ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM - - LÊ THỊ MAI QUỲNH ĐẶC TRƯNG CỦA MƠĐUN COHEN–MACAULAY DÃY QUA TÍNH CHẤT PHÂN TÍCH THAM SỐ Chuyên ngành: Đại số lý thuyết số Mã số: 60.46.05 LUẬN VĂN THẠC SỸ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: GS.TSKH NGUYỄN TỰ CƯỜNG THÁI NGUYÊN NĂM 2008 Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn ✶ ▼ô❝ ❧ô❝ ▼ô❝ ❧ô❝ ✶ ▲ê✐ ❝➯♠ ➡♥ ✷ P❤➬♥ ♠ë ➤➬✉ ✸ ❈❤➢➡♥❣ ■✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✺ ✶✳✶✳ ❍Ư t❤❛♠ sè ✺ ✶✳✷✳ ❉➲② ❝❤Ý♥❤ q✉② ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✼ ✶✳✸✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❈❤➢➡♥❣ ■■✳ P❤➞♥ tÝ❝❤ t❤❛♠ sè ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✶✵ ✶✹ ✷✳✶✳ ➜➷❝ tr➢♥❣ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛❧❛② ❞➲② ✶✹ ✷✳✷✳ ➜❛ t❤ø❝ ❍✐❧❜❡rt✲❙❛♠✉❡❧ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✷✼ ✷✳✸✳ ❱Ý ❞ơ ✸✶ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ✸✽ ✷ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ●❙✳❚❙❑❍ ◆❣✉②Ơ♥ ❚ù ❈➢ê♥❣✳ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❦Ý♥❤ trä♥❣ ✈➭ ❜✐Õt ➡♥ s➞✉ s➽❝ t ủ ì ế t tỏ ò ❜✐Õt ➡♥ tí✐ P●❙✳❚❙ ▲➟ ❚❤Þ ❚❤❛♥❤ ◆❤➭♥✱ P●❙✳❚❙ ◆❣✉②Ơ♥ ◗✉è❝ ❚❤➽♥❣ ❝ï♥❣ t♦➭♥ t❤Ó ❝➳❝ t❤➬② ❝➠ ❣✐➳♦ ë ❑❤♦❛ ❚♦➳♥ ✈➭ P❤ß♥❣ ➜➭♦ t➵♦ s❛✉ ➜➵✐ ❤ä❝ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ➤➲ t❐♥ t×♥❤ ❣✐➯♥❣ ❞➵② ✈➭ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ t➵✐ tr➢ê♥❣✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ sù ❣✐ó♣ ➤ì ♥❤✐Ưt t❤➭♥❤ ✈➭ ❝❤✉ ➤➳♦ ❝đ❛ ◆❈❙ ❚r➬♥ ◆❣✉②➟♥ ❆♥✱ ❜➵♥ ❍♦➭♥❣ ▲➟ ❚r➢ê♥❣ ♣❤ß♥❣ ➤➵✐ sè tr♦♥❣ q✉➳ tr×♥❤ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ✸ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤♦ R ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈í✐ ✐➤➟❛♥ tè✐ ➤➵✐ m ✈➭ M ❧➭ R− ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ dim M = d✳ ❈❤♦ x = x1 , , xd ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ M ✈➭ q = (x1 , , xd ) ❧➭ ✐➤➟❛♥ t❤❛♠ sè ủ M s x ỗ số n✱ ❦ý ❤✐Ö✉ d d Λd,n = {(α1 , , αd ) ∈ Z | αi ≥ 1, ∀1 ≤ i ≤ d, αi = d + n − 1} i=1 q(α) = (xα1 , , xαd d ) ✈í✐ ∀α = (α1 , , αd ) ∈ Λd,n ✳ ❚❛ ♥ã✐ r➺♥❣ ❤Ö t❤❛♠ sè x ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè ♥Õ✉ ➤➻♥❣ t❤ø❝ qn M = q(α)M ➤ó♥❣ ✈í✐ ∀n ≥ 1✳ ❱❐② ❦❤✐ ♥➭♦ ♠ét ❤Ư t❤❛♠ sè ✈➭ α∈Λd,n ❝❤♦ tr➢í❝ ❝đ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ❱✃♥ ➤Ò ♥➭② ❍❡✐♥③❡r✱ ❘❛t❧✐❢❢ ✈➭ ❙❤❛❤ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ♠ét ❞➲② ❝➳❝ ♣❤➬♥ tö R− ❝❤Ý♥❤ q✉② ❧✉➠♥ ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ❙❛✉ ➤ã✱ ●♦t♦ ✈➭ ❙❤✐♠♦❞❛ ➤➲ ❝❤Ø r r ề ợ ũ ú ỗ tư ❝đ❛ ❞➲② ❦❤➠♥❣ ❧➭ ➢í❝ ❝đ❛ ❦❤➠♥❣ tr♦♥❣ R✳ ữ ọ ò r ột tr ❝đ❛ R ✈í✐ dim R ≥ 2, tr♦♥❣ ➤ã ♠ä✐ ❤Ư t❤❛♠ sè ❝đ❛ R ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ❚❛ ♥ã✐ ♠➠➤✉♥ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè x ♥➭♦ ➤ã s❛♦ ❝❤♦ x ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ❇➞② ❣✐ê✱ t❛ ❤➵♥ ❝❤Õ sù q✉❛♥ t➞♠ ❝đ❛ ❝➞✉ ❤á✐ tr➟♥ ❝❤♦ ❤Ư t❤❛♠ sè tèt ❝ñ❛ M ✳ ❑❤✐ ➤ã ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❝ã t❤Ĩ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❜ë✐ tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè ❝đ❛ ♠ét ❤Ư t❤❛♠ sè tèt ♥❤➢ t❤Õ ♥➭♦✳ ộ ó ợ trì tr Prtr ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣♦✇❡rs ♦❢ ♣❛r❛♠❡t❡r ✐❞❡❛❧s ❛♥❞ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❝đ❛ t➳❝ ❣✐➯ ◆❣✉②Ơ♥ ❚ù ❈➢ê♥❣ ✈➭ ❍♦➭♥❣ ▲➟ ❚r➢ê♥❣✳ ❇➭✐ ❜➳♦ sÏ r❛ ë t➵♣ ❝❤Ý ✧ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✧ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr×♥❤ ❜➭② ❧➵✐ ♠ét ❝➳❝❤ ❤Ö t❤è♥❣ ✈➭ ❝❤✐ t✐Õt ❦Õt q✉➯ ❝đ❛ ❜➭✐ ❜➳♦ tr➟♥✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ❈❤➢➡♥❣ ❝❤➢➡♥❣✳ ✧❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ✧ ❧➭ ❝❤➢➡♥❣ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ➤➵✐ sè ❣✐❛♦ ❤♦➳♥ ♥❤➢ ❤Ö t❤❛♠ sè✱ ❞➲② ❝❤Ý♥❤ q✉②✱ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✱ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✹ ❈❤➢➡♥❣ ✧P❤➞♥ tÝ❝❤ t❤❛♠ sè ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✧ tr×♥❤ ❜➭② ♠ét sè ❜ỉ ➤Ị tõ ➤ã ➤✐ ➤Õ♥ ➤Þ♥❤ ❧ý ❝❤Ý♥❤ ❝đ❛ ❝❤➢➡♥❣ ♥ã✐ ✈Ị ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② q✉❛ ♣❤➞♥ tÝ❝❤ t❤❛♠ sè ✈➭ ❤Ư q✉➯ ❝đ❛ ♥ã✳ ➜Þ♥❤ ❧ý ♣❤➳t ❜✐Ĩ✉ r➺♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✻✳ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r✳ M ❧➭ R− ♠➠➤✉♥ ❤÷❛ ❤➵♥ s✐♥❤✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✭✐✐✮ ▼ä✐ ❤Ư t❤❛♠ sè tèt ❝đ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ✭✐✐✐✮ ❚å♥ t➵✐ ❤Ö t❤❛♠ sè tèt ❝ñ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ◆❣♦➭✐ r ò trì ố q ệ ữ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➲② M ✈➭ ❜✐Ĩ✉ t❤ø❝ ❝đ❛ ❤➭♠ ❍✐❧❜❡rt✲❙❛♠✉❡❧ t❤➠♥❣ q✉❛ ➤Þ♥❤ ❧ý ➜Þ♥❤ ❧ý ✷✳✷✳✸✳ ❈❤♦ D : D0 ⊂ D1 ⊂ ⊂ Dt = M ❧➭ ❧ä❝ ❝❤✐Ị✉ ❝đ❛ M ✈➭ ➤➷t Di = Di /Di−1 ✈í✐ ♠ä✐ i = 1, , t, D0 = D0 ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✭✐✐✮ ❱í✐ ❜✃t ❦ú ✐➤➟❛♥ t❤❛♠ sè tèt q ❝ñ❛ M ✱ ➤➻♥❣ t❤ø❝ t l(M/q n+1 M) = i=0 ➤ó♥❣ ✈í✐ ♠ä✐ n ≥ 0✳ ✭✐✐✐✮ ❚å♥ t➵✐ ✐➤➟❛♥ t❤❛♠ sè tèt q ❝ñ❛ M t l(M/q n+1 M) = i=0 ➤ó♥❣ ✈í✐ ♠ä✐ n + di l(Di /qDi ) di s❛♦ ❝❤♦ ➤➻♥❣ t❤ø❝ n + di l(Di /qDi ) di n ≥ 0✳ P❤➬♥ ❝✉è✐ ❝ï♥❣ ❝đ❛ ❝❤➢➡♥❣ sÏ ①➞② ❞ù♥❣ ✈Ý ❞ơ ♥❤➺♠ ❧➭♠ s➳♥❣ tá ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ ➤➲ ♥➟✉ ë tr➟♥✳ ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ➤➵✐ sè ❣✐❛♦ ❤♦➳♥ ➤➢ỵ❝ sư ❞ơ♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ➤Þ♥❤ ♥❣❤Ü❛✱ ❝➳❝ ♠Ư♥❤ ➤Ị ✈➭ ❜ỉ ➤Ị ✈Ị ❤Ư t❤❛♠ sè✱ ❞➲② ❝❤Ý♥❤ q✉②✱ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✶✳✶ ❍Ö t❤❛♠ sè ❚r♦♥❣ ♣❤➬♥ ♥➭② t❛ sÏ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈Ị ❤Ư t❤❛♠ sè✱ ➤➞② ❧➭ ♠ét ❦❤➳✐ ♥✐Ư♠ q✉❛♥ trä♥❣ ①✉②➟♥ s✉èt q✉➳ tr×♥❤ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ (R, m) ❧➭ ị tr M R ữ s✐♥❤ ✈í✐ dim M = d✳ ❚❐♣ ❝➳❝ ♣❤➬♥ tư x = (x1 , x2 , , xd )✱ xi ∈ m , ∀i = 1, , d t❤♦➯ ♠➲♥ lR (M/xM ) < ∞ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét t❤❛♠ sè ❝đ❛ ●✐➯ sư s✐♥❤ ✈í✐ ❤Ư M✳ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r✱ M ❧➭ R− ♠➠➤✉♥ ❤÷✉ ❤➵♥ dim M = d✳ ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ♥➟✉ ❧➟♥ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❤Ư t❤❛♠ sè✳ ✺ ✻ ✶✳✶✳✷ ▼Ư♥❤ ➤Ị✳ ❬✶✱ ▼Ư♥❤ ➤Ị ❆✳✹❪ ❈❤♦ x1 , x2 , , xt ∈ m ❦❤✐ ➤ã dim(M/(x1 , , xt )M ) ≥ dim M − t ➜➻♥❣ t❤ø❝ s➯② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝ñ❛ x1 , x2 , , xt ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè M✳ ✶✳✶✳✸ ▼Ư♥❤ ➤Ị✳ ❬✽✱ ❈❤ó ý ✶✺✳✷✵❪ ◆Õ✉ ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ ❝ñ❛ x1 , , xd ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ M α1 , , αd t❛ ❝ã xα1 , , xαd d t❤× ❝ị♥❣ ❧➭ ❤Ư t❤❛♠ sè M✳ ◆❤❐♥ ①Ðt✳ ✭✶✮ ❈❤♦ ❦❤✐ x ∈ m ❦❤✐ ➤ã x ❧➭ ♠ét ♣❤➬♥ tư ❝đ❛ ❤Ư t❤❛♠ sè ❝đ❛ M ❦❤✐ ✈➭ ❝❤Ø x ∈ p ✈í✐ ♠ä✐ p ∈ Ass R s❛♦ ❝❤♦ dimR/p = d✳ ✭✷✮ ❈❤♦ x1 , , xd ∈ m ①➳❝ ➤Þ♥❤ ❜ë✐ xi+1 ∈ p, ∀p ∈ Ass R(M/(x1 , , xi )M ), dim R/p = d − i ✈í✐ i = 0, , d − 1✳ ❑❤✐ ➤ã {x1 , , xd } ❧➭ ❤Ö t❤❛♠ sè ❝đ❛ M ✳ ❚✐Õ♣ t❤❡♦ t❛ sÏ ➤➢❛ r❛ ➤Þ♥❤ ♥❣❤Ü❛ ✈Ị ❤➭♠ ❍✐❧❜❡rt✲❙❛♠✉❡❧ ✈➭ ➤Þ♥❤ ❧ý ➤❛ t❤ø❝ ❍✐❧❜❡rt✱ ➤➞② ❧➭ ♠ét ➤Þ♥❤ ❧ý ♥ỉ✐ t✐Õ♥❣ ✈➭ ❝ã ø♥❣ ❞ơ♥❣ ♥❤✐Ị✉ tr♦♥❣ ➤➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② t❛ ❝❤Ø ♥❤➽❝ ❧➵✐ ➤Þ♥❤ ♥❣❤Ü❛ ✈➭ ➤Þ♥❤ ❧ý ❞ï♥❣ ❝❤♦ ❝❤➢➡♥❣ s❛✉ ♠➭ ❦❤➠♥❣ ❝❤ø♥❣ ♠✐♥❤✳ ✶✳✶✳✹ ➜Þ♥❤ ♥❣❤Ü❛✳ ◆♦❡t❤❡r ❈❤♦ M ❧➭ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ tr➟♥ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ (R, m) ✈í✐ dim M = d✱ q ❧➭ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ M ✭ tø❝ ❧➭ l(M/qM ) < ∞✮✳ ❑❤✐ ➤ã t❛ ➤Þ♥❤ ♥❣❤Ü❛ ♠ét ❤➭♠ sè ❣ä✐ ❧➭ ❙❛♠✉❡❧ Fq,M (n) = l(M/qn+1 M ) ❤➭♠ ❍✐❧❜❡r✲ ✼ ✶✳✶✳✺ ▼Ư♥❤ ➤Ị✳ ❬✼✱ ➜Þ♥❤ ❧ý ✶✸✳✷❪ ❈❤♦ ◆♦❡t❤❡r✳ sö r➺♥❣ R0 R = t≥0 Rt ❧➭ ✈➭♥❤ ♣❤➞♥ ❜❐❝ ❧➭ ✈➭♥❤ ❆rt✐♥ ✈➭ ▼ ❧➭ ❘✲ ♠➠➤✉♥ ♣❤➞♥ ❜❐❝ ❤÷❛ ❤➵♥ s✐♥❤✳ ●✐➯ R = R0 [x1 , , xr ] ✈➭ xi tû ❝đ❛ ♥ ❤➡♥ ♥÷❛ tå♥ t➵✐ ➤❛ t❤ø❝ ❜❐❝ di ❦❤✐ ➤ãFq,M (n) ❧➭ ♠ét ❤➭♠ ❤÷✉ Pq,M (n) ✈í✐ ❤Ư sè ❤÷✉ tû ❜❐❝ d s❛♦ ❝❤♦ n ➤đ ❧í♥ t❤× ✈í✐ Fq,M (n) = Pq,M (n) ✈➭ tå♥ t➵✐ ♥❤÷♥❣ sè ♥❣✉②➟♥ e0 (q, M )(> 0), e1 (q, M ), , ed (q, M ) s❛♦ ❝❤♦ Pq,M (n) = e0 (q, M ) n+d−1 +· · ·+ed (q, M ) d−1 n+d +e1 (q, M ) d ❙è e0 (q, M ) ➤➢ỵ❝ ❣ä✐ ❧➭ sè ❜é✐ ❩❛③✐s❦✐✲❙❛♠✉❡❧✳ ❑❤✐ q s✐♥❤ ❜ë✐ ♠ét ❤Ö t❤❛♠ sè x = {x1 , x2 , , xd } t❛ ❦ý ❤✐Ö✉ e0 (q, M ) = e( x, M )✳ ✶✳✷ ❉➲② ❝❤Ý♥❤ q✉② ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❚r♦♥❣ ♣❤➬♥ ♥➭② t❛ sÏ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈Ị ❞➲② ❝❤Ý♥❤ q✉②✱ ➤ã ❧➭ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ➤Ĩ ➤Þ♥❤ ♥❣❤Ü❛ ➤é s➞✉ ❝đ❛ ♠ét ♠➠➤✉♥ tõ ➤ã ➤➢❛ ➤Õ♥ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ✈➭♥❤ ✈➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✶✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ tư ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ✈➭ M ❧➭ R− ♠➠➤✉♥✳ ▼ét ♣❤➬♥ x ∈ R ➤➢ỵ❝ ❣ä✐ ❧➭ M − ❝❤Ý♥❤ q✉② ♥Õ✉ :M x = 0✱ tø❝ ❧➭ xa = ✈í✐ ∀a ∈ M, a = 0✳ ▼ét ❞➲② ❝➳❝ ♣❤➬♥ tö x1 , , xn ❝đ❛ R ➤➢ỵ❝ ❣ä✐ ❧➭ M −❞➲② ❝❤Ý♥❤ q✉② ♥Õ✉ q✉② ✈í✐ ♠ä✐ (x1 , , xn )M = M ✈➭ xi ❧➭ M/(x1 , , xi−1 )M − ❝❤Ý♥❤ i = 1, , n✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ➤➞② ♥➟✉ ❧➟♥ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ❞➲② ❝❤Ý♥❤ q✉②✳ ✶✳✷✳✷ ▼Ư♥❤ ➤Ị✳ ❬✽✱ ❇ỉ ➤Ị ✶✻✳✹❪ ❈❤♦ ➤Ò s❛✉ t➢➡♥❣ ➤➢➡♥❣✿ M ❧➭ R− ♠➠➤✉♥ ❦❤✐ ➤ã ❝➳❝ ♠Ö♥❤ ✽ ✭✐✮ ❉➲② x1 , , xn ❧➭ ❞➲② M − ❝❤Ý♥❤ q✉②✳ ✭✐✐✮ ❉➲② x1 , , xi ❧➭ ❞➲② M − ❝❤Ý♥❤ q✉② ✈➭ xi+1 , , xn ❧➭ ❞➲② M/(x1 , , xi )M − ❝❤Ý♥❤ q✉② ✈í✐ ♠ä✐ ≤ i ≤ n − ✶✳✷✳✸ ▼Ư♥❤ ➤Ị✳ ❬✼✱ ➜Þ♥❤ ❧ý ✶✻✳✶❪ ◆Õ✉ t❤× ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ α1 , , α n x1 , , xn t❛ ❝ã ❧➭ ❞➲② M− ❝❤Ý♥❤ q✉② {xα1 , , xαnn } ❝ò♥❣ ❧➭ ❞➲② M − ❝❤Ý♥❤ q✉②✳ ✶✳✷✳✹ ▼Ư♥❤ ➤Ị✳ ❬✽✱ ị ý ế tì ọ ị ủ ❝➳❝ ♣❤➬♥ tö x1 , , x n x1 , , x n ❧➭ ❞➲② M− ❝❤Ý♥❤ q✉② t❛ ✈➱♥ ➤➢ỵ❝ ♠ét ❞➲② M− ❝❤Ý♥❤ q✉②✳ ✶✳✷✳✺ ▼Ư♥❤ ➤Ị✳ ❬✶✱ ▼Ư♥❤ ➤Ị ✶✳✷✳✶✷❪ ◆Õ✉ tr➟♥ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈➭ x1 , , xt M ❧➭ x1 , , xt ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❝đ❛ R− ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❧➭ ❞➲② M− ❝❤Ý♥❤ q✉② tì M ị ĩ ề í q tr➟♥ ❝❤♦ ♣❤Ð♣ ➤✐ ➤Õ♥ ❦❤➳✐ ♥✐Ư♠ ➤é s➞✉ ❝đ❛ ♠ét ♠➠➤✉♥✱ ➤Ĩ tõ ➤ã ➤✐ ➤Õ♥ ❦❤➳✐ ♥✐Ư♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✶✳✷✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ s✐♥❤ s❛♦ ❝❤♦ ❈❤♦ I ❧➭ ✐➤➟❛♥ ❝đ❛ ✈➭♥❤ R✱ M ❧➭ R− ♠➠➤✉♥ ❤÷✉ ❤➵♥ M = IM ✳ ❑❤✐ ➤ã ➤é ❞➭✐ ❝ù❝ ➤➵✐ ❝ñ❛ ❞➲② M − ❝❤Ý♥❤ q✉② ❝ñ❛ I ❣ä✐ ❧➭ ➤é s➞✉ ❝đ❛ ✐➤➟❛♥ I ➤è✐ ✈í✐ R− ♠➠➤✉♥ M ✱ ❦Ý ❤✐Ö✉ depth R(I, M )✳ ◆Õ✉ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r✱ t❛ ❝ã t❤Ĩ ❦Ý ❤✐Ư✉ ➤é s➞✉ ❝ñ❛ R− ♠➠➤✉♥ M ❧➭ depthR M ❤♦➷❝ ❝ã t❤Ó ➤➡♥ ❣✐➯♥ ❤➡♥ ❧➭ depth M ✳ ✶✳✷✳✼ ▼Ư♥❤ ➤Ị✳ ❬✶✱ ▼Ư♥❤ ➤Ị ✶✳✷✳✶✸❪ ❈❤♦ ◆♦❡t❤❡r✱ M ❧➭ (R, m) ị R ữ s✐♥❤✳ ❚❛ ❝ã ❦❤➻♥❣ ➤Þ♥❤ s❛✉✳ depth M ≤ dim R/p ≤ dim M, ∀p ∈ Ass M ✳ ❱➭ t✐Õ♣ t❤❡♦ t❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ị ĩ M ợ ọ ❈♦❤❡♥✲▼❛❝❛✉❧❛② M = ❤♦➷❝ M = ✈➭ depth M = dim M ❱➭♥❤ R ❣ä✐ ❧➭ ▼❛❝❛✉❧❛② ♥Õ✉ ♥ã ❧➭ ♥Õ✉ ✈➭♥❤ ❈♦❤❡♥✲ R− ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ▼Ư♥❤ ➤Ị s❛✉ ♥➟✉ ❧➟♥ ❝➳❝ ➤➷❝ tr➢♥❣ ❝➡ ❜➯♥ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✶✳✷✳✾ ▼Ư♥❤ ➤Ị✳ ❬✼✱ ➜Þ♥❤ ❧ý ✶✼✳✸❪ ✭✶✮ ◆Õ✉ t❤× ✈í✐ ∀p ∈ Ass M ✭✷✮ ◆Õ✉ t❛ ❝ã x1 , , xd ∈ m ▼❛❝❛✉❧❛② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ M ❧➭ ❞➲② M− ❝❤Ý♥❤ q✉② t❤× M/(x1 , , xd )M ❧➭ ❞➲② ✈➭ M/N M M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× M − ❝❤Ý♥❤ q✉②✳ ✶✳✷✳✶✶ ❇ỉ ➤Ị✳ ❬✸✱ ❇ỉ ➤Ị ✷✳✷❪ ❈❤♦ dim N < dim M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② dim R/p = dim M ✳ ✶✳✷✳✶✵ ▼Ư♥❤ ➤Ị✳ ❬✼✱ ❈❤ó ý ✶✸✻❪ ◆Õ✉ ♠ä✐ ❤Ư t❤❛♠ sè ❝đ❛ M N ❧➭ ♠➠➤✉♥ ❝♦♥ ❝đ❛ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈❤♦ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❝ñ❛ M ❦❤✐ ➤ã (x1 , , xi )M ∩N ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ M t❤♦➯ ♠➲♥ x1 , , x i ❧➭ = (x1 , , xi )N ✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ ✐✳ i = t❛ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ x1 M ∩ N = x1 N ✳ ❚❛ ❧✉➠♥ ❝ã x1 N ⊆ x1 M ∩ N t❛ ❝❤ø♥❣ ♠✐♥❤ x1 M ∩ N ⊆ x1 N ✳ ❚❤❐t ✈❐②✱ ❧✃② y ∈ x1 M ∩ N ❦❤✐ ➤ã y ∈ x1 M ✈➭ y = x1 m ✈í✐ m ∈ M s✉② r❛ y = x1 m ∈ N ❤❛② x1 m + N = + N tr♦♥❣ M/N tø❝ x1 (m + N ) = s✉② r❛ m + N = ❤❛② m ∈ N ✳ ❉♦ ➤ã y = x1 m ∈ x1 N ●✐➯ sö ▲✃② i > 1✳ ❚❛ ❧✉➠♥ ❝ã (x1 , , xi )N ⊆ (x1 , , xi )M ∩ N (1) a ∈ (x1 , , xi )M ∩ N ❦❤✐ ➤ã a = x1 a1 + · · · + xi tr♦♥❣ ➤ã aj ∈ M ✈í✐ ♠ä✐ j = 1, , i ✈× a ∈ N ♥➟♥ ∈ (N + (x1 , , xi−1 )M ) : xi ✳ ▼➷t ❦❤➳❝✱ ✈× ❞➲② x1 , , xi ❧➭ M/N − ❝❤Ý♥❤ q✉② ✈➭ (N + (x1 , , xi−1 )M ) :M xi = N + (x1 , , xi−1 )M ✷✺ ❈❤♦ q(α)M ❦❤✐ ➤ã t❛ ❝ã x ∈ qn M , tr♦♥❣ ➤ã x ❧➭ ➯♥❤ ❝ñ❛ x x ∈ α∈Λd,n tr♦♥❣ M ❞♦ ➤ã x ∈ qn M + Dt−1 ✳ ❱❐② q(α)M ⊆ qn M + Dt−1 ✳ ❱× α∈Λd,n α1 αd x1 , , xd ❧➭ ❤Ö t❤❛♠ sè tèt ❝đ❛ M ✈í✐ ∀α ∈ Λd,n ✈➭ t❤❡♦ ❜ỉ ➤Ị αdt−1 q(α)M ∩ Dt−1 = (xα1 , , xdt−1 )Dt−1 ✳ ❉♦ ✈❐② q(α)M ] ∩ [qn M + Dt−1 ] q(α)M = [ α∈Λd,n 1.3.8 ❧➭ α∈Λd,n q(α)M ∩ qn M ] + [ =[ α∈Λd,n q(α)M ∩ Dt−1 ] α∈Λd,n = qn M + [q(α)M ∩ Dt−1 ] α∈Λd,n αd t−1 (xα1 , , xdt−1 )Dt−1 = qn M + α∈Λd,n ❚❛ ❧✉➠♥ ❝ã (β1 , , βdt−1 , 1, , 1) ∈ Λd,n ✈í✐ ❜✃t ❦ú (β1 , , βdt−1 ) ∈ Λdt−1 ,n ✈➭ ➤é ❞➭✐ ❝ñ❛ ❧ä❝ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② Dt−1 ❧➭ t − 1✳ ❉♦ ➤ã t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ t❛ ❝ã αd βd t−1 (xβ1 , , xdt−1 )Dt−1 t−1 (xα1 , , xdt−1 )Dt−1 ⊆ (α1 , ,αd )∈Λd,n (β1 , ,βdt−1 )∈Λdt−1 ,n = (x1 , x2 , , xdt−1 )n Dt−1 ⊆ qn M q(α)M = qn M ✳ ❙✉② r❛ α∈Λd,n ✭✐✐✮⇒✭✐✐✐✮✳ ❱× ♠ä✐ ❤Ư t❤❛♠ sè ❝ñ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè ♥➟♥ ❧✉➠♥ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè ♥➭♦ ➤ã ❝ñ❛ ✭✐✐✐✮ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ⇒ ✭✐✮✳ ❈❤♦ x = x1 , , xd ❧➭ ❤Ư t❤❛♠ sè tèt ❝đ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ❚❛ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ❝❤ø♥❣ ♠✐♥❤ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❤❛② Ds /Ds−1 , ∀s = 1, , t ❧➭ ♠➠❞✉♥ ❈♦❤❡♥✲ ✷✻ ▼❛❝❛✉❧❛② ✈í✐ D : D0 ⊂ D1 ⊂ ⊂ Dt = M ❧➭ ❧ä❝ ❝❤✐Ị✉ ❝đ❛ M ✳ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ị✉ ➤ã tr➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ (qi M + Ds ) : xi+1 = qi M + Ds ✈í✐ ∀i < ds+1 ✈➭ s = 0, , t − 1✳ ❚❤❐t ✈❐②✱ t❤❡♦ ❜ỉ ➤Ị 2.1.5 sÏ tå♥ t➵✐ sè ♥❣✉②➟♥ k s❛♦ ❝❤♦ qi M : xki+1 = qi M + :M xki+1 k qi M : xk+1 ds+1 = qi M + :M xds+1 ❍➡♥ ♥÷❛ t❤❡♦ ❜ỉ ➤Ị 1.3.5 ❝ã :M xki+1 ⊆ :M xkds+1 ✳ ❑❤✐ ➤ã t❛ ❝ã (qi M + :M xds+1 ) : xki+1 ⊆ qi M : xds+1 xki+1 = (qi M + :M xki+1 ) : xds+1 ⊆ qi M : xk+1 ds+1 = qi M + :M xkds+1 ♠➭ t❤❡♦ ❜ỉ ➤Ị 1.3.5 ❝ã Ds = : xkds+1 ❞♦ ➤ã (qi M + Ds ) : xki+1 = qi M ⊆ (qi M + Ds ) : xi+1 ✈í✐ ∀i < ds+1 s✉② r❛ (qi M + Ds ) : xi+1 = qi M + Ds ✳ ❚❛ ❝ã depth M/Ds ≥ ds+1 ✈í✐ s = 0, , t − 1✳ ♥➟♥ tõ ❞➲② ❦❤í♣ ♥❣➽♥ −→ Ds /Ds−1 −→ M/Ds−1 −→ M/Ds −→ ❦Ð♦ t❤❡♦ Ds /Ds−1 ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ∀s = 1, , t ❤❛② M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✷✳✶✳✼ ❍Ö q✉➯✳ ❈❤♦ ♣❤➢➡♥❣ t❤ø dim M ≥ ❝ñ❛ M ✈➭ Hm0 (M ) ø♥❣ ✈í✐ ✐➤➟❛♥ tè✐ ➤➵✐ ❧➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ m✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M/Hm0 (M ) ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ mHm0 (M ) = 0✳ ✭✐✐✮ ▼ä✐ ❤Ö t❤❛♠ sè ❝ñ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ✷✼ ✭✐✮⇒ ✭✐✐✮✳ ❚❤❡♦ ❣✐➯ t❤✐Õt M/Hm (M ) ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❈❤ø♥❣ ♠✐♥❤✳ ♥➟♥ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈í✐ ❧ä❝ ❝❤✐Ị✉ D : Hm0 (M ) ⊂ M ✳ ❍➡♥ ♥÷❛✱ t❤❡♦ ❜ỉ ➤Ị 1.2.11 t❛ ❝ã (x1 , , xd )M ∩ Hm0 (M ) = (x1 , , xd )Hm0 (M ) ♠➷t ❦❤➳❝ (x1 , , xd )Hm0 (M ) ⊆ mHm0 (M ) = ✈í✐ ❜✃t ❦ú ❤Ư t❤❛♠ sè x1 , , xd ❝ñ❛ M ✳ ❙✉② r❛ (x1 , , xd )M ∩ Hm0 (M ) = 0✳ ➜✐Ị✉ ♥➭② ❝ã ♥❣❤Ü❛ r➺♥❣ ♠ä✐ ❤Ư t❤❛♠ sè ❝đ❛ M ❧➭ tèt✱ ❞♦ ➤ã t❤❡♦ ➤Þ♥❤ ❧ý ❝❤Ý♥❤ ♥ã ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè✳ ✭✐✐✮⇒ ✭✐✮✳ ❱× ♠ä✐ ❤Ư t❤❛♠ sè ❝đ❛ M ❝ã tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè ♥➟♥ t❤❡♦ ➤Þ♥❤ ❧ý ❝❤Ý♥❤ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❤❛② t❛ ❝ã M/Hm0 (M ) ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❚❛ ❝ß♥ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ ❝❤ø♥❣ ♠✐♥❤ ♣❤➬♥ tö x1 mDt−1 = 0✳ ❚❤❐t ✈❐② ❣✐➯ sư ♥❣➢ỵ❝ ❧➵✐✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ∈ m s❛♦ ❝❤♦ x1 Dt−1 = ✈➭ dim M/x1 M = d−1✳ ❱× d ≥ ♥➟♥ t❛ ❝ã t❤Ó ❝❤ä♥ x2 ∈ m s❛♦ ❝❤♦ x2 Dt−2 = ✈➭ dim M/(x1 , x2 )M = d − 2✳ ❚❛ ❞Ơ t❤✃② r➺♥❣ ❞➲② ❝đ❛ mHm0 (M ) = 0✳❚❛ sÏ x1 , x2 ✈➭ x1 , x1 + x2 ❧➭ ❝➳❝ ♣❤➬♥ tư ❝đ❛ ❤Ư t❤❛♠ sè M ✳ ❉♦ ➤ã✱ t❤❡♦ ❣✐➯ t❤✐Õt ✈➭ ❜ỉ ➤Ị 2.1.3✭✐✮ t❛ ❝ã (x21 , x1 + x2 )M ∩ (x1 , (x1 + x2 )2 )M = (x1 , x1 + x2 )2 M = (x1 , x2 )2 M = (x21 , x2 )M ∩ (x1 , x22 )M ❱× M/Dt−1 ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ tõ ❜ỉ ➤Ị 1.2.11 ❝ã x1 Dt−1 = (x21 , x1 + x2 )Dt−1 ∩ (x1 , (x1 + x2 )2 )Dt−1 = (x21 , x2 )Dt−1 ∩ (x1 , x22 )Dt−1 = x21 Dt−1 ❚❤❡♦ ❜ỉ ➤Ị ◆❛❦❛②❛♠❛ t❛ ❝ã x1 Dt−1 = 0✳ ❙✉② r❛ mDt−1 = 0✳ ✷✽ ✷✳✷ ➜❛ t❤ø❝ ❍✐❧❜❡rt✲❙❛♠✉❡❧ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② P❤➬♥ tr➟♥ ➤➲ ❝❤♦ t❛ t❤✃② ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② M ❝ã t❤Ĩ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❜ë✐ tÝ♥❤ ❝❤✃t ♣❤➞♥ tÝ❝❤ t❤❛♠ sè ❝ñ❛ ❤Ö t❤❛♠ sè tèt ♥❤➢ t❤Õ ♥➭♦✱ tr♦♥❣ ♣❤➬♥ ♥➭② t❛ sÏ ❝❤Ø r❛ r➺♥❣ ✈í✐ ❤➭♠ ❍✐❧❜❡rt✲❙❛♠✉❡❧ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② t❤× Fq,M (n) = l(M/q n+1 M ) ❧➭ ♠ét ❜✐Ĩ✉ t❤ø❝ ➤➷❝ ❜✐Ưt ✈í✐ ❤Ư sè ❦❤➠♥❣ ➞♠✱ ♥ã ❝ã t❤Ĩ tÝ♥❤ t♦➳♥ ➤➢ỵ❝ ❜➺♥❣ ❧ä❝ ❝❤✐Ị✉ ✈➭ ❤➭♠ ♥➭② trï♥❣ ✈í✐ ➤❛ t❤ø❝ ❍✐❧❜❡rt✲❙❛♠✉❡❧ ♥➭♦ ❝đ❛ Pq,M (n) ✈í✐ ❜✃t ❦× ✐➤➟❛♥ t❤❛♠ sè tèt q M ✈➭ ✈í✐ ♠ä✐ n ≥ 1✳ ❍➡♥ ♥÷❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② M ❝ã t❤Ĩ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❜ë✐ ❜✐Ĩ✉ t❤ø❝ ♥➭② ❝đ❛ ❤➭♠ ❍✐❧❜❡rt✲❙❛♠✉❡❧✳ ❚r➢í❝ t✐➟♥ t❛ ❜➽t ➤➬✉ ❜➺♥❣ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ❤❛✐ ❜ỉ ➤Ị s❛✉✳ ✷✳✷✳✶ ❇ỉ ➤Ị✳ ❈❤♦ q ❧➭ ✐➤➟❛♥ t❤❛♠ sè tèt ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② M ✳ ❑❤✐ ➤ã qn M ∩ Di = qn Di ✈í✐ ∀n ≥ ✈➭ i = 0, , t✳ ❈❤ø♥❣ ♠✐♥❤✳ sè tèt ❝ñ❛ ❈❤♦ q ❧➭ ✐➤➟❛♥ t❤❛♠ sè tèt ❝ñ❛ M ✈➭ x1 , , xd ❧➭ ❤Ö t❤❛♠ m t❛ ❦ý ❤✐Ö✉ qM = (x1 , , xd )M ✳ ❱í✐ ∀n ≥ ✈➭ i = 0, , t t❛ ❧✉➠♥ ❝ã qn Di ⊆ qn M ∩ Di ✳ ❚❛ ❝ß♥ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ qn M ∩ Di ⊆ qn Di ✳ ❚❤❐t ✈❐② t❛ ❝ã qn M ∩ Di = [ q(α)M ] ∩ Di α∈Λd,n (q(α)M ∩ Di ) = α∈Λd,n α (xα1 , , xdidi )Di = α∈Λd,n ✷✾ ▼➷t ❦❤➳❝ t❛ ❧✉➠♥ ❝ã (β1 , , βdi , 1, , 1) ∈ Λd,n ✈í✐ ∀(β1 , , βdi ) ∈ Λdi ,n ✳ ❉♦ ➤ã t❤❡♦ ➤Þ♥❤ ❧ý 2.1.6 t❛ ❝ã α β (xβ1 , , xdidi )Di (xα1 , , xdidi )Di ⊆ α∈Λd,n (β1 , ,βdi )∈Λdi ,n = (x1 , , xdi )n Di ❙✉② r❛ qn M ∩ Di ⊆ (x1 , , xdi )n Di ⊆ qn Di ✳ ❱❐② t❛ ❝ã qn M ∩ Di = qn Di ✈í✐ ∀n ≥ ✈➭ i = 0, , t✳ ✷✳✷✳✷ ❇ỉ ➤Ị✳ ❈❤♦ q ❧➭ ✐➤➟❛♥ t❤❛♠ sè ❝ñ❛ ♠➠➤✉♥ M.