖♥ t❤❡ ❧❡♥❣❤t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✳❏✳ ❆❧❣❡❜r❛ ✷✻✺✿✶✵✵✲✶✶✸✳ ❬❍❪ ❍❡❧❧✉s✱ ▼✳ ✭✷✵✵✶✮✳ ❖♥ t❤❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❏✳ ❆❧❣❡❜r❛ ✷✼✸❀✹✵✻✲✹✶✾✳ ❬❑❪ ❑✐r❜②✱ ❉✳ ✭✶✾✾✵✮✳ ❉✐♠❡♥s✐♦♥ ❛♥❞ ❧❡♥❣❤t ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✳ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ✹✶✿✹✶✾✲✹✷✾✳ ❬▼❆❈❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞✱ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠✲ ♠✉t❛t✐✈❡ r✐♥❣✱ ❙②♠♣♦s✐❛ ▼❛t❤❡r♠❛t✐❝❛✱ ✶✶ ✭✶✾✼✸✮✱ ✷✸✲✹✸✳ ❬▼❆❚❪ ▼❛ts✉♠✉r❛✱ ❍✳ ✭✶✾✽✻✮ ❈♦♠♠✉t❛t✐✈❡ ❘✐♥❣ ❚❤❡♦r②✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✳ ❬◆❪ ◆❤❛♥✱ ▲✳ ❚✳ ✭✷✵✵✺✮✳ ❖♥ ❣❡♥❡r❛❧✐③❡❞ r❡❣✉❧❛r s❡q✉❡♥❝❡s ❛♥❞ t❤❡ ❢✐♥✐t❡♥❡ss ❢♦r ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✸✿✼✾✸✲✽✵✻✳ ❬◆▼❪ ◆❤❛♥✱ ▲✳ ❚✳✱ ▼♦r❛❧❡s✳ ▼✳ ✭✷✵✵✻✮✳ ●❡♥❡r❛❧✐③❡❞ ❋✲▼♦❞✉❧❡s ❛♥❞ t❤❡ ❛s✲ s♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✹✿✽✻✸✲ ✽✼✽✳ ❬❙❪ ❙❝❤❡♥③❡❧✱ P✳ ✭✶✾✽✷✮✳ ❉✉❛❧✐s✐❡r❡♥❞❡ ❑♦♠♣❧❡①❡ ✐♥ ❞❡r ❧♦❦❛❧❡♥ ❆❧❣❡❜r❛ ✉♥❞ ❇✉❝❤s❜❛✉♠ ❘✐♥❣❡✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤❡♠❛t✐❝s ✾✵✼✳ ❇❡r❧✐♥✱ ❍❡✐❞❡❧✲ ❜❡r❣✱ r rr tu ăr s ❘✐♥❣s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✧✳ ❇❡r❧✐♥✿ ❲❊❇ ❉❡✉ts❡❝❤❡r ❱❡r❧❛❣ ❞❡r ❲✐ss❡♥s❝❤❛❢t❡♥✳ ✹✸ ❬❚❪ ❚r✉♥❣✱ ◆✳ ❱✳ ✭✶✾✽✻✮✳ ❚♦✇❛r❞ ❛ t❤❡♦r② ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✳ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✶✵✷✿✶✲✹✾✳ ❬❩❪ ❩❛♠❛♥✐✱ ◆✳ ✭✷✵✵✸✮✳ ❆ ♥♦t❡ ♦♥ t❤❡ s❡t ♦❢ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s♠♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✸✶✿✶✷✵✸✲✶✷✵✻✳ 2013 ... tr♦♥❣ f ✲ depth(I, M ), ợ ị ĩ ộ ủ ột f ✲❞➲② ❝ù❝ tr♦♥❣ I ✳ ë ➤➞②✱ ♥Õ✉ ❦❤➠♥❣ tå♥ t➵✐ f ✲❞➲② ❝ù❝ ➤➵✐ tr♦♥❣ I t❤× t❛ ể ị ĩ M ợ ọ ♥Õ✉ ♠ä✐ ❤Ư t❤❛♠ sè ❝đ❛ M ❧➭ f ✲❞➲②✳ ▼ét ✈➭♥❤ ➤➢ỵ❝ ❣ä✐ ❧➭ f ✲✈➭♥❤... ❣ä✐ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ ♥Õ✉ ♠ä✐ ❤Ư t❤❛♠ sè ❝đ❛ ❧➭ ❞➲② ❝❤Ý♥❤ q✉② s✉② ré♥❣✳ ▼ét ✈➭♥❤ ➤➢ỵ❝ ❣ä✐ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ tr➟♥ ❝❤Ý♥❤ ♥ã✳ f ✲✈➭♥❤ s✉② ré♥❣ ♥Õ✉ ♥ã ✶✼ ❱Ý ❞ô ✷✳✶✳✷✳ ✭✐✮ ▼ä✐ f ✲♠➠➤✉♥ ❧➭ f ✲♠➠➤✉♥... ❧➭ ➤ó♥❣✳ ✭✐✮ ◆Õ✉ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤× M ❝ị♥❣ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ✭✐✐✮ ●✐➯ sư r➺♥❣ R ❧➭ ✈➭♥❤ t❤➢➡♥❣ ❝đ❛ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ◆Õ✉ M ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣ t❤× M ❝ò♥❣ ❧➭ f ✲♠➠➤✉♥ s✉② ré♥❣✳ ❈❤ø♥❣