www.VNMATH.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM - TRẦN THỊ HƯỜNG CHIỀU NOETHER CỦA MÔĐUN ARTIN LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC THÁI NGUYÊN – 2008 Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM - TRẦN THỊ HƯỜNG CHIỀU NOETHER CỦA MÔĐUN ARTIN Chuyên ngành: Đại số lý thuyết số Mã số: 60.46.05 LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: TS NGUYỄN THỊ DUNG THÁI NGUYÊN – 2008 Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM - TRẦN THỊ HƯỜNG CHIỀU NOETHER CỦA MÔĐUN ARTIN Chuyên ngành: Đại số lý thuyết số Mã số: 60.46.05 TÓM TẮT LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: TS NGUYỄN THỊ DUNG THÁI NGUYÊN – 2008 Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com Cơng trình hồn thành Người hướng dẫn khoa học: TS Nguyễn Thị Dung Phản biện 1: Phản biện 2: Luận văn bảo vệ trước Hội đồng chấm luận văn họp tại: Trường Đại học sư phạm - ĐHTN Ngày tháng 10 năm 2008 Số hóa Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn www.VNMATH.com ✸ ▼ë ➤➬✉ ❈❤♦ ♥❤✃t (R, m) ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ➤Þ❛ ♣❤➢➡♥❣✱ ◆♦❡t❤❡r ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❞✉② m; M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥✳ ◆❤➢ ❝❤ó♥❣ t❛ ➤➲ ❜✐Õt✱ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✱ ❝❤✐Ị✉ r ữ ệ ủ ì ọ ➤➵✐ sè ✈➭ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ♠➭ t❤➠♥❣ q✉❛ ➤ã ♥❣➢ê✐ t❛ ❝ã t❤Ĩ ♥ã✐ ❧➟♥ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ➤❛ t➵♣ ➤➵✐ sè ❤♦➷❝ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ✈➭♥❤ ◆♦❡t❤❡r ✈➭ ❝➳❝ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ tr➟♥ ❝❤ó♥❣✳ ❈❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠ét ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ M ✱ ý ệ dim M ợ ị ĩ ề ❑r✉❧❧ ❝đ❛ ✈➭♥❤ R/ Ann M ✈➭ t❛ ❝ã ➤Þ♥❤ ❧ý ❝➡ ❜➯♥ ❝đ❛ ❧ý t❤✉②Õt ❝❤✐Ị✉ ♥❤➢ s❛✉ δ(M ) = dim M = d(M ), tr♦♥❣ ➤ã δ(M ) ❧➭ sè ♥❣✉②➟♥ t ♥❤á ♥❤✃t s❛♦ ❝❤♦ tå♥ t➵✐ ♠ét ❞➲② ❝➳❝ ♣❤➬♥ tö a1 , , at ∈ m ➤Ĩ ➤é ❞➭✐ ❝đ❛ ♠➠➤✉♥ M/(a1 , , at )M ❧➭ ❜❐❝ ❝đ❛ ➤❛ t❤ø❝ ❍✐❧❜❡rt ❧➭ ❤÷✉ ❤➵♥ ✈➭ d(M ) PM,I (n) ø♥❣ ✈í✐ ✐➤➟❛♥ ➤Þ♥❤ ♥❣❤Ü❛ I ✳ ❑❤➳✐ ♥✐Ư♠ ➤è✐ ♥❣➱✉ ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ ❝❤♦ ♠ét ♠➠➤✉♥ ❆rt✐♥ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❘✳ ◆✳ ❘♦❜❡rt ❬✶✻❪ ✈➭ s❛✉ ➤ã ❉✳ ❑✐r❜② ❬✼❪ ➤ỉ✐ t➟♥ t❤➭♥❤ ❝❤✐Ị✉ ◆♦❡t❤❡r✱ ❦ý ❤✐Ư✉ ❧➭ N-dim ➤Ĩ tr➳♥❤ ♥❤➬♠ ❧➱♥ ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ ợ ị ĩ tr ột số ❦Õt q✉➯ ♠➭ t❤❡♦ ♠ét ♥❣❤Ü❛ ♥➭♦ ➤ã ➤➢ỵ❝ ①❡♠ ❧➭ ➤è✐ ♥❣➱✉ ✈í✐ ❝➳❝ ❦Õt q✉➯ ✈Ị ❝❤✐Ị✉ ❑r✉❧❧ ữ s ợ r ❜✐Öt✱ ❘✳ ◆✳ ❘♦❜❡rts ❬✶✻❪ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ❦Õt q✉➯ ✈Ị tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈í✐ ❜❐❝ ❝đ❛ ➤❛ t❤ø❝ ❍✐❧❜❡rt ❝ñ❛ ♠➠➤✉♥ ❆rt✐♥ tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✱ s❛✉ ➤ã ❉✳ ❑✐r❜② ❬✼❪ ✈➭ ◆✳ ❚ ✳ ❈➢ê♥❣ ✲ ▲✳ ❚✳ ◆❤➭♥ ❬✸❪ ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ tr➟♥ ❝ñ❛ ❘♦❜❡rts ❝❤♦ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❜✃t ❦ú N-dim A = deg( R (0 :A mn )) = inf{t : ∃a1 , , at ∈ m : R (0 :A (a1 , , at )R) < ∞} ❚õ ❦Õt q✉➯ tr➟♥✱ ♠ét ❝➳❝❤ tù ♥❤✐➟♥ ❝ã t❤Ĩ ➤Þ♥❤ ♥❣❤Ü❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❤Ư t❤❛♠ sè✱ ❤Ư ❜é✐ ❝❤♦ ♠➠➤✉♥ ❆rt✐♥ t❤➠♥❣ q✉❛ ❝❤✐Ị✉ ◆♦❡t❤❡r✳ www.VNMATH.com ✹ ❚✐Õ♣ t❤❡♦✱ ♥❤✐Ị✉ t➳❝ ❣✐➯ ❝ị♥❣ ➤➲ ❞ï♥❣ ❝❤✐Ị✉ ◆♦❡t❤❡r ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ ✭①❡♠ ❬✺❪✱ ❬✼❪✱ ❬✶✾❪✱✳✳✳✮✳ ➜➷❝ ❜✐Öt✱ t➳❝ ❣✐➯ ◆✳ ❚✳ ❈➢ê♥❣ ✈➭ ▲✳ ❚✳ ◆❤➭♥ ❬✹❪ ➤➲ ❝ã ♥❤÷♥❣ ♥❣❤✐➟♥ ❝ø✉ s➞✉ ❤➡♥ ✈Ò ❝❤✐Ò✉ ◆♦❡t❤❡r✱ q✉❛♥ t➞♠ ➤➷❝ ❜✐Ưt tí✐ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❦❤✐ ❝❤ó♥❣ ❧➭ ❆rt✐♥ ✈➭ ➤➲ ➤➵t ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ t❤ó ✈Þ✱ ❝❤ø♥❣ tá ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ◆♦❡t❤❡r t❤❡♦ ♠ét ♥❣❤Ü❛ ♥➭♦ ➤ã ❧➭ ♣❤ï ❤ỵ♣ ✈í✐ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❚➢➡♥❣ tù ♥❤➢ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ ♠ét ❝➳❝❤ tù ♥❤✐➟♥✱ ố ỗ rt ủ A R/ AnnR A✳ ❝❤✐Ị✉ ❑r✉❧❧ dimR A ❝ị♥❣ ➤➢ỵ❝ ❤✐Ĩ✉ ❧➭ ❝❤✐Ị✉ ❑r✉❧❧ ▼ét ❦Õt q✉➯ q✉❛♥ trä♥❣ tr♦♥❣ ❬✹❪ ❧➭ ♥❣❤✐➟♥ ❝ø✉ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✿ N-dimR A N-dimR A < dimR A✳ dimR A✱ ❤➡♥ ♥÷❛ ❝❤Ø r ữ trờ ợ r ệt ết q ❦❤➳ ❜✃t ♥❣ê tr♦♥❣ ❬✹❪ ❝❤♦ t❛ ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤Ĩ ❦❤✐ ♥➭♦ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠ét ♠➠➤✉♥ ❆rt✐♥ ❜➺♥❣ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♥ã ❧➭ AnnR (0 :A p) = p, ∀p ∈ V (AnnR A) (∗) ❈➬♥ ❝❤ó ý r ố ỗ R ữ s M t❤❡♦ ❇ỉ ➤Ị ◆❛❦❛②❛♠❛✱ t❛ ❧✉➠♥ ❝ã tÝ♥❤ ❝❤✃t AnnR M ✳ A✱ AnnR M/pM = p, ❘â r➭♥❣ r➺♥❣✱ R ủ tì ỗ t ➤è✐ ♥❣➱✉ ▼❛t❧✐s✱ t❛ ❝ã ❧✉➠♥ ❝ã ♥❣✉②➟♥ tè (∗)✱ UsuppR M ❝❛♦ ♥❤✃t p ❝❤ø❛ R✲♠➠➤✉♥ ❆rt✐♥ AnnR (0 :A p) = p✱ ✈í✐ ♠ä✐ ✐➤➟❛♥ p ❝❤ø❛ AnnR A✱ t✉② ♥❤✐➟♥ tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❜✃t ❦ú✱ ❦❤➠♥❣ ♣❤➯✐ ♠ä✐ ♠➠➤✉♥ ❆rt✐♥ ➤✐Ị✉ ❦✐Ư♥ ✈í✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè A ➤Ị✉ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ▼ét ➤✐Ị✉ tú ị ữ t ó tể tr ➤➢ỵ❝ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ❣✐➳ ❦❤➠♥❣ tré♥ ❧➱♥ ❝đ❛ ♠➠➤✉♥ Hmd (M ) M t❤➠♥❣ q✉❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ✭①❡♠ ❬✷❪✮❀ tÝ♥❤ ❦❤➠♥❣ tré♥ ❧➱♥ ✈➭ tÝ♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ Hmi (M ) ✭①❡♠ ❬✶✺❪✮✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ tr×♥❤ ❜➭② ❧➵✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤➲ ❣✐í✐ t❤✐Ư✉ ë tr➟♥ tr♦♥❣ ❜➭✐ ❜➳♦ ❝ñ❛ ◆✳ ❚✳ ❈➢ê♥❣ ✲ ▲✳ ❚✳ ◆❤➭♥ ✭✷✵✵✷✮ ✈➭ ♠ét ♣❤➬♥ ❦Õt q✉➯ ❝ñ❛ ❝➳❝ ❜➭✐ ❜➳♦ ❝ñ❛ ❘✳ ◆✳ ❘♦❜❡rts ✭✶✾✼✺✮❀ ❉✳ ❑✐r❜② ✭✶✾✾✵✮ www.VNMATH.com ✺ ✈➭ ◆✳ ❚✳ ❈➢ê♥❣ ✲ ▲✳ ❚✳ ◆❤➭♥ ✭✶✾✾✾✮✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ✸ ❝❤➢➡♥❣✱ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➬♥ t❤✐Õt ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥é✐ ❞✉♥❣ ❝đ❛ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ♥❤➽❝ ❧➵✐ ①❡♥ ❦Ï tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ✈Ị ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✱ ➤➷❝ ❜✐Ưt ❧➭ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈í✐ ❜❐❝ ❝đ❛ ➤❛ t❤ø❝ ❍✐❧❜❡rt ❝đ❛ ♠ét ♠➠➤✉♥ ❆rt✐♥✳ ❈❤➢➡♥❣ ❞➭♥❤ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ ❧➵✐ ❝➳❝ ❦Õt q✉➯ ✈Ị ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❦❤✐ ❝❤ó♥❣ ❧➭ ❆rt✐♥❀ ♠è✐ q✉❛♥ ❤Ư ữ ề tr ủ ố ề ị t❤ø i ✈í✐ ❝❤Ø sè i ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❜❛♥ ➤➬✉✳ ❈❤➢➡♥❣ trì ố q ệ ữ ề tr ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✿ N-dimR A dimR A ỉ r ữ trờ ợ r❛ ❞✃✉ ♥❤á ❤➡♥ t❤ù❝ sù ✈➭ ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤Ĩ ❦❤✐ ♥➭♦ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠ét ♠➠➤✉♥ ❆rt✐♥ ❜➺♥❣ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♥ã✳ P❤➬♥ ❦Õt ❧✉❐♥ ❝đ❛ ❧✉❐♥ ✈➝♥ tæ♥❣ ❦Õt ❧➵✐ t♦➭♥ ❜é ❝➳❝ ❦Õt q✉➯ ➤➲ ➤➵t ➤➢ỵ❝✳ www.VNMATH.com ✻ ❈❤➢➡♥❣ ✶ ❈❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ➤❛ t❤ø❝ ❍✐❧❜❡rt ❚r♦♥❣ t♦➭♥ ❜é ❝❤➢➡♥❣ ♥➭②✱ t❛ ❧✉➠♥ ❦ý ❤✐Ö✉ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r ❦❤➠♥❣ ♥❤✃t t❤✐Õt ➤Þ❛ tết ị ợ tr♦♥❣ tõ♥❣ tr➢ê♥❣ ❤ỵ♣ ❝ơ t❤Ĩ✮✱ M ❧➭ R✲♠➠➤✉♥✱ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❝❤➢➡♥❣ ♥➭② ❧➭ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝❤♦ ♠ét ♠➠➤✉♥ t✉ú ý ✈➭ ♠ét sè ❦Õt q✉➯ ✈Ò ❝❤✐Ò✉ ◆♦❡t❤❡r ❝❤♦ ♠➠➤✉♥ ❆rt✐♥✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❝❤➢➡♥❣ ❧➭ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈í✐ ❜❐❝ ❝đ❛ ➤❛ t❤ø❝ ❍✐❧❜❡rt ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✳ ❑Õt q✉➯ ♥➭② ➤➲ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❘✳ ◆✳ ❘♦❜❡rts ❬✶✻❪ ❝❤♦ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ s❛✉ ➤ã ❉✳ ❑✐r❜② ❬✽❪✱ ◆✳ ❚✳ ❈➢ê♥❣ ✲ ▲✳ ❚✳ ◆❤➭♥ ❬✸❪ ♠ë ré♥❣ ❝❤♦ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✳ ✶✳✶ ❈❤✐Ò✉ ◆♦❡t❤❡r ❑❤➳✐ ♥✐Ư♠ ➤è✐ ♥❣➱✉ ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ ❝❤♦ ♠ét ♠➠➤✉♥ t✉ú ý ✭❑❞✐♠✮ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❘✳ ◆✳ ❘♦❜❡rts ❬✶✻❪ ✈➭ ë ➤ã✱ ➠♥❣ ❝ò♥❣ ➤➢❛ r❛ ♠ét sè ❦Õt q✉➯ ✈Ò ❝❤✐Ò✉ ❑r✉❧❧ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ❆rt✐♥✳ ❙❛✉ ➤ã ❉✳ ❑✐r❜② tr♦♥❣ ❬✽❪ ➤➲ ➤ỉ✐ t❤✉❐t ♥❣÷ ❝đ❛ ❘♦❜❡rts ✈➭ ➤Ị ♥❣❤Þ t❤➭♥❤ ❝❤✐Ị✉ ◆♦❡t❤❡r ✭N-dim✮ ➤Ĩ tr ề r