Tài liệu Tóm tắt luận án tiến sĩ toán học tính chẻ ra của môđun đối đồng điều địa phương và ứng dụng

VIỆN KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM VIỆN TOÁN HỌC -----oOo----- Phạm Hùng Quý TÍNH CHẺ RA CỦA MÔĐUN Đ I Đ NG ĐI U ĐỊA PHƯƠNG VÀ ỨNG DỤNG Chuyên ngành: Đại s và lý thuy t s Mã s : 62. 46. 01. 04 TÓM TẮT LUẬN ÁN TI N SĨ TOÁN HỌC HÀNỘI-2013 Công trình được hoàn thành tại: Viện Toán học, Viện khoa học và Công nghệ Việt Nam Tập thể hướng dẫn khoa học: GS. TSKH. Nguyễn Tự Cường Phản biện 1: Phản biện 2: Phản biện 3: Luận án đã được bảo vệ trước Hội đồng chấm luận án cấp Viện họp tại: Viện Toán học – Viện Khoa học và Công nghệ Việt Nam vào hồi …… giờ ngày …… tháng …… năm 201…. Có thể tìm hiểu luận án tại: - Thư viện Quốc gia - Thư viện Viện Toán học ✶ ▼ë ➤➬✉ ❚Ý♥❤ ❝❤❰ r❛ ❝ñ❛ ❝➳❝ ❞➲② ❦❤í♣ ♥❣➽♥ ❧✉➠♥ ➤➢î❝ ❝❤ó ý tr♦♥❣ ➜➵✐ sè ➜å♥❣ ➤✐Ò✉✳ ❇ë✐ ❦❤✐ ➤ã ❝✃✉ tró❝ ❝ñ❛ ❝➳❝ t❤➭♥❤ ♣❤➬♥ tr♦♥❣ ♥ã trë ♥➟♥ râ r➭♥❣ ❤➡♥✳ ❉♦ ➤ã ♥❣➢ê✐ t❛ t❤➢ê♥❣ ❝è ❣➽♥❣ ➤➷❝ t➯ ✈➭ ♣❤➳t ❤✐Ö♥ tÝ♥❤ ❝❤✃t ♥➭②✳ ❇➯♥ ❧✉❐♥ ➳♥ ♥➭② q✉❛♥ t➞♠ ➤Õ♥ tÝ♥❤ ❝❤✃t ❝❤❰ r❛ ❝ñ❛ ❞➲② ❦❤í♣ ♥❣➽♥ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❚r♦♥❣ t♦➭♥ ❜é ❧✉❐♥ ➳♥ t❛ ❧✉➠♥ ①Ðt ♠ét ✈➭♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦➳♥ ❝ã ➤➡♥ ✈Þ✳ ❳Ðt R ❧➭ a ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R✳ ❍➭♠ tö ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ Hai (•) ✈í✐ ❣✐➳ a S i ❤➭♠ tö ①♦➽♥ Γa(•) Γa (M ) = 0 :M a∞ = n≥1 (0 :M an ) ➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ❧➭ ❤➭♠ tö ❞➱♥ s✉✃t ♣❤➯✐ t❤ø ✈í✐ M ❝ñ❛ ❧➭ ♠ét ✱ ë ➤➞② R✲♠➠➤✉♥✳ ▲Ý t❤✉②Õt ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ➤➢î❝ ❣✐í✐ t❤✐Ö✉ ❜ë✐ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✈➭♦ ♥❤÷♥❣ ♥➝♠ ✶✾✻✵✳ ❇ë✐ tÝ♥❤ ❧✐♥❤ ❤♦➵t tr♦♥❣ sö ❞ô♥❣ ❝ï♥❣ ✈í✐ ❦❤➯ ♥➝♥❣ ➤➷❝ t➯ ♥❤✐Ò✉ ❝✃✉ tró❝ t♦➳♥ ❤ä❝ ❝ñ❛ ♥ã✱ ♥❣➭② ♥❛② ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ô q✉❛♥ trä♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥❤✐Ò✉ ❧Ý t❤✉②Õt t♦➳♥ ❤ä❝ tr♦♥❣ ➤ã ❝ã ➜➵✐ sè ●✐❛♦ ❤♦➳♥✳ ▼ét ❦Ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ q✉❛♥ trä♥❣ tr♦♥❣ ➜➵✐ sè ●✐❛♦ ❤♦➳♥ ❧➭ ❝❤ä♥ ♠ét ♣❤➬♥ tö ❝❤Ý♥❤ q✉② M x ∈ a ❝ñ❛ ✈➭ ①Ðt ❞➲② ❦❤í♣ ♥❣➽♥ x 0 → M → M → M/xM → 0. ❚➳❝ ➤é♥❣ ❤➭♠ tö ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ Hai (•) ✈➭♦ ❞➲② ❦❤í♣ tr➟♥ t❛ t❤✉ ➤➢î❝ ❞➲② ❦❤í♣ ❞➭✐ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ s❛✉ · · · → Hai (M ) → Hai (M ) → Hai (M/xM ) → Hai+1 (M ) → · · · . ❚r♦♥❣ ❧✉❐♥ ➳♥ ♥➭② ❝❤ó♥❣ t➠✐ t×♠ ➤✐Ò✉ ❦✐Ö♥ ➤Ó ❞➲② ❦❤í♣ ❞➭✐ tr➟♥ ❝❤♦ t❛ ♥❤÷♥❣ ❞➲② ❦❤í♣ ♥❣➽♥ 0 → Hai (M ) → Hai (M/xM ) → Hai+1 (M ) → 0, ✈➭ ❦❤✐ ♥➭♦ t❤× ❞➲② ❦❤í♣ ♥❣➽♥ ♥➭② ❧➭ ❝❤❰ r❛✱ tø❝ ❧➭ t❛ ❝ã Hai (M/xM ) ∼ = Hai (M ) ⊕ Hai+1 (M ). ✷ ➜é♥❣ ❧ù❝ ❝❤♦ ✈✐Ö❝ ①❡♠ ①Ðt tÝ♥❤ ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❝ñ❛ ❜➯♥ ❧✉❐♥ ➳♥ ♥➭② ①✉✃t ♣❤➳t tõ ❝➞✉ ❤á✐ ➤➷t ❞➢í✐ ➤➞② ✈Ò ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❈➞✉ ❤á✐ ✶✳ ❈❤♦ (R, m) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ M ▼❛❝❛✉❧❛② s✉② ré♥❣ ❝❤✐Ò✉ ❞➢➡♥❣ n d > 0✳ ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲ ❑❤✐ ➤ã ♣❤➯✐ ❝❤➝♥❣ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ♣❤➬♥ tö t❤❛♠ sè x ❝ñ❛ M ❝❤ø❛ tr♦♥❣ mn t❛ ❝ã Hmi (M/xM ) ∼ = Hmi (M ) ⊕ Hmi+1 (M ) ✈í✐ ♠ä✐ i < d − 1❄ ❈➞✉ ❤á✐ tr➟♥ ❝ã t❤Ó ➤➢î❝ ①❡♠ ①Ðt ❞➢í✐ ❞➵♥❣ ♠➵♥❤ ❤➡♥ ❝❤♦ ✐➤➟❛♥ ➤✐Ò✉ ❦✐Ö♥ t Hai (M ) ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ ♠ä✐ ♥➭♦ ➤ã✳ ◆❤➽❝ ❧➵✐ r➺♥❣ x∈a i a ❜✃t ❦× ✈í✐ ♥❤á ❤➡♥ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ❧➭ ♠ét ♣❤➬♥ tö a✲❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ M ♥Õ✉ x∈ / p ✈í✐ ♠ä✐ p ∈ AssM, a * p✳ ❈➞✉ ❤á✐ ✷✳ R✲♠➠➤✉♥ ❈❤♦ a ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ ✈➭♥❤ ◆♦❡t❤❡r ❤÷✉ ❤➵♥ s✐♥❤✳ ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ ♠ä✐ ❳Ðt t i < t✳ R ✭❜✃t ❦×✮ ✈➭ M ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ ❧➭ ♠ét Hai (M ) ❑❤✐ ➤ã ♣❤➯✐ ❝❤➝♥❣ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ n s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ♣❤➬♥ tö a✲❧ä❝ ❝❤Ý♥❤ q✉② x ❝ñ❛ ❝❤ø❛ tr♦♥❣ an H i (M/xM ) ∼ = H i (M ) ⊕ H i+1 (M ) ✈í✐ ♠ä✐ i < t − 1❄ ❞➢➡♥❣ a a t❛ ❝ã a ▲✉❐♥ ➳♥ ➤➢î❝ ❝❤✐❛ ❧➭♠ ❜è♥ ❝❤➢➡♥❣✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✶ ❝ñ❛ ❧✉❐♥ ➳♥ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ❝➞✉ tr➯ ❧ê✐ ➤➬② ➤ñ ❝❤♦ ❝➳❝ ❝➞✉ ❤á✐ tr➟♥ ✭➜Þ♥❤ ❧Ý ✶✳✹✳✹✱ ❍Ö q✉➯ ✶✳✹✳✺✮✳ ❍➡♥ ♥÷❛✱ ❝❤ó♥❣ t➠✐ ❝ß♥ tr×♥❤ ❜➭② ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤Þ♥❤ ❧Ý ❝❤❰ r❛ ❞➢í✐ ❞➵♥❣ tæ♥❣ q✉➳t ➤Ó ➳♣ ❞ô♥❣ ✈➭♦ ♥❤✐Ò✉ ❤♦➭♥ ❝➯♥❤ ❦❤➳❝ ♥❤❛✉✳ ❈❤ó♥❣ t➠✐ ❝♦✐ ♠ç✐ ❞➲② ❦❤í♣ ♥❣➽♥ 0 → A → B → C → 0 ♥❤➢ ❧➭ ➤➵✐ ❞✐Ö♥ ❝ñ❛ ♠ét ♣❤➬♥ tö tr♦♥❣ ♠➠➤✉♥ ♠ë ré♥❣ Ext1R (C, A)✳ ➜Ó ❝❤ø♥❣ ♠✐♥❤ ♠ét ❞➲② ❦❤í♣ ♥❣➽♥ ❧➭ ❝❤❰ r❛ t❛ ❝❤ø♥❣ ♠✐♥❤ ♥ã ❧➭ ➤➵✐ ❞✐Ö♥ ❝ñ❛ ♣❤➬♥ tö ❦❤➠♥❣ ❝ñ❛ Ext1R (C, A)✳ ❈➳❝ ➜Þ♥❤ ❧Ý ✶✳✸✳✸ ✈➭ ✶✳✸✳✹ ➤ã♥❣ ✈❛✐ trß q✉②Õt ➤Þ♥❤ tr♦♥❣ t✃t ❝➯ ❝➳❝ ➤Þ♥❤ ❧Ý ❝❤❰ r❛ ❝ñ❛ ❝❤ó♥❣ t➠✐✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✷✱ ❝❤ó♥❣ t➠✐ ➳♣ ❞ô♥❣ tÝ♥❤ ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ✸ ♣❤➢➡♥❣ ➤Ó ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t æ♥ ➤Þ♥❤ ❝ñ❛ ❤Ö t❤❛♠ sè tèt ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✭❝➳❝ ➜Þ♥❤ ❧Ý ✷✳✷✳✺ ✈➭ ✷✳✷✳✽✮✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✸✱ ❝❤ó♥❣ t➠✐ ❧✉➠♥ ①Ðt ✈➭♥❤ ❝➡ së (R, m) ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ ♠ét ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ➤Þ❛ ♣❤➢➡♥❣✳ ❈❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ t❤❡♦ ❝➳❝ ♣❤➬♥ tö t❤❛♠ sè x ∈ b(M )3 ✱ ë ➤➞② b(M ) = ∩dx;i=1 Ann(0 : xi )M/(x1 ,...,xi−1 )M , ✈í✐ x = x1 , ..., xd ❝❤➵② tr♦♥❣ t✃t ❝➯ ❝➳❝ ❤Ö t❤❛♠ sè ❝ñ❛ M ✭➜Þ♥❤ ❧Ý ✸✳✷✳✹✮✳ ➜Þ♥❤ ❧Ý ✸✳✷✳✹ ❧➭ ❤÷✉ Ý❝❤ ❦❤✐ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ♠➠➤✉♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❱í✐ ♠ç✐ M ❝ã ❝❤✐Ò✉ ♥❤á ❤➡♥ ❤✐Ö✉ ❧➭ UM (0)✳ dim M t❤á❛ ♠➲♥ ❧➭ t❛ ❣ä✐ ♠➠➤✉♥ ❝♦♥ ❧í♥ ♥❤✃t ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ M ✈➭ ❦Ý ❑❤✐ ➤ã ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ tå♥ t➵✐ ♠ét ❞➲② ♠➠➤✉♥ Ui (M ), 0 ≤ i ≤ d − 1, M R✲♠➠➤✉♥ M ✱ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ö t❤❛♠ sè xi ∈ b(M/(xi+1 , ..., xd )M )3 ✈í✐ ♠ä✐ x = x1 , ..., xd i ≤ d t❛ ❝ã ❝ñ❛ Ui (M ) ∼ = UM/(xi+2 ,...,xd )M (0) ✈í✐ ♠ä✐ 0 ≤ i ≤ d − 1 ✭➜Þ♥❤ ❧Ý ✸✳✷✳✾✮✳ ❚õ ❞➲② ♠➠➤✉♥ ♥➭② ❝❤ó♥❣ t➠✐ ➤➲ ①➞② ❞ù♥❣ ❦❤➳✐ ♥✐Ö♠ ❜❐❝ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ♠➠➤✉♥ ✭➜Þ♥❤ ♥❣❤Ü❛ ✸✳✸✳✼✮✳ ➜å♥❣ t❤ê✐ ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❜❐❝ ❦❤➠♥❣ tré♥ ❧➱♥ ❧➭ ♠ét ❧♦➵✐ ❜❐❝ ♠ë ré♥❣ t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❲✳ ❱❛s❝♦♥❝❡❧♦s ✭➜Þ♥❤ ❧Ý ✸✳✸✳✽✱ ▼Ö♥❤ ➤Ò ✸✳✸✳✾✱ ➜Þ♥❤ ❧Ý ✸✳✸✳✶✼✮✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✹✱ ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ❤÷✉ ❤➵♥ ❝ñ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ➤➬✉ t✐➟♥ ❦❤➠♥❣ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ ❝ã t❐♣ ❣✐➳ ✈➠ ❤➵♥ ✭➜Þ♥❤ ❧Ý ✹✳✶✳✽✮✳ ❈❤ó♥❣ t➠✐ ❝ò♥❣ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❤÷✉ ❤➵♥ ❝ñ❛ ♠ét sè t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝❤✐Ò✉ ❤÷✉ ❤➵♥ ❝ñ❛ M t➢➡♥❣ ø♥❣ ✈í✐ a ✭➜Þ♥❤ ❧Ý ✹✳✷✳✾✮✳ ❈➳❝ ❦Õt q✉➯ t❤✉é❝ ❝➳❝ ❈❤➢➡♥❣ ✶✱ ✷✱ ✈➭ ✹ ➤➢î❝ ✈✐Õt t❤➭♥❤ ❜è♥ ❜➭✐ ❜➳♦ ➤➲ ➤➢î❝ ➤➝♥❣ ✈➭ ♥❤❐♥ ➤➝♥❣ t➵✐ ✵✸ t➵♣ ❝❤Ý tr♦♥❣ ❞❛♥❤ s➳❝❤ ❙❈■ ✈➭ ✵✶ t➵♣ ❝❤Ý tr♦♥❣ ❞❛♥❤ s➳❝❤ ❙❈■❊✳ ❈➳❝ ❦Õt q✉➯ tr♦♥❣ ❈❤➢➡♥❣ ✸ sÏ ➤➢î❝ t➳❝ ❣✐➯ t✐Õ♣ tô❝ ♣❤➳t tr✐Ó♥ tr♦♥❣ t➢➡♥❣ ❧❛✐✳ ✹ ❈❤➢➡♥❣ ✶ ❚Ý♥❤ ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ▼ô❝ t✐➟✉ ❝ñ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ ①➞② ❞ù♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ✈➭ ➤➢❛ r❛ ♠ét ➤Þ♥❤ ❧Ý ❝❤❰ r❛ ❝ñ❛ ❞➲② ❦❤í♣ ♥❣➽♥ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❝ï♥❣ ✈í✐ ♠ét sè ➳♣ ❞ô♥❣ trù❝ t✐Õ♣ ❝ñ❛ ♥ã✳ ❑Ü t❤✉❐t ❞ï♥❣ ➤Ó ❝❤ø♥❣ ♠✐♥❤ ❧➭ ①❡♠ ♠ç✐ ❞➲② ❦❤í♣ ♥❣➽♥ ♥❤➢ ➤➵✐ ❞✐Ö♥ ❝ñ❛ ♠ét ♣❤➬♥ tö ❝ñ❛ ♠➠➤✉♥ ♠ë ré♥❣ Ext1R (•, •)✳ ❑❤✐ ➤ã ♠ét ❞➲② ❦❤í♣ ♥❣➽♥ ❧➭ ❝❤❰ r❛ ♥Õ✉ ♥ã ➤➵✐ ❞✐Ö♥ ❝❤♦ ♣❤➬♥ tö 0 ❝ñ❛ Ext1R (•, •)✳ ▼è✐ ❧✐➟♥ ❤Ö ❣✐÷❛ tæ♥❣ ✈➭ tÝ❝❤ ❝ñ❛ ❝➳❝ ♣❤➬♥ tö tr♦♥❣ R ✈➭ ❝➳❝ ❞➲② ❦❤í♣ ♥❣➽♥ t➢➡♥❣ ø♥❣ sÏ ➤➢î❝ tr×♥❤ ❜➭② tr♦♥❣ ❚✐Õt ✶✳✸✳ ❚r♦♥❣ ❤❛✐ t✐Õt ➤➬✉ ❝❤ó♥❣ t➠✐ ♥❤➽❝ ❧➵✐ ✈Ò ♠ét sè ❦Õt q✉➯ ❝➡ së ✈Ò ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣✱ ✈➭ ♣❤Ð♣ t♦➳♥ tr♦♥❣ ♠➠➤✉♥ ♠ë ré♥❣ ✶✳✶ Ext1R (•, •)✳ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❚r♦♥❣ t✐Õt ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♥❤÷♥❣ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ❝ñ❛ ❧Ý t❤✉②Õt ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ t❤❡♦ ❝✉è♥ s➳❝❤ ✈Ò ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❝ñ❛ ▼✳ ❇r♦❞♠❛♥♥ ✈➭ ❘✳❨✳ ❙❤❛r♣✳ ✶✳✷ P❤Ð♣ t♦➳♥ tr♦♥❣ ♠➠➤✉♥ Ext(C, A) ❚r♦♥❣ t✐Õt ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ✈Ò ❝✃✉ tró❝ ❝ñ❛ ♠➠➤✉♥ ♠ë ré♥❣ t❤❡♦ ❝✉è♥ s➳❝❤ ✧❍♦♠♦❧♦❣②✧ ❝ñ❛ ❙✳ ▼❛❝▲❛♥❡✳ ✶✳✸ ▼➠➤✉♥ Ext(Hai+1(M ), Hai (M )) ❚r♦♥❣ t✐Õt ♥➭② ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ❦Ü t❤✉❐t ❝❤Ý♥❤ ➤Ó ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❝ñ❛ ❝❤ó♥❣ t➠✐✳ ❳Ðt ❤➵♥ s✐♥❤ ✈➭ M ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ a ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R✳ ❳Ðt t ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ✈➭ U ❧➭ ♠ét ✺ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ ❦✐Ö♥ M ✳ ➜➷t M = M/U ✳ ❚❛ ♥ã✐ ♠ét ♣❤➬♥ tö x ❧➭ t❤á❛ ♠➲♥ ➤✐Ò✉ (♯) ♥Õ✉ 0 :M x = U ✱ ✈➭ ❞➲② ❦❤í♣ ♥❣➽♥ x 0 → M → M → M/xM → 0 ❝➯♠ s✐♥❤ ❝➳❝ ❞➲② ❦❤í♣ ♥❣➽♥ 0 → Hai (M ) → Hai (M/xM ) → Hai+1 (M ) → 0 ✈í✐ ♠ä✐ i < t − 1✳ ◆Õ✉ x ❧➭ ♠ét ♣❤➬♥ tö t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ (♯) t❤× t❛ ❦Ý Exi ❧➭ ♣❤➬♥ tö tr♦♥❣ Ext(Hai+1 (M ), Hai (M )) ➤➵✐ ❞✐Ö♥ ❜ë✐ ❞➲② ❦❤í♣ ♥❣➽♥ t tr➟♥✳ ❍➡♥ ♥÷❛ ♥Õ✉ H (M ) ∼ = H t (M )✱ t❛ ❝ã ❞➲② ❦❤í♣ ♥❣➽♥ s❛✉ ❤✐Ö✉ a a 0 → Hat−1 (M ) → Hat−1 (M/xM ) → 0 :Hat (M ) x → 0. ❳Ðt b ❧➭ ♠ét ✐➤➟❛♥ s❛♦ ❝❤♦ x ∈ b✳ Ext(0 :Hat (M ) b, 0 :Hat−1 (M ) b) ❚❛ ❣ä✐ Fxt−1 ❧➭ ♣❤➬♥ tö tr♦♥❣ ♠➠➤✉♥ ➤➵✐ ❞✐Ö♥ ❜ë✐ ❞➲② ❦❤í♣ ♥❣➽♥ ❞➢í✐ ➤➞② ♥Õ✉ ♥ã tå♥ t➵✐ 0 → 0 :Hat−1 (M ) b → 0 :Hat−1 (M/xM ) b → 0 :Hat (M ) b → 0. ❱í✐ ♥❤÷♥❣ ❦Ý ❤✐Ö✉ ♥➟✉ tr➟♥ ❝❤ó♥❣ t➠✐ ➤➲ ❝❤Ø sù ❧✐➟♥ ❤Ö ♠❐t t❤✐Õt ❣✐÷❛ tæ♥❣ ✈➭ tÝ❝❤ ❝ñ❛ ❝➳❝ ♣❤➬♥ tö t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ (♯) ✈➭ ❝➳❝ ♠ë ré♥❣ t➢➡♥❣ ø♥❣ ♥❤➢ tr♦♥❣ ❤❛✐ ➤Þ♥❤ ❧Ý s❛✉✳ ➜Þ♥❤ ❧ý ✶✳✸✳✸✳ ❈❤♦ t ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ✈➭ M ✳ ➜➷t M = M/U ✳ ●✐➯ sö x ✈➭ y U ❧➭ ♠ét ♠➠➤✉♥ ❝♦♥ ❝ñ❛ ❧➭ ❝➳❝ ♣❤➬♥ tö t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ (♯) ✈➭ 0 :M (x + y) = U ✱ ❦❤✐ ➤ã ✭✐✮ ✭✐✐✮ i x + y ❝ò♥❣ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ (♯) ✈➭ Ex+y = Exi + Eyi ✈í✐ ♠ä✐ i < t − 1✳ ◆Õ✉ Hat (M ) ∼ = Hat (M ) ➤Þ♥❤ ✈➭ ➜➷t Fxt−1 , Fyt−1 ❧➭ ①➳❝ ➤Þ♥❤✱ t❤× t−1 Fx+y ❝ò♥❣ ①➳❝ t−1 Fx+y = Fxt−1 + Fyt−1 ✳ ➜Þ♥❤ ❧ý ✶✳✸✳✹✳ ❈❤♦ M✳ ✈➭ t M = M/U ✳ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ✈➭ ●✐➯ sö x ✈➭ y U ❧➭ ♠ét ♠➠➤✉♥ ❝♦♥ ❝ñ❛ ❧➭ ❝➳❝ ♣❤➬♥ tö ❝ñ❛ R s❛♦ ❝❤♦ x (♯) ✈➭ 0 :M xy = U ✳ ❈➳❝ ❦❤➻♥❣ ➤Þ♥❤ ❞➢í✐ ➤➞② ❧➭ ➤ó♥❣ t❤á❛ ✻ ✭✐✮ xy ✈➭ i = yExi Exy ✈í✐ ♠ä✐ Hat (M ) ∼ = Hat (M )✳ ❑❤✐ ➤ã ♥Õ✉ Fxt−1 t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ t❤➟♠ r➺♥❣ ❝ò♥❣ ❧➭ ①➳❝ ➤Þ♥❤ ✈➭ ✭✐✐✮ ●✐➯ sö (♯)✱ ●✐➯ sö ❧➭ ①➳❝ ➤Þ♥❤✱ t❤× t−1 Fxy t−1 Fxy = yFxt−1 ✳ Hat (M ) ∼ = Hat (M ) ✈➭ yHai (M ) = 0 i t−1 Exy = 0 ✈í✐ ♠ä✐ i < t − 1✳ ❍➡♥ ♥÷❛✱ Fxy ✶✳✹ i < t − 1✳ ✈í✐ ♠ä✐ i < t✳ ❧➭ ①➳❝ ➤Þ♥❤ ✈➭ ❑❤✐ ➤ã t−1 Fxy = 0✳ ➜Þ♥❤ ❧Ý ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❚r♦♥❣ ➜Þ♥❤ ❧Ý ✶✳✸✳✹ ❝❤ó♥❣ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t ❝❤❰ r❛ t❤❡♦ ♥❤÷♥❣ ♣❤➬♥ tö ❝ã ❞➵♥❣ xy t❤á❛ ♠➲♥ ♠ét sè ➤✐Ò✉ ❦✐Ö♥ t❤Ý❝❤ ❤î♣✳ ❇æ ➤Ò ❞➢í✐ ➤➞② ❝❤♦ ♣❤Ð♣ t❛ ❝❤✉②Ó♥ ♠ét ♣❤➬♥ tö tæ♥❣ q✉➳t ✈Ò ❞➵♥❣ ➤➷❝ ❜✐Öt ♥➭②✳ ❇æ ➤Ò ✶✳✹✳✶✳ ❈❤♦ ✈➭ x p1 , ..., pn (R, m) ❧➭ ♠ét ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣✱ a✱ b ❧➭ ❝➳❝ ✐➤➟❛♥ ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè s❛♦ ❝❤♦ ❧➭ ♠ét ♣❤➬♥ tö ♥➺♠ tr♦♥❣ tå♥ t➵✐ ❝➳❝ ♣❤➬♥ tö ab ♥❤➢♥❣ a1 , ..., ar ∈ a x = a1 b1 + · · · + ar br s❛♦ ❝❤♦ i ≤ r, j ≤ n✳ ✈➭ ab * pj x ∈ / pj ✈í✐ ♠ä✐ b1 , ..., br ∈ b ai bi ∈ / pj ✈➭ ✈í✐ ♠ä✐ j ≤ n✳ j ≤ n✳ ❳Ðt ❑❤✐ ➤ã ➤Ó t❛ ❝ã t❤Ó ❜✐Ó✉ ❞✐Ô♥ a1 b1 + · · · + ai bi ∈ / pj ✈í✐ ♠ä✐ ❚õ ❝➳❝ ➜Þ♥❤ ❧Ý ✶✳✸✳✸ ✈➭ ✶✳✸✳✹ ✈➭ ❇æ ➤Ò ✶✳✹✳✶ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ❝➞✉ tr➯ ❧ê✐ ❦❤➻♥❣ ➤Þ♥❤ ❝❤♦ ❝➯ ❤❛✐ ❝➞✉ ❤á✐ ➤➲ ♥➟✉ ë ♣❤➬♥ ♠ë ➤➬✉ ♥❤➢ s❛✉✳ ➜Þ♥❤ ❧ý ✶✳✹✳✹✳ ❈❤♦ ✈➭ a M ❧➭ ♠ét ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ tr➟♥ ✈➭♥❤ ◆♦❡t❤❡r ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ an0 Hai (M ) = 0 x ∈ a2n0 ❝ñ❛ ✈í✐ ♠ä✐ R✳ ❳Ðt i < t✳ t ✈➭ n0 ❧➭ ❝➳❝ sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ ♣❤➬♥ tö a✲❧ä❝ M ✱ t❛ ❝ã Hai (M/xM ) ∼ = Hai (M ) ⊕ Hai+1 (M ), ✈í✐ ♠ä✐ R i < t − 1✱ ✈➭ 0 :Hat−1 (M/xM ) an0 ∼ = Hat−1 (M ) ⊕ 0 :Hat (M ) an0 . ❝❤Ý♥❤ q✉② ✼ ▼ét tr♦♥❣ ♥❤÷♥❣ ➳♣ ❞ô♥❣ ➤➳♥❣ ❝❤ó ý ❝ñ❛ ➜Þ♥❤ ❧Ý ✶✳✹✳✹ ♠➭ ❝❤ó♥❣ t➠✐ t❤✉ ➤➢î❝ ❧➭ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❝❤✃t æ♥ ➤Þ♥❤ ❝ñ❛ ❝❤Ø sè ❦❤➯ q✉② ❝ñ❛ ✐➤➟❛♥ t❤❛♠ sè ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ◆❤➽❝ ❧➵✐ r➺♥❣ ❝❤Ø sè ❦❤➯ q✉② ❝ñ❛ ♠ét ♠➠➤✉♥ ❝♦♥ N ❝ñ❛ M ❜✃t ❦❤➯ q✉② rót ❣ä♥ ❝ñ❛ ❝❤Ø sè ❦❤➯ q✉② ❝ñ❛ M q ❧➭ sè ♠➠➤✉♥ ❝♦♥ ❜✃t ❦❤➯ q✉② tr♦♥❣ ♠ét ❜✐Ó✉ ❞✐Ô♥ N ✳ ❳Ðt q ❧➭ ♠ét ✐➤➟❛♥ t❤❛♠ sè ❝ñ❛ M M tr➟♥ ❧➭ ❝❤Ø sè ❦❤➯ q✉② ❝ñ❛ ♠➠➤✉♥ ❝♦♥ ✈➭ ➤➢î❝ tÝ♥❤ ❜➺♥❣ ❝➠♥❣ t❤ø❝ qM NR (q, M ) = dimR/m Soc(M/qM )✱ Soc(N ) ∼ = 0 :N m ∼ = HomR (R/m, N ) ✈í✐ ♠ét ❦Õt q✉➯ q✉❡♥ ❜✐Õt ❦❤➻♥❣ ➤Þ♥❤ r➺♥❣ ♥Õ✉ M NR (q, M ) t❛ ➤Þ♥❤ ♥❣❤Ü❛ R✲♠➠➤✉♥ ❜✃t ❦× ❝ñ❛ ë ➤➞② N✳ ▼ét ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ t❤× ❧➭ ♠ét ❤➺♥❣ sè ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ✈✐Ö❝ ❝❤ä♥ ✐➤➟❛♥ t❤❛♠ sè q✳ ◆✳❚✳ ❈➢ê♥❣ ✈➭ ❍✳▲✳ ❚r➢ê♥❣ ✭✷✵✵✾✮ ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ tr➟♥ ❝❤♦ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❚õ ➜Þ♥❤ ❧Ý ✶✳✹✳✹ ❝❤ó♥❣ t➠✐ t❤✉ ➤➢î❝ ❞➵♥❣ ♠➵♥❤ ❝ñ❛ ❦Õt q✉➯ ❝ñ❛ ❤ä ♥❤➢ s❛✉✳ ❇æ ➤Ò ✶✳✹✳✼✳ ❈❤♦ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❝❤✐Ò✉ tr➟♥ ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣ (R, m)✱ ✈➭ n0 d>0 ❧➭ sè ♥❣✉②➟♥ ❞➢➡♥❣ ♥❤á ♥❤✃t mn0 Hmi (M ) = 0 ✈í✐ ♠ä✐ i < d✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ ✐➤➟❛♥ t❤❛♠ sè q ❝ñ❛
M ❝❤ø❛ tr♦♥❣ m2n0 ✈➭ k ≤ n0 ✱ ➤é ❞➭✐ ℓR (qM :M mk )/qM ❧➭ ♠ét ❤➺♥❣ sè s❛♦ ❝❤♦ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ✈✐Ö❝ ❝❤ä♥ q ✈➭
k ℓR (qM :M m )/qM = ◆ã✐ r✐➟♥❣✱ ❝❤Ø sè ❦❤➯ q✉② ✈✐Ö❝ ❝❤ä♥ d   X d i=0 NR (q, M ) i ℓR (0 :Hmi (M ) mk ). ❧➭ ♠ét ❤➺♥❣ sè ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ q ✈➭ NR (q, M ) = d   X d i=0 i dimR/m Soc(Hmi (M )). ✽ ❈❤➢➡♥❣ ✷ ❚Ý♥❤ ❝❤✃t æ♥ ➤Þ♥❤ ❝ñ❛ ❤Ö t❤❛♠ sè tèt ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② t❛ ❧✉➠♥ ①Ðt M ❧➭ ♠ét (R, m) ❧➭ ♠ét ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣✱ ✈➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d > 0✳ ▼ô❝ t✐➟✉ ❝ñ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t æ♥ ➤Þ♥❤ ❝ñ❛ ❤Ö t❤❛♠ sè tèt ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ✷✳✶ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✈➭ ❤Ö t❤❛♠ sè tèt ✷✳✶✳✶ ▲ä❝ ❝❤✐Ò✉ ✈➭ ❤Ö t❤❛♠ sè tèt ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✶✳ ✭✐✮ ❚❛ ♥ã✐ ♠ét ❧ä❝ ❤÷✉ ❤➵♥ F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M M ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ dim M1 < · · · < dim Mt ✱ ❧➭ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❝❤✐Ò✉ ♥Õ✉ ✈➭ ❦❤✐ ➤ã t❛ ♥ã✐ ➤é ❞➭✐ ❝ñ❛ ❧ä❝ t❤✉❐♥ t✐Ö♥✱ t❛ ❧✉➠♥ ❣✐➯ sö r➺♥❣ ✭✐✐✮ ▼ét ❧ä❝ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❣ä✐ ❧➭ ❧ä❝ ❝❤✐Ò✉ ❝ñ❛ ✭❛✮ M dim M1 > 0✳ D : D0 ⊆ D1 ⊆ · · · ⊆ Dt = M dim M0 < F ❝ñ❛ ❧➭ M t✳ ➜Ó ➤➢î❝ ♥Õ✉ ❤❛✐ ➤✐Ò✉ ❦✐Ö♥ s❛✉ t❤á❛ ♠➲♥✿ Di−1 ❧➭ ♠➠➤✉♥ ❝♦♥ ❧í♥ ♥❤✃t ❝ñ❛ Di ♠➭ dim Di−1 < dim Di ✈í✐ ♠ä✐ i = t, t − 1, ..., 1✳ ✭❜✮ D0 = Hm0 (M )✳ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✷✳ ❈❤♦ F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❝❤✐Ò✉✳ ➜➷t di = dim Mi ✈í✐ ♠ä✐ i ≤ t✳ ❧➭ ♠ét ❧ä❝ t❤á❛ ▼ét ❤Ö t❤❛♠ sè ✾ x = x1 , ..., xd ❧ä❝ F ♥Õ✉ ❝ñ❛ M ➤➢î❝ ❣ä✐ ❧➭ ♠ét ❤Ö t❤❛♠ sè tèt ❝ñ❛ ✷✳✶✳✷ M ➤➢î❝ ❣ä✐ ➤➡♥ ❣✐➯♥ ❧➭ ♠ét ❤Ö t❤❛♠ sè M✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❈❤♦ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✹✳ F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❝❤✐Ò✉✳ ➜➷t di = dim Mi ❧➭ ♠ét ❤Ö t❤❛♠ sè tèt ❝ñ❛ M ❤Ö t❤❛♠ sè ❝ñ❛ Mi ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ t➢➡♥❣ ø♥❣ ✈í✐ ❧ä❝ e(x1 , ..., xdi ; Mi ) i ≤ t✱ F✳ ❧➭ ♠ét ❧ä❝ t❤á❛ ✈➭ ①Ðt ❑❤✐ ➤ã x = x1 , ..., xd x1 , ..., xdi ❧➭ ♠ét i ≤ t✳ ❈❤♦ ♥➟♥ t❛ ❝ã t❤Ó ➤Þ♥❤ ♥❣❤Ü❛ IF,M (x) = ℓ(M/(x)M ) − ë ➤➞② t➢➡♥❣ ø♥❣ ✈í✐ Mi ∩ (xdi +1 , ..., xd )M = 0 ✈í✐ ♠ä✐ i = 0, 1, ..., t − 1✳ ▼ét ❤Ö t❤❛♠ sè tèt t➢➡♥❣ ø♥❣ ✈í✐ ❧ä❝ ❝❤✐Ò✉ ❝ñ❛ tèt ❝ñ❛ M t X e(x1 , ..., xdi ; Mi ), i=0 ❧➭ ❜é✐ ❙❡rr❡ ✈➭ e(x1 , ..., xd0 ; M0 ) = ℓ(M0 ) ♥Õ✉ dim M0 = 0✳ F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✻✳ ❈❤♦ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ M ✳ ❚❤× ❧ä❝ F ❧➭ ♠ét ❧ä❝ ➤➢î❝ ❣ä✐ ❧➭ ♠ét ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭s✉② ré♥❣✮ ♥Õ✉ ♥ã t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❝❤✐Ò✉✱ dim M0 = 0 ✈➭ M1 /M0 , ..., Mt /Mt−1 ❧➭ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭s✉② ré♥❣✮✳ ❚❛ ♥ã✐ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ✭s✉② ré♥❣✮ ❞➲② ♥Õ✉ ♥ã ❝ã ♠ét ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭s✉② ré♥❣✮✳ ❳Ðt F ❧➭ ♠ét ❧ä❝ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ❝❤✐Ò✉ ❝ñ❛ supx IF,M (x)✱ ✈í✐ x = x1 , ..., xd M✳ ➜➷t IF (M ) = ❝❤➵② tr➟♥ t✃t ❝➯ ❝➳❝ ❤Ö t❤❛♠ sè tèt ❝ñ❛ M t➢➡♥❣ ø♥❣ ✈í✐ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✈í✐ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ✷✳✷ F F ✳ ◆✳❚✳ ❈➢ê♥❣ ✈➭ ➜✳❚✳ ❈➢ê♥❣ ✭✷✵✵✼✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ IF (M ) < ∞✳ ▼ét sè tÝ♥❤ ❝❤✃t æ♥ ➤Þ♥❤ ❑Ý ❤✐Ö✉ ✷✳✷✳✶✳ ❚r♦♥❣ t✐Õt ♥➭② t❛ ❧✉➠♥ ①Ðt ✶✵ ✭✐✮ M ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❝❤✐Ò✉ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ di = dim Mi ✈í✐ ♠ä✐ ✭✐✐✮ ▲ä❝ ❝❤✐Ò✉ ❝ñ❛ ✭✐✐✐✮ ➜➷t M ✭✐✈✮ ❚❛ ❝ã M F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M ✱ D : Hm0 (M ) = D0 ⊆ D1 ⊆ · · · ⊆ Dt = M ✳ x = x1 , ..., xd ✈í✐ t➢➡♥❣ ø♥❣ ✈í✐ Hmj (M/Mi ) ✈í✐ i = 0, ..., t✳ IF (M ) = supx IF,M (x)✱ t❤❛♠ sè tèt ❝ñ❛ ♠ä✐ ❧➭ d > 0 ❝❤➵② tr➟♥ t✃t ❝➯ ❝➳❝ ❤Ö F✳ i ≤ t−1 ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ✈í✐ ♠ä✐ j ≤ di+1 − 1✱ ♥➟♥ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ n0 ✈➭ ✈í✐ s❛♦ ❝❤♦ mn0 Hmj (M/Mi ) = 0 ✈í✐ ♠ä✐ i ≤ t − 1 ✈➭ ✈í✐ ♠ä✐ j ≤ di+1 − 1 ✭✈✮ ➜➷t ci = AnnMi ✈í✐ ♠ä✐ i = 0, ..., t✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❈❤➢➡♥❣ ✷ ❧➭ ❝➳❝ ➤Þ♥❤ ❧Ý ❞➢í✐ ➤➞②✳ ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ❈❤♦ d>0 M ✈í✐ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ di = dim Mi ✈í✐ ♠ä✐ mn0 Hmj (M/Mi ) = 0 ci = AnnMi ✭✐✮ ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❝❤✐Ò✉ ❳Ðt ✈í✐ ♠ä✐ ❳Ðt n0 i ≤ t−1 ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ ✈➭ ✈í✐ ♠ä✐ xj ∈ m3n0 ci j ≤ di+1 − 1✳ ➜➷t i = 0, ..., t✳ ❈➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ ➤ó♥❣ x = x1 , ..., xd ❧➭ ♠ét ❤Ö t❤❛♠ sè tèt ❝ñ❛ M ❝❤♦ ➤ã ✈í✐ ♠ä✐ i = 0, ..., t✳ F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M ✱ ✈í✐ ♠ä✐ IF,M (x) = IF (M ) ✈➭ 0 ≤ i ≤ t−1 t➢➡♥❣ ø♥❣ ✈í✐ ❧ä❝ ✈➭ ✈í✐ ♠ä✐ F di < j ≤ di+1 ✳ s❛♦ ❑❤✐ IF (M ) = ℓ(Hm0 (M/M0 ))    −1  t−1 di+1 X X di+1 − 1 di − 1 − ℓ(Hmj (M/Mi )). + j j i=0 j=1 ✭✐✐✮ IF,M (x) = IF (M ) ø♥❣ ✈í✐ ❧ä❝ F ✈í✐ ♠ä✐ ❤Ö t❤❛♠ sè tèt ♥➺♠ tr♦♥❣ mn ✈í✐ n ≫ 0✳ x = x1 , ..., xd ❝ñ❛ M t➢➡♥❣ ✶✶ ❈❤ó ý r➺♥❣ ◆✳❚✳ ❈➢ê♥❣ ✈➭ ➜✳❚✳ ❈➢ê♥❣ ✭✷✵✵✼✮ ❝ò♥❣ ➤➵t ➤➢î❝ ❦Õt q✉➯ t➢➡♥❣ tù ♥❤➢ tr➟♥ ❝❤♦ ♥❤÷♥❣ ❤Ö t❤❛♠ sè tèt ❝ã ❞➵♥❣ xn1 1 , ..., xnd d ✈í✐ ni ≫ 0✳ ➜Þ♥❤ ❧Ý ❞➢í✐ ➤➞② ❧➭ ♠ë ré♥❣ ❝ñ❛ ❍Ö q✉➯ ✶✳✹✳✼ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳ ➜Þ♥❤ ❧ý ✷✳✷✳✽✳ ❈❤♦ d>0 M ✈í✐ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ di = dim Mi ✈í✐ ♠ä✐ mn0 Hmj (M/Mi ) = 0 ci = AnnMi ✭✐✮ ❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❝❤✐Ò✉ ✈í✐ ♠ä✐ i = 0, ..., t✳ ✈í✐ ♠ä✐ xj ∈ m3n0 +1 ci ❦❤➯ q✉② ❝ñ❛ n0 i ≤ t−1 ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ ✈➭ ✈í✐ ♠ä✐ j ≤ di+1 − 1✳ ➜➷t i = 0, ..., t✳ ❈➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ ➤ó♥❣ ❱í✐ ♠ä✐ ❤Ö t❤❛♠ sè tèt ♠➲♥ ❳Ðt F : M0 ⊆ M1 ⊆ · · · ⊆ Mt = M ✱ x = x1 , ..., xd ✈í✐ ♠ä✐ (x) tr➟♥ M ❝ñ❛ M t➢➡♥❣ ø♥❣ ✈í✐ ❧ä❝ F t❤á❛ 0 ≤ i ≤ t − 1 ✈➭ ♠ä✐ di < j ≤ di+1 ✱ ❝❤Ø sè ❧➭ ♠ét ❤➺♥❣ sè ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ✈✐Ö❝ ❝❤ä♥ x✱ ✈➭ NR ((x), M ) = dimR/m Soc(Hm0 (M ))    di+1  t−1 X X di di+1 + dimR/m Soc(Hmj (M/Mi )). − j j i=0 j=1 ✭✐✐✮ ❚å♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ sè tèt (x) ⊆ mn ❝ñ❛ t❤✉é❝ ✈➭♦ ✈✐Ö❝ ❝❤ä♥ M x✳ n s❛♦ ❝❤♦ ❝❤Ø sè ❦❤➯ q✉② ❝ñ❛ ♠ä✐ ❤Ö t❤❛♠ t➢➡♥❣ ø♥❣ ✈í✐ ❧ä❝ F ❧➭ ♠ét ❤➺♥❣ sè ❦❤➠♥❣ ♣❤ô ✶✷ ❈❤➢➡♥❣ ✸ ❚Ý♥❤ ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ ❜❐❝ ❝ñ❛ ♠ét ♠➠➤✉♥ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② t❛ ❧✉➠♥ ①Ðt ▼❛❝❛✉❧❛② ✈➭ M ❧➭ ♠ét (R, m) ❧➭ ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ ♠ét ✈➭♥❤ ❈♦❤❡♥✲ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d > 0✳ ▼ô❝ ➤Ý❝❤ ❝ñ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ ①➞② ❞ù♥❣ ➜Þ♥❤ ❧Ý ❝❤❰ r❛ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ HIi (M ) tr♦♥❣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ➤➷❝ ❜✐Öt ❦❤✐ I = m✳ ❈❤ó♥❣ t➠✐ sö ❞ô♥❣ ➜Þ♥❤ ❧Ý ❝❤❰ r❛ t❤✉ ➤➢î❝ ➤Ó ♥❣❤✐➟♥ ❝ø✉ ✈Ò ❝✃✉ tró❝ ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❚õ ➤ã ❝❤ó♥❣ t➠✐ ①➞② ❞ù♥❣ ❦❤➳✐ ♥✐Ö♠ ❜❐❝ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ♠ét ♠➠➤✉♥✳ ✸✳✶ ▲✐♥❤ ❤♦➳ tö ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② t❛ t❤➢ê♥❣ ①✉②➟♥ ❞ï♥❣ ❝➳❝ ❦Ý ❤✐Ö✉ s❛✉✳ ❑Ý ❤✐Ö✉ ✸✳✶✳✶✳ ❈❤♦ (R, m) ❧➭ ♠ét ✈➭♥❤ ◆♦❡t❤❡r ➤Þ❛ ♣❤➢➡♥❣✱ ✈➭ M ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d > 0✳ ❚❛ ➤Þ♥❤ ♥❣❤Ü❛ ✭✐✮ ❱í✐ ♠ç✐ i ✭✐✐✮ ➜➷t < d ❦Ý ❤✐Ö✉ ai (M ) = AnnHmi (M )✱ ✈➭ a(M ) = a0 (M )...ad−1 (M )✳ b(M ) = Td x;i=1 Ann(0 : xi )M/(x1 ,...,xi−1 )M tr♦♥❣ t✃t ❝➯ ❝➳❝ ❤Ö t❤❛♠ sè ❝ñ❛ ✈í✐ x = x1 , ..., xd ❝❤➵② M✳ ▲✐♥❤ ❤♦➳ tö ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ♣❤➯♥ ➳♥❤ s➞✉ s➽❝ ❝✃✉ tró❝ ❝ñ❛ ♠➠➤✉♥ ❈❤ó ý ✸✳✶✳✷✳ M ✳ ❉➢í✐ ➤➞② ❧➭ ♠ét sè tÝ♥❤ ❝❤✃t ❤÷✉ Ý❝❤ ❝ñ❛ a(M )✳ ✭✐✮ ❙❝❤❡♥③❡❧ ✭✶✾✽✷✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ a(M ) ⊆ b(M ) ⊆ a0 (M ) ∩ · · · ∩ ad−1 (M ). ✶✸ ✭✐✐✮ ❚❤❡♦ ◆✳❚✳ ❈➢ê♥❣ ✈➭ ➜✳❚✳ ❈➢ê♥❣ ♥Õ✉ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ t❤× R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ ♠ét ✈➭♥❤ dim R/ai (M ) ≤ i ✈í✐ ♠ä✐ i < d✳ ❍➡♥ ♥÷❛ dim R/ai (M ) = i ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ p ∈ AssM ✱ dim R/p = i✳ ✸✳✷ ➜Þ♥❤ ❧Ý ❝❤❰ r❛ ❝ñ❛ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❚r➢í❝ ❤Õt ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ö✉ ✈Ò ❦❤➳✐ ♥✐Ö♠ t❤➭♥❤ ♣❤➬♥ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ♠ét ♠➠➤✉♥✳ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✶✳ ❚❛ ♥ã✐ ♠➠➤✉♥ ❝♦♥ ❧í♥ ♥❤✃t ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ❇æ ➤Ò ✸✳✷✳✸✳ ❳Ðt ❝ñ❛ I M ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ M ❝ã ❝❤✐Ò✉ ♥❤á ❤➡♥ d ❧➭ UM (0)✳ R ✈➭ x, y ∈ b(M ) ❧➭ ❝➳❝ ♣❤➬♥ tö t❤❛♠ sè M ✳ ➜➷t M = M/UM (0) ✈➭ t = d − dim R/I ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ i < t − 1 t❛ ❧✉➠♥ ❝ã ❞➲② ❦❤í♣ ♥❣➽♥ 0 → HIi (M ) → HIi (M/xyM ) → HIi+1 (M ) → 0. ❍➡♥ ♥÷❛✱ ♥Õ✉ HIt (M ) ∼ = HIt (M ) t❤× t❛ ❝ã ❞➲② ❦❤í♣ ♥❣➽♥ 0 → HIt−1 (M ) → HIt−1 (M/xyM ) → 0 :HIt (M ) xy → 0. ❱í✐ ❝➳❝ ❦Ý ❤✐Ö✉ ♥❤➢ tr➟♥✱ ①Ðt x ∈ b(M )2 ✱ ✈í✐ ♠ä✐ i < t − 1✱ t❛ ❦Ý ❤✐Ö✉ Exi ❧➭ ♣❤➬♥ tö tr♦♥❣ Ext(HIi+1 (M ), HIi (M )) ➤➵✐ ❞✐Ö♥ ❜ë✐ ❞➲② ❦❤í♣ ♥❣➽♥ ❞➢í✐ ➤➞② ♥Õ✉ ♥ã tå♥ t➵✐ 0 → HIi (M ) → HIi (M/xM ) → HIi+1 (M ) → 0. ❚r♦♥❣ tr➢ê♥❣ ❤î♣ i = t − 1✱ ❣✐➯ sö HIt (M ) ∼ = HIt (M )✱ t➳❝ ➤é♥❣ ❤➭♠ tö Hom(R/b(M ), •) ✈➭♦ ❞➲② ❦❤í♣ ♥❣➽♥ 0 → HIt−1 (M ) → HIt−1 (M/xM ) → 0 :HIt (M ) x → 0 t❛ t❤✉ ➤➢î❝ ❞➲② ❦❤í♣ tr➳✐ ❞➢í✐ ➤➞② 0 → HIt−1 (M ) → 0 :HIt−1 (M/xM ) b(M ) → 0 :HIt (M ) b(M ). ✶✹ ❚❛ ❦Ý ❤✐Ö✉ Fxt−1 ❧➭ ♣❤➬♥ tö ❝ñ❛ Ext(0 :HIt (M ) b(M ), HIt−1 (M )) ➤➵✐ ❞✐Ö♥ ❜ë✐ ❞➲② ❦❤í♣ ❞➢í✐ ➤➞② ♥Õ✉ ♥ã tå♥ t➵✐ 0 → HIt−1 (M ) → 0 :HIt−1 (M/xM ) b(M ) → 0 :HIt (M ) b(M ) → 0. ❑❤✐ ➤ã ➜Þ♥❤ ❧Ý ❝❤❰ r❛ ➤➢î❝ ♣❤➳t ❜✐Ó✉ ♥❤➢ s❛✉✳ ❈❤♦ ➜Þ♥❤ ❧ý ✸✳✷✳✹✳ I ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R ✈➭ x ❧➭ ♠ét ♣❤➬♥ tö t❤❛♠ sè ❝ñ❛ M ✳ ➜➷t M = M/UM (0) ✈➭ t = d − dim R/I ✳ ❑❤✐ ➤ã ✭✐✮ ◆Õ✉ x ∈ b(M )2 t❤× Exi ❧➭ ①➳❝ ➤Þ♥❤ ✈í✐ ♠ä✐ x ∈ b(M )3 t❤× Exi = 0 H t (M ) ∼ = H t (M ) t❤× F t−1 = 0✳ ✭✐✐✮ ◆Õ✉ I M ❝ñ❛ ❤Ö t❤❛♠ sè i < t − 1✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ x I ❈❤ó ý r➺♥❣ ✈í✐ ♠ä✐ i < t − 1✳ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ ♣❤➬♥ xi , ..., xd ✱ 2 ≤ i ≤ d✱ UM/(xi ,...,xd )M (0) = 0✳ ➜Þ♥❤ ❧Ý ❞➢í✐ ➤➞② ❝❤♦ t❛ ❤✐Ó✉ râ ❤➡♥ ✈Ò ❝✃✉ tró❝ ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ➜Þ♥❤ ❧Ý ♥➭② ❧➭ ❝➡ së ➤Ó ❝❤ó♥❣ t➠✐ t❤✐Õt ❧❐♣ ♠ét ❧♦➵✐ ❜❐❝ ♠í✐ ❝❤♦ ♠➠➤✉♥ tr♦♥❣ t✐Õt s❛✉✳ ➜Þ♥❤ ❧ý ✸✳✷✳✾✳ ❳Ðt x = x1 , ..., xd xi ∈ b(M/(xi+1 , ..., xd )M )3 M t❤á❛ ♠➲♥ 1 ≤ i ≤ d✱ ❝➳❝ ♠➠➤✉♥ ❧➭ ♠ét ❤Ö t❤❛♠ sè ❝ñ❛ ✈í✐ ♠ä✐ i ≤ d✳ ❱í✐ ♠ä✐ UM/(xi+1 ,...,xd )M (0) ❧➭ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ✈✐Ö❝ ❝❤ä♥ ❤Ö t❤❛♠ sè x ✭s❛✐ ❦❤➳❝ ♠ét ➤➻♥❣ ❝✃✉✮✳ ❑Ý ❤✐Ö✉ ✸✳✷✳✶✵✳ ❱í✐ ♠ç✐ 0 ≤ i ≤ d − 1 t❛ ❦Ý ❤✐Ö✉ Ui (M ) ❧➭ ♠ét x = x1 , ..., xd ❝ñ❛ M t❤á❛ ♠➲♥ xi ∈ b(M/(xi+1 , ..., xd )M )3 ✈í✐ ♠ä✐ i ≤ d t❛ ❝ã Ui (M ) ∼ = UM/(xi+2 ,...,xd )M (0) ✈í✐ ♠ä✐ 0 ≤ i ≤ d − 1✳ ❈❤ó ý r➺♥❣ ❦❤✐ ➤ã Ud−1 (M ) ∼ = UM (0)✳ ♠➠➤✉♥ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ö t❤❛♠ sè ✸✳✸ ❇❐❝ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ♠ét ♠➠➤✉♥ ▼ô❝ ➤Ý❝❤ ❝ñ❛ t✐Õt ♥➭② ❧➭ ①➞② ❞ù♥❣ ♠ét ❧♦➵✐ ❜❐❝ ♠í✐ ❝❤♦ ♠➠➤✉♥ ❝➳❝ ♠➠➤✉♥ Ui (M ) ➤➲ t❤✉ ➤➢î❝ tr♦♥❣ t✐Õt tr➢í❝✳ ❳Ðt I M ❧➭ ♠ét ✐➤➟❛♥ ❞ù❛ tr➟♥ m✲♥❣✉②➟♥ ✶✺ s➡✱ tr♦♥❣ t✐Õt ♥➭② t❛ ❞ï♥❣ deg(I, M ) ➤Ó ❦Ý ❤✐Ö✉ ❜é✐ ❍✐❧❜❡rt✲❙❛♠✉❡❧ t❤➠♥❣ t❤➢ê♥❣✳ ❈➠♥❣ t❤ø❝ ❜é✐ ❧✐➟♥ ❦Õt ❞➢í✐ ➤➞② ❝❤Ø r❛ r➺♥❣ deg(I, M ) ❝❤Ø ♣❤ô t❤✉é❝ ✈➭♦ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ã ❝❤✐Ò✉ ❝❛♦ ♥❤✃t ❝ñ❛ deg(I, M ) = X M ℓRp (Mp )deg(I, R/p). (⋆) p∈AsshM ❈❤ó ý r➺♥❣ ♥Õ✉ ❞➭✐ ❤÷✉ ❤➵♥ ✈➭ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt tè✐ t✐Ó✉ ❝ñ❛ M✱ t❤× Mp ❝ã ➤é 0 (Mp )✳ ◆➟♥ ❝➠♥❣ t❤ø❝ (⋆) ❝ã t❤Ó ➤➢î❝ ✈✐Õt ❧➵✐ ♥❤➢ Mp = HpR p s❛✉ X deg(I, M ) = 0 (Mp ))deg(I, R/p). (⋆⋆) ℓRp (HpR p p∈AsshM ✳ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✸✳✶✳ ❳Ðt ❧ä❝ ❝❤✐Ò✉ D : D0 ⊆ D1 ⊆ · · · ⊆ Dt = M ❝ñ❛ M ✈➭ di = dim Di ✈í✐ ♠ä✐ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt p i ≤ t✳ ❚❛ ❝ã deg(I, M ) ❝ã ❝❤✐Ò✉ ❝❛♦ ♥❤✃t tø❝ ❧➭ ❝❤Ø ❧✐➟♥ q✉❛♥ ➤Õ♥ p ∈ AssM/Dt−1 ✳ M t➢➡♥❣ ø♥❣ ✈í✐ I ✱ adeg(I, M )✱ ➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ♥❤➢ s❛✉ Pt adeg(I, M ) = i=0 deg(I, Di )✳ ❳Ðt p ∈ AssDi ✱ dim R/p = di ✳ ❚❛ ❝ã ❇❐❝ sè ❤ä❝ ❝ñ❛ p∈ / AssM/Di ✳ ❉♦ ➤ã tõ ❞➲② ❦❤í♣ ♥❣➽♥ 0 → Di → M → M/Di → 0 t❛ ❝ã 0 0 HpR ((Di )p ) ∼ (Mp )✳ ❱❐② = HpR p p X 0 adeg(I, M ) = (Mp ))deg(I, R/p). (⋆ ⋆ ⋆) ℓRp (HpR p p∈AssM ❚❛ ❜✐Õt r➺♥❣ deg(M ), adeg(M ) ♣❤➯♥ ➳♥❤ ❝✃✉ tró❝ ♣❤ø❝ t➵♣ ❝ñ❛ ♠➠➤✉♥ M ✳ ◆❤➺♠ ❦✐Ó♠ s♦➳t tèt ❤➡♥ ❝➳❝ ➤é ♣❤ø❝ t➵♣ ♥➭② ✈➭ t❤✉❐♥ t✐Ö♥ tr♦♥❣ sö ❞ô♥❣ ▲✳ ❉♦❡r✐♥❣✱ ❚✳ ●✉♥st♦♥ ✈➭ ❲✳ ❱❛s❝♦♥❝❡❧♦s ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❜❐❝ ♠ë ré♥❣ ❝ñ❛ ♠➠➤✉♥ ♣❤➞♥ ❜❐❝ M ✈➭ ♣❤➳t tr✐Ó♥ ♠ét tr➢ê♥❣ ❤î♣ ➤➷❝ ❜✐Öt ❝ñ❛ ❜❐❝ ♠ë ré♥❣ ❧➭ ❜❐❝ ➤å♥❣ ➤✐Ò✉ ❝ñ❛ ♠➠➤✉♥ ♣❤➞♥ ❜❐❝ M ✳ ❇❐❝ ♠ë ré♥❣ ❝ñ❛ ♠ét ♠➠➤✉♥ tr➟♥ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ➤➢î❝ ①❡♠ ①Ðt ❜ë✐ ▼✳❊✳ ❘♦ss✐✱ ◆✳❱✳ ❚r✉♥❣ ✈➭ ●✳ ❱❛❧❧❛✳ ✶✻ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✸✳✸✳ ❜❐❝ ♠ë ré♥❣ tr➟♥ M(R) ❧➭ ♣❤➵♠ trï ❝➳❝ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ❳Ðt M(R) t➢➡♥❣ ø♥❣ ✈í✐ ✐➤➟❛♥ I ▼ét ❧➭ ♠ét ❤➭♠ sè Deg(I, •) : M(R) → R t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ ✭✐✮ ✭✐✐✮ Deg(I, M ) = Deg(I, M ) + ℓ(Hm0 (M ))✱ ✈í✐ M = M/Hm0 (M )✳ Deg(I, M ) ≥ Deg(I, M/xM ) ❝ñ❛ ✭✐✐✐✮ ◆Õ✉ M✳ M ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤× ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✸✳✹✳ ♣❤➢➡♥❣ ✈í✐ ♠ä✐ ♣❤➬♥ tö tæ♥❣ q✉➳t Deg(I, M ) = deg(I, M )✳ (R, m) ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ ♠ét ✈➭♥❤ ●♦r❡♥st❡✐♥ ➤Þ❛ ❳Ðt (S, n) ❝❤✐Ò✉ n✱ ✈➭ M ❜❐❝ ➤å♥❣ ➤✐Ò✉✱ x ∈ I \ mI hdeg(I, M )✱ ❝ñ❛ M ♥❤➢ s❛✉ hdeg(I, M ) = deg(M ) + R✲❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d✳ ❑❤✐ ➤ã ❧➭ ♠ét ♠➠➤✉♥ t➢➡♥❣ ø♥❣ ✈í✐  n X i=n−d+1 I ➤➢î❝ ➤Þ♥❤ ♥❣❤Ü❛ ➤Ö q✉②  d−1 hdeg(I, ExtiS (M, S)). i−n+d−1 ❈❤♦ ➤Õ♥ ♥❛② ❜❐❝ ➤å♥❣ ➤✐Ò✉ ❧➭ ❧♦➵✐ ❜❐❝ ♠ë ré♥❣ ❞✉② ♥❤✃t ➤➢î❝ ①➞② ❞ù♥❣ râ r➭♥❣✳ ◆ã✐ ❝❤✉♥❣ ❜❐❝ ➤å♥❣ ➤✐Ò✉ ❝ñ❛ ♠ét ♠➠➤✉♥ ❧➭ ❦❤ã tÝ♥❤ t♦➳♥ ✈➭ t❤➢ê♥❣ ❧➭ ❝➳❝ ❣✐➳ trÞ sè ❧í♥✳ ❉➢í✐ ➤➞② ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❜❐❝ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ♠ét ♠➠➤✉♥ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♥ã ❧➭ ♠ét ❧♦➵✐ ❜❐❝ ♠ë ré♥❣✳ ❑Ý ❤✐Ö✉ ✸✳✸✳✻✳ ❳Ðt UM (0) ➤Þ♥❤ ♥❣❤Ü❛ g UM (0)) = deg(I, ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✸✳✼✳ ♥❣❤Ü❛ ( ❧➭ t❤➭♥❤ ♣❤➬♥ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ♠➠➤✉♥ deg(I, UM (0)) 0 ❳Ðt M ❧➭ ♠ét ❜❐❝ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ M ♥Õ✉ ♥Õ✉ ❚❛ dim UM (0) = dim M − 1 dim UM (0) < dim M − 1. R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ t➢➡♥❣ ø♥❣ ✈í✐ udeg(I, M ) = deg(I, M ) + M✳ d−1 X i=0 d✳ ❚❛ ➤Þ♥❤ I ✱ udeg(I, M )✱ ♥❤➢ s❛✉ g Ui (M )), deg(I, ✶✼ ë ➤➞② Ui (M ) ➤➢î❝ ❤✐Ó✉ ♥❤➢ ❧➭ UM/(xi+2 ,...,xd )M (0) ✈í✐ x1 , ..., xd ❧➭ ♠ét ❤Ö xi ∈ b(M/(xi+1 , ..., xd )M )3 ✈í✐ ♠ä✐ i ≤ d✳ g Ui (M ))✳ ◆❣♦➭✐ r❛✱ ✈í✐ ♠ç✐ 0 ≤ i ≤ d−1 t❛ ➤Þ♥❤ ♥❣❤Ü❛ udegi (I, M ) = deg(I, t❤❛♠ sè ❜✃t ❦× ❝ñ❛ M t❤á❛ ♠➲♥ udeg(I, •) t❤á❛ ♠➲♥ ❜❛ ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ ❈➳❝ ❦Õt q✉➯ ❞➢í✐ ➤➞② ❝❤ø♥❣ tá r➺♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✸✳✸ ❞♦ ➤ã ♥ã ❧➭ ♠ét ❜❐❝ ♠ë ré♥❣ tr➟♥ ➜Þ♥❤ ❧ý ✸✳✸✳✽✳ ❳Ðt M ❧➭ ♠ét M(R)✳ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d✳ ❚❛ ❝ã deg(I, M ) ≤ adeg(I, M ) ≤ udeg(I, M ). ❍➡♥ ♥÷❛ ✭✐✮ deg(I, M ) = udeg(I, M ) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ M ✭✐✐✮ adeg(I, M ) = udeg(I, M ) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ M ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳ ▼Ö♥❤ ➤Ò ✸✳✸✳✾✳ ❳Ðt N ❧➭ ♠ét ♥♦❞✉❧❡ ❝♦♥ ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ❝ñ❛ M ✳ ❚❛ ❝ã udeg(I, M ) = udeg(I, M/N ) + ℓ(N ). ➜Þ♥❤ ❧ý ✸✳✸✳✶✼✳ ❈❤♦ M ❧➭ ♠ét ♣❤➬♥ tö s✐➟✉ ❜Ò ♠➷t ❝ñ❛ ❝➯ I ✳ ❚❛ ❝ã M R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d✳ ❳Ðt x ❧➭ ♠ét ✈➭ ❝➳❝ Ui (M )✱ 1 ≤ i ≤ d − 1✱ t➢➡♥❣ ø♥❣ ✈í✐ udeg(I, M/xM ) ≤ udeg(I, M ). ❉➢í✐ ➤➞② ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ♠ét ✈Ý ❞ô ë ➤ã ♥➟♥ ❤❛✐ ❦❤➳✐ ♥✐Ö♠ ♥➭② ❧➭ ♣❤➞♥ ❜✐Öt✳ udeg(m, M ) < udeg(m, M )✱ ➜Ó ➤➡♥ ❣✐➯♥ t❛ ❞ï♥❣ ❝➳❝ ❦Ý ❤✐Ö✉ deg(M ), adeg(M ), hdeg(M ) ✈➭ udeg(M ) t❤❛② ❝❤♦ deg(m, M ), adeg(m, M )✱ hdeg(m, M ) ✈➭ udeg(m, M )✱ t➢➡♥❣ ø♥❣✳ ❱Ý ❞ô ✸✳✸✳✷✵✳ ♠ét tr➢ê♥❣ ✈➭ ❳Ðt R = k[[X1 , ..., X7 ]]/(X1 , X2 , X3 ) ∩ (X4 , X5 , X6 ) ✈í✐ k Xi , 1 ≤ i ≤ 7, ❧➭ ❝➳❝ ❜✐Õ♥✳ ❚❛ ❝ã deg(R) = adeg(R) = 2 < udeg(R) = 4 < hdeg(R) = 5. ❧➭ ✶✽ ❈❤➢➡♥❣ ✹ ❚Ý♥❤ ❤÷✉ ❤➵♥ ❝ñ❛ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ❧✉➠♥ ①Ðt a ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R ✈➭ M ❧➭ ♠ét R✲♠➠➤✉♥✳ ▼ô❝ t✐➟✉ ♥❣❤✐➟♥ ❝ø✉ ❝ñ❛ ❝❤➢➡♥❣ ♥➭② ①✉✃t ♣❤➳t tõ ❝➞✉ ❤á✐ s❛✉ ❝ñ❛ ❈✳ ❍✉♥❡❦❡ tr♦♥❣✿ P❤➯✐ ❝❤➝♥❣ AssHai (M ) ♠ét ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ ♠ä✐ ❧✉➠♥ ❧➭ ♠ét t❐♣ ❤÷✉ ❤➵♥ ❦❤✐ i ≥ 0❄ M ❧➭ ◆ã✐ ❝❤✉♥❣ ❝➞✉ ❤á✐ ❝ñ❛ ❍✉♥❡❦❡ ❦❤➠♥❣ ❝ß♥ ➤ó♥❣ ❞♦ ❝➳❝ ✈Ý ❞ô ❝ñ❛ ❆✳ ❙✐♥❣❤ ✭✷✵✵✵✮ ✈➭ ▼✳ ❑❛t③♠❛♥ ✭✷✵✵✷✮✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ q✉❛♥ t➞♠ ➤Õ♥ ❝➞✉ ❤á✐ ❝ñ❛ ❍✉♥❡❦❡ tr♦♥❣ ♥❤÷♥❣ ➤✐Ò✉ ❦✐Ö♥ ♥❤✃t ➤Þ♥❤✳ ✹✳✶ ▼➠➤✉♥ ❋❙❋ ▼ô❝ ➤Ý❝❤ ❝ñ❛ t✐Õt ♥➭② ❧➭ ➤➢❛ r❛ ♠ét ❦Õt q✉➯ tæ♥❣ ❤î♣ ❝➳❝ ❦Õt q✉➯ ❞➢í✐ ➤➞② ❝ñ❛ ❇r♦❞♠❛♥♥ ✈➭ ❋❛❣❤❛♥✐ ✭✷✵✵✵✮✱ ✈➭ ❝ñ❛ ❑❤❛s❤②❛r♠❛♥❡s❤ ✈➭ ❙❛❧❛r✐❛♥ ✭✶✾✾✾✮✳ ➜Þ♥❤ ❧ý ✹✳✶✳✶✳ ❈❤♦ ❦❤➠♥❣ ➞♠✳ ❑❤✐ ➤ã M ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ t ❧➭ ♠ét sè ♥❣✉②➟♥ AssHat (M ) ❧➭ ♠ét t❐♣ ❤î♣ ❤÷✉ ❤➵♥ ♥Õ✉ ♠ét tr♦♥❣ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ t❤á❛ ♠➲♥ ✭✐✮ ✭✐✐✮ Hai (M ) ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ ✈í✐ ♠ä✐ i < t✳ supp(Hai (M )) ❧➭ ♠ét t❐♣ ❤÷✉ ❤➵♥ ✈í✐ ♠ä✐ i < t✳ ❈❤ó♥❣ t➠✐ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❋❙❋ ♠➠➤✉♥ ❞➢í✐ ➤➞②✳ ➜Þ♥❤ ♥❣❤Ü❛ ✹✳✶✳✷✳ ▼ét R✲♠➠➤✉♥ M ♠ét ♠➠➤✉♥ ❝♦♥ ❤÷✉ ❤➵♥ s✐♥❤ N ❝ñ❛ ➤➢î❝ ❣ä✐ ❧➭ ♠ét ♠➠➤✉♥ ❋❙❋ ♥Õ✉ tå♥ t➵✐ M s❛♦ ❝❤♦ supp(M/N ) ❧➭ ♠ét t❐♣ ❤÷✉ ❤➵♥✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ t✐Õt ♥➭② ❧➭ ❤❛✐ ❦❤➻♥❣ ➤Þ♥❤ ❞➢í✐ ➤➞②✳