❑❤✐ ➤ã l(M/qn+1 M ) ≤ n+d l(M/qM ) d ❍➡♥ ♥÷❛✱ ❜✃t ➤➻♥❣ t❤ø❝ trë t❤➭♥❤ ➤➻♥❣ t❤ø❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö q = (x1 , , xd ) ❧➭ ✐➤➟❛♥ t❤❛♠ sè ❝ñ❛ M ✳ ❚❛ ➤➷t ∞ N = (M/qM )[X1 , , Xd ] ✈➭ grq(M ) = qi M/qi+1 M ❦❤✐ ➤ã t❛ ❝ã t♦➭♥ i=0 ϕ : N −→ grq(M ) ①➳❝ ➤Þ♥❤ ❜ë✐ ϕ(Xi ) = xi = xi + q2 M ∈ qM/q2 M ✳ ➜➷t Q = Ker ϕ✳ ❚❤❡♦ ➤Þ♥❤ ❧ý ➤å♥❣ ❝✃✉ ♠➠➤✉♥ ❝ã N/Q ∼ = grq(M ) ✳ ●ä✐ ❝✃✉ J ❧➭ ✐➤➟❛♥ s✐♥❤ ❜ë✐ X1 , , Xd s✉② r❛ N/JN ∼ = M/qM ✈➭ M/qn M ∼ = N/J n N + Q✳ ❉♦ ➤ã l(M/qn+1 M ) = l(N/J n+1 N + Q) = l(N/J n+1 N ) − l(J n+1 N + Q/J n+1 N ) ≤ l(N/J n+1 N ) ✸✵ ♠➷t ❦❤➳❝ t❛ ❝ã n+d l(N/JN ) d n+d l(M/qM ) d l(N/J n+1 N ) = = ❙✉② r❛ n+d l(M/qM ) d l(M/qn+1 M ) ≤ ❍➡♥ ♥÷❛✱ ❜✃t ➤➻♥❣ t❤ø❝ trë t❤➭♥❤ ➤➻♥❣ t❤ø❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❤❛② ϕ ❧➭ ➤➻♥❣ ❝✃✉ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✷✳✷✳✸ ➜Þ♥❤ ❧ý✳ ❈❤♦ ✈➭ ➤➷t D : D0 ⊂ D1 ⊂ ⊂ Dt = M Di = Di /Di−1 ✈í✐ ♠ä✐ ❧➭ ❧ä❝ ❝❤✐Ị✉ ❝đ❛ M i = 1, , t, D0 = D0 ✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✭✐✐✮ ❱í✐ ❜✃t ❦ú ✐➤➟❛♥ t❤❛♠ sè tèt q ❝ñ❛ M ✱ ➤➻♥❣ t❤ø❝ t l(M/q n+1 M) = i=0 ➤ó♥❣ ✈í✐ ♠ä✐ n + di l(Di /qDi ) di n ≥ 0✳ ✭✐✐✐✮ ❚å♥ t➵✐ ✐➤➟❛♥ t❤❛♠ sè tèt q ❝ñ❛ M t l(M/q n+1 M) = i=0 s❛♦ ❝❤♦ ➤➻♥❣ t❤ø❝ n + di l(Di /qDi ) di ➤ó♥❣ ✈í✐ ♠ä✐ n ≥ 0✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮⇒ ✭✐✐✮✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ ➤é ❞➭✐ ❝❤✐Ị✉ t ❝đ❛ ❧ä❝ D ❝đ❛ M ❚r➢ê♥❣ ❤ỵ♣ ❜ỉ ➤Ị tr➟♥ t = ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥ ✈× M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♥➟♥ t❤❡♦ l(M/qn+1 M ) = n+d d l(M/qM ) = n+d0 d0 l(D0 /qD0 ) ✸✶ ●✐➯ sö t > 0✳ ❚❛ ❧✉➠♥ ❝ã ❞➲② ❦❤í♣ ♥❣➽♥ s❛✉ −→ qn+1 M +Dt−1 /qn+1 M −→ M/qn+1 M −→ M/qn+1 M +Dt−1 −→ ❚❤❡♦ ➤Þ♥❤ ❧ý ➤å♥❣ ❝✃✉ ♠➠➤✉♥ t❛ ❝ã qn+1 M + Dt−1 /qn+1 M ∼ = Dt−1 /qn+1 M ∩ Dt−1 2.2.1 t❛ ❝ã qn+1 M ∩ Dt−1 = qn+1 Dt−1 ♥➟♥ s✉② r❛ qn+1 M + Dt−1 /qn+1 M ∼ = Dt−1 /qn+1 Dt−1 ✳ ❉♦ ➤ã t❛ ❝ã ❞➲② ❦❤í♣ ♥❣➽♥ ▼➷t ❦❤➳❝ t❤❡♦ ❜ỉ ➤Ị −→ Dt−1 /qn+1 Dt−1 −→ M/qn+1 M −→ M/qn+1 M + Dt−1 −→ 0, s✉② r❛ t❛ ❝ã l(M/qn+1 M ) = l(Dt−1 /qn+1 Dt−1 ) + l(Dt /qn+1 Dt ) ❱× Dt−1 ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❧ä❝ ❝❤✐Ị✉ ❝đ❛ ♥ã ❝ã ➤é ❞➭✐ t−1 t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ t❛ ❝ã t−1 l(Dt−1 /q n+1 Dt−1 ) = i=0 ♠➷t ❦❤➳❝ n + di l(Di /qDi ) di Dt ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ❝❤✐Ị✉ d = dt ✱ t❛ ❝ã n+d l(Dt /qDt ) d l(Dt /qn+1 Dt ) = ❙✉② r❛ t l(M/q n+1 M) = i=0 ➤ó♥❣ ✈í✐ ♠ä✐ n ≥ 0✳ ✭✐✐✮⇒ ✭✐✐✐✮ ❧➭ ❤✐Ó♥ ♥❤✐➟♥✳ n + di l(Di /qDi ) di ✸✷ ✭✐✐✐✮⇒ ✭✐✮✳ ❱× ❞➲② s❛✉ ❧➭ ❦❤í♣ Dt−1 /qn+1 Dt−1 −→ M/qn+1 M −→ M/qn+1 M + Dt−1 −→ 0, ♥➟♥ t❛ ❝ã l(M/qn+1 M ) ≤ l(Dt−1 /qn+1 Dt−1 ) + l(Dt /qn+1 Dt ) ❉♦ ➤ã✱ tõ sù q✉② ♥➵♣ t❤❡♦ ➤é ❞➭✐ ❝ñ❛ ❧ä❝ ❝❤✐Ị✉ t❛ ❝ã t❤Ĩ ❝❤Ø r❛ r➺♥❣ t l(M/q n+1 l(Di /qn+1 Di ) M) ≤ i=0 ▼➷t ❦❤➳❝ t❤❡♦ ❜ỉ ➤Ị 2.2.2 ❝ã l(Di /qn+1 Di ) ≤ ✈í✐ n + di l(Di /qDi ) di ∀i = 0, , t✱ ♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt (iii) t❛ ❝ã t l(M/q n+1 M) ≤ t l(Di /q n+1 i=0 ❞♦ ➤ã l(Di /qDi ) = n+di di Di ) ≤ i=0 n + di l(Di /qDi ) di l(Di /qDi ) ✈í✐ ∀i = 0, , t✳ ❉♦ ✈❐② Di ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈í✐ ∀i = 0, , t ✭❝ò♥❣ t❤❡♦ ❜ỉ ➤Ị 2.2.2✮✳ ❉♦ ➤ã M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ✷✳✸ ❈❤♦ ❱Ý ❞ơ S ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❝❤Ý♥❤ q✉② ✈í✐ dim S = 3✱ m ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐ ❝đ❛ S ✈➭ ❣✐➯ sư m = (X, Y, Z) ✈í✐ X, Y, Z ∈ S ➜➷t R = S/(X, Y )∩(Z)✳ ●ä✐ x, y, z t➢➡♥❣ ø♥❣ ❧➭ ➯♥❤ ❝ñ❛ X, Y, Z tr♦♥❣ R✱ ➤å♥❣ t❤ê✐ ➤➷t Q = (x+z, y)✳ ❑❤✐ ➤ã t❛ ❝ã ✸✸ ✭✶✮ Qn = (x + z, y; α) ✈➭ lR (R/Qn ) = α∈Λ2,n ✭✷✮ ➜➷t b1 n2 +3n ✈í✐ ∀n ≥ = x + z ✈➭ b2 = x + y + z ❦❤✐ ➤ã Q = (b1 , b2 ) ✈➭ ✈í✐ ∀n ≥ t❤× n2 + 2n ♥Õ✉ n = 2q, q ∈ Z lR (R/ (b; α)) = (n + 1)2 α∈Λ2,n ♥Õ✉ n = 2q + 1, q ∈ Z ❞♦ ➤ã ❤➭♠ lR (R/ (b; α)) ❦❤➠♥❣ trï♥❣ ✈í✐ ♠ét ➤❛ t❤ø❝ ❝đ❛ n ♥➟♥ α∈Λ2,n n (b; α)]/Qn ) = ∞ (b; α) ✈í✐ n ≥ ♥➭♦ ➤ã ✈➭ supn>0 lR ([ Q = α∈Λ2,n α∈Λ2,n ❈❤ø♥❣ ♠✐♥❤✳ ✭✶✮ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ t❤✐Õt ❜➭✐ t♦➳♥ t❛ ❝ã dim R = 2✳ ❚❤❐t ✈❐②✱ tõ ❣✐➯ R/(X, Y ), R/(Z) ị í q ì ♠✐Ò♥ ♥❣✉②➟♥✱ s✉② r❛ (X, Y ), (Z) ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ✈❐② Ass R = Ass(S/(X, Y ) ∩ (Z)) = {(X, Y ), (Z)}✳ ❉♦ S/(0) ✈➭ S ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ♥➟♥ (0) ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ S ✳ ●✐➯ sư P0 ⊂ P1 ⊂ ⊂ Pd ❧➭ ❞➲② ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ S ❝❤ø❛ P ∈ Ass R t❤× ❧✉➠♥ ❝ã (0) ⊂ P1 ⊂ Pd ❧➭ ❞➲② ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ S ✳ ❱❐② dim R < dim S ❤➡♥ ♥÷❛ (Z) ⊂ (Z, X) ⊂ (Z, X, Y ) = m ❧➭ ♠ét ❞➲② ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤ø❛ (Z) ∈ Ass R ✈➭ ❝ã ➤é ❞➭✐ ❜➺♥❣ ✷ ✈❐② t❛ ❝ã dim R = ❚✐Õ♣ t❤❡♦✱ ➤➷t a1 = x + z, a2 = y, I = (z)✳ ➜Ó ❝❤ø♥❣ ♠✐♥❤ Qn = (a1 , a2 ; α) t❛ sÏ α∈Λ2,n ❝❤ø♥❣ ♠✐♥❤ ✭✐✮ ❱➭♥❤ R = S/(X, Y ) ∩ (Z) ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈í✐ ❧ä❝ (0) ⊂ (Z)/(X, Y ) ∩ (Z) ⊂ R✳ ✭✐✐✮ (a1 , a2 ) ❧➭ ❤Ư t❤❛♠ sè tèt ❝đ❛ R✳ ❚❤❐t ✈❐②✱ t❛ ❝ã ❉♦ R/I = S/((X, Y ) ∩ (Z))/(Z)/((X, Y ) ∩ (Z)) ∼ = S/(Z)✳ S ❧➭ ✈➭♥❤ ❝❤Ý♥❤ q✉② ✈➭ Z ❧➭ ♠ét ♣❤➬♥ ❝đ❛ ❤Ư t❤❛♠ sè ❝❤Ý♥❤ q✉② ♥➟♥ ✸✹ S/(Z) ❝ò♥❣ ❧➭ ✈➭♥❤ ❝❤Ý♥❤ q✉②✱ ✈❐② S/(Z) ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ dim R/I = ❤❛② R/I ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭✯✮✳ ❚✐Õ♣ t❤❡♦ t❛ ❝ã I = (Z)/((X, Y ) ∩ (Z)) ∼ = (X, Y, Z)/(X, Y ) ❞♦ X, Y, Z ❧➭ R− ❝❤Ý♥❤ q✉② ♥➟♥ Z ❧➭ S/(X, Y )− ❝❤Ý♥❤ q✉② tõ ➤➞② t❛ ❝ã S− ➤å♥❣ ❝✃✉ θ : S/(X, Y ) −→ S/(X, Y ) ①➳❝ ➤Þ♥❤ ❜ë✐ θ(u) = uZ Ker(θ) = AnnS/(X,Y ) (Z) = ✈➭ Im(θ) = Z(S/(X, Y )) = (X, Y, Z)/(X, Y ) = m/(X, Y ) ❙✉② r❛ S/(X, Y ) ∼ = I ✳ ▼➷t ❦❤➳❝ S/(X, Y ) ❧➭ ✈➭♥❤ ❈♦❤❡♥✲ = m/(X, Y ) ∼ ▼❛❝❛✉❧❛② ✈➭ dim S/(X, Y ) = ♥➟♥ I ❧➭ R− ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ dimR I = dim S/(X, Y ) = ✭✯✯✮✳ ❚õ ✭✯✮ ✈➭ ✭✯✯✮ s✉② r❛ ❚❛ ❝ã R ❧➭ ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ (a1 , a2 ) ❧➭ ❤Ö t❤❛♠ sè tèt ❝ñ❛ R✳ ❚❤❐t ✈❐②✱ ❣ä✐ a1 , a2 ❧➭ ➯♥❤ ❝ñ❛ a1 , a2 tr♦♥❣ R/(z)✳ ❚❛ ❝ã (a1 , a2 )R/(z) = (x+z, y, z)/(z) = (x, y, z)/(z) ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ R/(z)✱ ▼➷t ❦❤➳❝ dim R ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ ♥÷❛ t❛ ❝ã = dim R/(z) = ♥➟♥ (a1 , a2 ) R/(z)✈➭ ❝ò♥❣ s✉② r❛ (a1 , a2 ) ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ R ❤➡♥ a2 I = (yz) = ♥➟♥ (a1 , a2 ) ❧➭ ❤Ö t❤❛♠ sè tèt ❝đ❛ R✳ ❚õ ✭✐✮ ✈➭ ✭✐✐✮ t❤❡♦ ➤Þ♥❤ ❧ý 2.1.5 t❛ ❝ã Qn = (a1 , a2 ; α) α∈Λ2,n ❈✉è✐ ❝ï♥❣ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ lR (R/Qn ) t❛ t❤✃② ♥Õ✉ = n2 +3n , ∀n ≥ ❚r➢í❝ ❤Õt ϕ : M −→ N ❧➭ R− ➤å♥❣ ❝✃✉ ♠➠➤✉♥ t❤× t❛ ❝ã ❞➲② ❦❤í♣ −→ M/ Ker ϕ −→ N −→ N/ Im ϕ −→ ✈í✐ α : M/ Ker ϕ −→ N ①➳❝ ➤✐♥❤ ❜ë✐ α(m + Ker ϕ) = ϕ(m) ✈➭ β ❧➭ t♦➭♥ ❝✃✉ tù ♥❤✐➟♥✳ ❚❛ ❝ã R− ➤å♥❣ ❝✃✉ ♠➠➤✉♥ ϕ : I −→ R/(al1 , am ) tr♦♥❣ ➤ã ϕ = pi ✈í✐ i : I −→ R ❧➭ ➤➡♥ ❝✃✉ ❝❤Ý♥❤ t➽❝ ✈➭ p : R −→ R/(al1 , am ) ❧➭ t♦➭♥ ❝✃✉ tù ♥❤✐➟♥ ✈❐② t❛ ❝ã ❞➲② ❦❤í♣ l m −→ I/ Ker ϕ −→ R/(al1 , am ) −→ (R/(a1 , a2 ))/ Im ϕ −→ ✸✺ m l ị ĩ tr tì t ó Im ϕ = (I + (al1 , am ))/(a1 , a2 ) ✈➭ Ker ϕ = I ∩ (al , am ) ❉♦ R/I ∼ = S/(Z) ♥➟♥ t❛ ❝ã ✈í✐ l m ∼ (R/(al1 , am ))/ Im ϕ = R/(I + (a1 , a2 )) ∼ = (S/Z)/((X + Z)l , Y m , Z)S/(Z) ∼ = S/((X + Z)l , Y m , Z) = S/(X l , Y m , Z) m l Ker ϕ = I ∩ (al1 , am ) ♥❤➢♥❣ (a1 , a2 ) ❧➭ R/I− ❝❤Ý♥❤ q✉② ♥➟♥ I ∩ (al , am ) = (al , am )I ✳ ▼➷t ❦❤➳❝ I ∼ = S/(X, Y ) ♥➟♥ ❚❛ ❝ã 2 I/ Ker ϕ = I/(al1 , am )I ∼ = S/(X, Y )/((X + Z)l , Y m )S/(X, Y ) ∼ = S/((X + Z)l , Y m , X, Y ) = S/(X, Y, Z l ) ❱❐② t❛ ❝ã ❞➲② ❦❤í♣ l m −→ S/(X, Y, Z l ) −→ R/(al1 , am ) −→ S/(X , Y , Z) −→ 0, ∀l, m ≥ 1✳ ❉♦ ➤ã t❛ ❝ã l l m lR (R/(al1 , am )) = lR (S/(X, Y, Z )) + lR (S/(X , Y , Z)) = e(X, Y, Z l ; S) + e(X l , Y m , Z; S) = l.e(X, Y, Z; S) + ml.e(X, Y, Z; S) = l(m + 1) ✸✻ ❱❐② t❛ ❝ã lR (R/Qn ) = lR (R/ (a1 , a2 ; α)) α∈Λ2,n n n−1 lR (R/(an+1−i , ai2 )) = i lR (R/(an−i , a2 )) − i=1 n i=1 n−1 (n + − i)(i + 1) − = i=1 n = (n − 1)(i + 1) i=1 (i + 1) i=1 n + 3n ✭✷✮ ❉Ô t❤✃② Q = (x + z, x + y + z) = (x + z, y) ❤❛② Q = (b1 , b2 )✳ ●ä✐ = b1 , b2 ❧➭ ➯♥❤ ❝ñ❛ b1 , b2 tr♦♥❣ R/(z)✳ ❑❤✐ ➤ã ❝ã (b1 , b2 )R/(z) = (x + z, x + y + z, z)/(z) = (x, y, z)/(z) ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❝ñ❛ t❤❛♠ sè ❝ñ❛ R/(z) ♥➟♥ b1 , b2 ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ R/(z) ❤❛② ❧➭ ❤Ö R/I ✳ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ✭✶✮ ✈í✐ ϕ = pi tr♦♥❣ ➤ã i : I −→ R ❧➭ ➤➡♥ ❝✃✉ ❝❤Ý♥❤ t➽❝ ✈➭ p : R −→ R/(bl1 , bm ) ❧➭ t♦➭♥ ❝✃✉ tù ♥❤✐➟♥ t❛ ❝ã l m (R/(bl1 , bm ))/ Im ϕ = R/(I + (b1 , b2 )) ∼ = (S/Z)/((X + Z)l , (X + Y + Z)m , Z)S/(Z) ∼ = S/(X l , (X + Y )m , Z) ▼➷t ❦❤➳❝ l m Ker ϕ = I ∩ (bl1 , bm ) = (b1 , b2 )I ✈➭ I/ Ker ϕ = I/(bl1 , bm )I ∼ = S/(X, Y )/((X + Z)l , (X + Y + Z)m , X, Y )S/(X, Y ) ∼ = S/(X, Y, (Z l , Z m )) ✸✼ ❱❐② t❛ ❝ã ❞➲② ❦❤í♣ ♥❣➽♥ l m −→ S/(X, Y, (Z l , Z m )) −→ R/(bl1 , bm ) −→ S/(X , (X+Y ) , Z) −→ ❉♦ ➤ã l m l m lR (R/(bl1 , bm )) = lR (S/(X, Y, (Z , Z ))) + lR (S/(X , (X + Y ) , Z)) = e(X, Y, (Z l , Z m ); S) + e(X l , (X + Y )m , Z; S) ❱❐② lR (R/(bl1 , bm )) ❝ã lR (R/ = lm + min{l, m}✳ ❑❤✐ ➤ã t❤❡♦ ❬✹✱ ▼Ư♥❤ ➤Ị ✹✳✸❪ t❛ (b1 , b2 ; α)) = α∈Λ2+n n n−1 lR (R/(b1n+1−i , bi2 )) = i lR (R/(bn−i , b2 )) − i=1 n i=1 n−1 (n + − i)i + min{n + − i, i} − = i=1 (n − i)i + min{n − i, i} i=1 n−1 = n + + (n − 1)n/2 + (min{n + − i, i} − min{n − i, i}) i=1 ◆Õ✉ n ❝❤➼♥ tø❝ ❧➭ n = 2q, q ∈ Z t❤× lR (R/ (b1 , b2 ; α)) = α∈Λ2+n ◆Õ✉ n ❧❰ tø❝ ❧➭ n = 2q + 1, q ∈ Z t❤× lR (R/ (b1 , b2 ; α)) = α∈Λ2+n ❚õ ➤ã t❛ t❤✃② lR (R/ n2 +n (n+1)2 (b1 , b2 ; α)) ❦❤➠♥❣ ♣❤➯✐ ❧➭ ➤❛ t❤ø❝ ❝ñ❛ n ♠➭ t❛ α∈Λ2+n ❜✐Õt ♣❤➯✐ tå♥ t➵✐ sè tù ♥❤✐➟♥ t❤ø❝ ➮♥ N ➤đ ❧í♥ s❛♦ ❝❤♦ lR (R/Qn ) trï♥❣ ✈í✐ ♠ét ➤❛ (b1 , b2 ; α) = Qn ✳ ❈❤♦ n = 2q, q ≥ n ✈í✐ ∀n ≥ N ✳ ❉♦ ➤ã α∈Λ2,n ✸✽ t❤× t❛ ❝ã (b1 , b2 ; α)/Qn ) = lR (Qn ) − lR (R/ lR (R/ α∈Λ2,n (b1 , b2 ; α)) α∈Λ2,n n2 + 3n n2 + 2n − 2 n = = q = ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá supn>0 lR ([ α∈Λ2,n (b; α)]/Qn ) < ∞ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❲✳ ❇r✉♥s ❛♥❞ ❏✳ ❍❡r③♦❣✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡r✲ s✐t② Pr❡ss✳ ❬✷❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❉✳ ❚✳ ❈✉♦♥❣✱ ❖♥ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✲ ✉❧❡s✱ ❬✸❪ ❑♦❞❛✐ ▼❛t❤✳ ❏✱ ✸✵ ✭✷✵✵✼✮✱ ✹✵✾✲✹✷✽✳ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ❍✳ ▲✳ ❚r✉♦♥❣✱ P❛r❛♠❡tr✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣♦✇❡rs ♦❢ ♣❛r❛♠❡t❡r ✐❞❡❛❧s ❛♥❞ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱ t♦ ❛♣♣❡❛r ✐♥ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✷✵✵✽✳ ❬✹❪ ❙✳ ●♦t♦ ❛♥❞ ❨✳ ❙❤✐♠♦❞❛✱ P❛r❛♠❡tr✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣♦✇❡rs ♦❢ ✐❞❡❛❧s ✈❡rs✉s r❡❣✉❧❛r✐t② ♦❢ s❡q✉❡♥❝❡s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✸✷ ✭✷✵✵✸✮✱ ✷✷✾✲✷✸✸✳ ❬✺❪ ❙✳ ●♦t♦ ❛♥❞ ❨✳ ❙❤✐♠♦❞❛ ❖♥ t❤❡ ♣❛r❛♠❡tr✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣♦✇❡rs ♦❢ ♣❛r❛♠❡t❡r ✐❞❡❛❧s ✐♥ ❛ ◆♦❡t❤❡r✐❛♥ ❧♦❝❛❧ r✐♥❣✱ ❚♦❦②♦ ❏✳ ▼❛t❤✱ ✷✼ ✭✷✵✵✹✮✱ ✶✷✺✲✶✸✹✳ ❬✻❪ ❲✳ ❍❡✐♥③❡r✱ ▲✳ ❏✳ ❘❛t❧✐❢❢ ❛♥❞ ❑✳ ❙❤❛❤✱ P❛r❛♠❡tr✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♠♦♥♦♠✐❛❧ ✐❞❡❛❧s ✭■✮✱ ❬✼❪ ❍♦✉st♦♥ ❏✳ ▼❛t❤✳✱ ✷✶ ✭✶✾✾✺✮✱ ✷✾✲✺✷✳ ❍✳ ▼❛ts✉♠✉r❛✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✽✻✳ ❬✽❪ ❘✳ ❨✳ ❙❤❛r♣✱ ❙t❡♣s ✐♥ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✽✵✳ ✸✾ ... x = x1 , , xd ❧➭ ❤Ư t❤❛♠ sè ❝đ❛ M ✈➭ q = (x1 , , xd ) ❧➭ ✐➤➟❛♥ t❤❛♠ sè ❝ñ❛ M s✐♥❤ ❜ë✐ x ỗ số n ý ệ d d Λd,n = {(α1 , , αd ) ∈ Z | αi ≥ 1, ∀1 ≤ i ≤ d, αi = d + n − 1} i=1 q(α) = (xα1