ợ ị ĩ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ◆♦❡t❤❡r✳ ➜Þ♥❤ ♥❣❤Ü❛ s❛✉ t❤❡♦ t❤❡♦ t❤✉❐t ữ ủ r www.VNMATH.com ị ĩ ề ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ M✱ ❦ý ❤✐Ư✉ ❜ë✐ N-dimR M, ➤➢ỵ❝ ➤Þ♥❤ ♥❣❤Ü❛ ❜➺♥❣ q✉② ♥➵♣ ♥❤➢ s❛✉✿ ❑❤✐ M = 0, ➤➷t N-dimR M = −1 ❱í✐ M = 0, ột số 0, s ỗ ❞➲② t➝♥❣ N-dimR M < d ❝♦♥ ❝ñ❛ d M, tå♥ t➵✐ sè ♥❣✉②➟♥ n0 s❛♦ ❝❤♦ t❛ ➤➷t N-dimR M = d M0 ⊆ M1 ⊆ ♥Õ✉ ❝➳❝ ♠➠➤✉♥ N-dimR (Mn+1 /Mn ) < d, ✈í✐ ♠ä✐ n > n0 ❈❤♦ ❱Ý ❞ô ✶✳✶✳✷✳ M ❧➭ ◆♦❡t❤❡r ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ R✲♠➠➤✉♥ M n > n0 ✳ ❉♦ ➤ã✱ ♠ä✐ Mn+1 /Mn = 0✱ ❱× N-dimR M = 0✳ t➝♥❣ ❜✃t ❦ú M = M ❧➭ R✲♠➠➤✉♥ ❚❤❐t ✈❐②✱ ❣✐➯ sö M ❧➭ R✲♠➠➤✉♥ ♥➟♥ n0 ∈ N ✈× t❤Õ s❛♦ ❝❤♦ Mn = Mn+1 ✱ ❉♦ ➤ã✱ s❛♦ ❝❤♦ Nk+1 = Nk ✱ N-dimR M = 0✳ n > n0 ✈í✐ ❑❤✐ ➤ã✱ ❧✃② ♠ét ❞➲② ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛✱ N-dimR Nk+1 /Nk = −1 < 0✱ ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ ✈➭ ❞♦ ➤ã t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛✱ N0 ⊆ N1 ⊆ ⊆ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ M ✳ n0 ❝➳❝ ♠➠➤✉♥ N-dimR Mn+1 /Mn = −1 < 0, N-dimR M ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư tå♥ t➵✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ k > n0 ✳ ❧➭ M0 ⊆ M1 ⊆ ⊆ Mn ⊆ ➤Ò✉ ❞õ♥❣ ♥➟♥ tå♥ t➵✐ n > n0 ✳ M N-dimR M = ◆♦❡t❤❡r✳ ❱× ♠ä✐ ❞➲② t➝♥❣ ❝♦♥ ❝đ❛ ❦❤➳❝ ❦❤➠♥❣✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ ❤❛② ❞➲② tr➟♥ ❧➭ ❞õ♥❣✱ ♥❣❤Ü❛ ❧➭ R✲♠➠➤✉♥ ◆♦❡t❤❡r✳ ▼Ư♥❤ ➤Ị ✶✳✶✳✸✳ ◆Õ✉ −→ M −→ M −→ M ” −→ ❧➭ ❞➲② ❦❤í♣ ❝➳❝ R✲♠➠➤✉♥ t❤× N-dimR M = max{N-dimR M , N-dimR M ”} ❈❤ø♥❣ ♠✐♥❤✳ ❑❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t✱ t❛ ❝ã t❤Ĩ ❣✐➯ sư M ⊂ M M = M/M ◆Õ✉ M = t❤× M = M ” = M = 0✱ s✉② r❛ N-dimR M = N-dimR M = N-dimR M = −1 ✈➭ www.VNMATH.com ✽ ❉♦ ➤ã t❛ ❧✉➠♥ ❝ã t❤Ó ❣✐➯ t❤✐Õt M = ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ N-dimR M = d ●✐➯ sö d = ❚❤❡♦ ✈Ý ❞ơ tr➟♥✱ M ❝ị♥❣ ❧➭ ❝➳❝ ●✐➯ sư R✲♠➠➤✉♥ ◆♦❡t❤❡r✳ ❱× ✈❐②✱ M ,M R✲♠➠➤✉♥ ◆♦❡t❤❡r ♥➟♥ s✉② r❛ N-dimR M = N-dimR M = d > sù ♥❤á ❤➡♥ ❧➭ d ✈➭ ♠Ư♥❤ ➤Ị ➤ó♥❣ ✈í✐ ♠ä✐ ♠➠➤✉♥ ❝ã ❝❤✐Ò✉ ◆♦❡t❤❡r t❤ù❝ ❈❤♦ M0 ⊆ M1 ⊆ ⊆ Mn ⊆ = ♠➠➤✉♥ ❝♦♥ ❝ñ❛ = = ❧➭ ♠ét ①Ý❝❤ t➝♥❣ ❜✃t ❦ú ❝➳❝ = M ❑❤✐ ➤ã✱ t❛ ❝ò♥❣ ❝ã ❝➳❝ ❞➲② M0 ∩ M ⊆ M1 ∩ M ⊆ ⊆ Mn ∩ M ⊆ = = = (1) = (M + M0 )/M ⊆(M + M1 )/M ⊆ ⊆(M + Mn )/M ⊆ = = = t➢➡♥❣ ø♥❣ ❧➭ ①Ý❝❤ t➝♥❣ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ ❉♦ N-dimR M = d N-dimR Mn+1 /Mn < d, M ✈➭ M = M/M ♥➟♥ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛✱ tå♥ t➵✐ ✈í✐ ♠ä✐ n > n0 ✳ (2) = n0 ∈ N s❛♦ ❝❤♦ ❱× ✈❐②✱ ➳♣ ❞ơ♥❣ ❣✐➯ t❤✐Õt q✉② ♥➵♣ ✈➭♦ ❞➲② ❦❤í♣ −→ Mn+1 M + Mn+1 M ∩ Mn+1 −→ −→ −→ 0, M ∩ Mn Mn M + Mn t❛ ❝ã N-dimR (Mn+1 /Mn ) = max{N-dimR ❱× t❤Õ✱ ✈í✐ ♠ä✐ M ∩ Mn+1 M + Mn+1 , N-dimR } M ∩ Mn M + Mn n > n0 ✱ t❛ ❝ã ❤♦➷❝ N-dimR M ∩ Mn+1 = N-dimR (Mn+1 /Mn ) < d M ∩ Mn ❤♦➷❝ M + Mn+1 = N-dimR (Mn+1 /Mn ) < d M + Mn ♥❣❤Ü❛ ❝❤✐Ò✉ ◆♦❡t❤❡r t❛ ❝ã ❤♦➷❝ N-dimR M = d N-dimR ❉♦ ➤ã✱ t❤❡♦ ➤Þ♥❤ N-dimR M = d ❤❛② N-dimR M = max{N-dimR M , N-dimR M } ❤♦➷❝ www.VNMATH.com ✷✻ ✸✳✶ ❈❤✐Ò✉ r ủ rt ố ỗ rt ột ❝➳❝❤ tù ♥❤✐➟♥ t❛ ❝ị♥❣ ❝ã ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ♥❤➢ s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✶✳✶✳ ❈❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ✈➭♥❤ A✱ ❦ý ❤✐Ư✉ ❜ë✐ dimR A✱ ❧➭ R/ AnnR A ❚❛ q✉② ➢í❝ dimR A = −1 ♥Õ✉ A = 0✳ ▲ý t❤✉②Õt ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ❝đ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❬✶✵❪ ➤➲ ➤➢ỵ❝ ♥❤➽❝ ❧➵✐ ë ❈❤➢➡♥❣ 2✳ ❱× ♠ä✐ ♠➠➤✉♥ ❆rt✐♥ ➤Ị✉ ❝ã ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ó✉ ❝đ❛ AttR A ♥➟♥ dimR A AnnR A ❝ị♥❣ ❝❤Ý♥❤ ❧➭ t❐♣ ❝➳❝ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ ❝❤Ý♥❤ ❧➭ ❝❐♥ tr➟♥ ❝ñ❛ ❝➳❝ sè dim R/p ❦❤✐ p ❝❤➵② ❦❤➽♣ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt dimR A = max{dim R/p | p ∈ AttR A} ❑Õt q✉➯ s❛✉ ❝❤Ø r❛ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ▼Ư♥❤ ➤Ị ✸✳✶✳✷✳ ✭✐✮ ❈➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ❧➭ ➤ó♥❣ N-dimR A = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ dimR A = ❚r♦♥❣ tr➢ê♥❣ ợ A ó ộ ữ ✭✐✐✮ N-dimR A ✈➭ dimR A N-dimR A dimR A ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ●✐➯ sö R (A) < ∞ dimR A = 0✳ ❝❤ø❛ R/ AnnR A ❧➭ ❆rt✐♥✳ N-dimR A = 0✱ ❦❤✐ ➤ã ❱× ✈❐②✱ t❤❡♦ ❬✶✷✱ ✶✷✳❇❪✱ t❛ ❝ã ✈➭♥❤ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư dimR A = 0✳ AnnR A ➤Ò✉ ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐✳ ●ä✐ J ❝❤ø❛ tr♦♥❣ ❧➭ R✲♠➠➤✉♥ ◆♦❡t❤❡r ✈➭ R/ AnnR A ❧➭ ❆rt✐♥ ✈➭ ❑❤✐ ➤ã✱ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧➭ ❣✐❛♦ ❝ñ❛ t✃t ❝➯ ✐➤➟❛♥ ♥❣✉②➟♥ tè AttR A✱ ❦❤✐ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✷✱ ✭✐✮ ✈➭ ❦ý ❤✐Ư✉ JA J= ❱× t❤Õ✱ tå♥ t➵✐ ❉♦ ➤ã A ∩ p= p∈AttR A ∩ p∈AttR A,p tè✐ t❤✐Ó✉ n ∈ N s❛♦ ❝❤♦ JAn A = 0✳ N-dim A = 0✳ p= ∩ t❛ ❝ã p = JA p∈V (AnnR A) ❙✉② r❛ R A < ∞ t❤❡♦ ❇ỉ ➤Ị ✶✳✷✳✶✳ www.VNMATH.com ✷✼ ✭✐✐✮ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ d = dimR A ◆Õ✉ d = t❤× N-dim A = 0✱ t❤❡♦ (i) ●✐➯ sö d > ✈➭ p1 , , pk ❧➭ t✃t ❝➯ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ t❐♣ AttR A s❛♦ ❝❤♦ dim R/pi = d, ♠➠➤✉♥ ❆rt✐♥ ♥➟♥ t❤❡♦ ▼Ö♥❤ ➤Ị ✶✳✶✳✹✱ t❐♣ ❝ù❝ ➤➵✐ ❝đ❛ ✈í✐ ♠ä✐ i = 1, , k ✳ ❧➭ R✳ ❈❤♦ JA ❧➭ ❣✐❛♦ ❝ñ❛ t✃t ❝➯ ✐➤➟❛♥ ❝ù❝ ➤➵✐ tr♦♥❣ t❐♣ Supp A ♥❤➢ i = 1, , k ❱× t❤Õ dimR (0 :A xR) q✉② ♥➵♣✱ t❛ ❝ò♥❣ ❝ã N-dim A A Supp A ❝❤Ø ❣å♠ ❤÷✉ ❤➵♥ ❝➳❝ ✐➤➟❛♥ tr♦♥❣ ❑ý ❤✐Ư✉ ✶✳✶✳✺✳ ❑❤✐ ➤ã t ó tể ọ ợ tử ọ ì N-dim(0 :A xR) d − 1✳ d − x ∈ JA ✈➭ x∈ / pi , ❉♦ ➤ã✱ t❤❡♦ ❣✐➯ t❤✐Õt ❚❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✷✳✼ t❛ ❝ã d ❑Õt q✉➯ s❛✉ ❧➭ ❤Ư q✉➯ trù❝ t✐Õ♣ ❝đ❛ ▼Ư♥❤ ➤Ị ✸✳✶✳✷ ❍Ư q✉➯ ✸✳✶✳✸✳ ◆Õ✉ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ➤➬② ➤đ t❤× t❛ ❧✉➠♥ ❝ã N-dimR A = dimR A ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✶✳✷ ➤➲ ❝ã N-dimR A ♠✐♥❤ ◆Õ✉ dimR A ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ d = N-dim A✳ N-dim A d=0 t❤× dimR A✱ ❝❤Ø ❝➬♥ ❝❤ø♥❣ dimR A = ❧➭ ♠ét ♣❤➬♥ tư t❤❛♠ sè ❝đ❛ t❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✶✳✷✳ ●✐➯ sư A✳ d>0 ✈➭ ❑❤✐ ➤ã ➳♣ ❞ơ♥❣ ▼Ư♥❤ ➤Ị ✶✳✷✳✼ t❛ ➤➢ỵ❝ N-dim(0 :A x) = d − 1✱ t❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ t❛ ❝ã dimR (0 :A x) ì R ị ➤đ ♥➟♥ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✸✳✸✱ ❧➭ x∈m d − HomR ((0 :A x); E) R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ❱× ✈❐②✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✸✳✸✱ ✭✐✐✐✮✱ t❛ ❝ã d−1 dimR (0 :A x) = dimR (HomR ((0 :A x); E)) = dimR (HomR (A; E)/x HomR (A; E)) dimR (Hom(A; E)) − = dimR A − ❱❐②✱ t❛ ❝ã ➤✐Ị✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ✶✳✷✳✺ ë ❈❤➢➡♥❣ ❞ô s❛✉ ❝❤♦ t❤✃② r➺♥❣ ♥Õ✉ 1✱ N-dimR A ❧✉➠♥ ❤÷✉ ❤➵♥✳ ❚✉② ♥❤✐➟♥✱ ✈Ý R ❦❤➠♥❣ ❧➭ ị tì ữ N-dim A ✈➭ dimR A ❝ã t❤Ó ❧➭ ✈➠ ❤➵♥✳ www.VNMATH.com ✷✽ ❱Ý ❞ô ✸✳✶✳✹✳ s❛♦ ❝❤♦ ❚å♥ t➵✐ ♠➠➤✉♥ ❆rt✐♥ A tr➟♥ ✈➭♥❤ ◆♦❡t❤❡r✱ ❦❤➠♥❣ ➤Þ❛ ♣❤➢➡♥❣ dimR A = ∞ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ ❧➭ ♠ét ✈➭♥❤ ➤❛ t❤ø❝ ✈➠ ❤➵♥ ❜✐Õ♥ T = k[x1 , , xn , ] x1 , , xn , ❧✃② ❤Ö sè tr➟♥ tr➢ê♥❣ k ❈❤♦ m1 , , mn , ❧➭ ❝➳❝ sè ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ tè ❝ñ❛ T mi − mi−1 < mi+1 − mi ➤➢ỵ❝ s✐♥❤ ❜ë✐ t✃t ❝➯ ♣❤➬♥ tư S= xj Ti ✈í✐ ♠ä✐ i✳ ❈❤♦ s❛♦ ❝❤♦ ✈➭ ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ pi mi < j < mi+1 ✳ ➜➷t R = TS Ti =T \pi ❑❤✐ ➤ã✱ t❤❡♦ ❬✶✹✱ ❆✶✱❱Ý ❞ô ✶❪✱ t❛ ❝ã ➜➷t A = E(R/m) ❧➭ ❜❛♦ ♥é✐ ①➵ ❝ñ❛ tr➢ê♥❣ t❤➷♥❣ ❞➢ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ♥➭♦ ➤ã ❝đ❛ N-dimR A = ht(m)✳ ✈× t❤Õ R ❧➭ ♠ét ✈➭♥❤ ◆♦❡t❤❡r ✈➭ dim R = ∞ R✳ ❑❤✐ ➤ã A R/m✱ ✈í✐ m ❧➭ ♠ét ❧➭ ❆rt✐♥ ✈➭ t❛ ❝ã t❤Ĩ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ ❚❛ ❝ị♥❣ ❝ã t❤Ĩ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ r➺♥❣ R ❧➭ ♠✐Ị♥ ♥❣✉②➟♥✱ AnnR A = ✈➭ ❞♦ ➤ã dimR A = dim R = ∞ ❈❤♦ A ❧➭ ♠ét R✲♠➠➤✉♥ ❆rt✐♥✳ ❑❤✐ ➤ã✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✹ ✈➭ ▼Ư♥❤ ➤Ị ✶✳✶✳✻✱ ❆rt✐♥ ✈➭ A ể ễ ợ ữ A ó tró❝ tù ♥❤✐➟♥ ❝đ❛ Rmj ✲♠➠➤✉♥ Rmj ✲♠➠➤✉♥ ❆rt✐♥✱ ✈í✐ mj ∈ Supp A, j = 1, , r✳ ❚õ ➤ã t❛ ❝ã ❝➳❝ ❦Õt q✉➯ s❛✉ ✭①❡♠ ❬✶✼✱ ❇ỉ ➤Ị ✶✳✽✱ ❍Ư q✉➯ ✶✳✶✷✱ ❍Ư q✉➯ ✷✳✼❪✮✳ ▼Ư♥❤ ➤Ị ✸✳✶✳✺✳ ✭✐✮ ✭✐✐✮ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ ➤ó♥❣✳ AttRmj A = {pRmj : p ∈ AttR A} AttRmj A = {q ∩ R : q ∈ AttRm A} j ▼Ư♥❤ ➤Ị ✸✳✶✳✷ ❝❤Ø r❛ r➺♥❣ ♥❤×♥ ❝❤✉♥❣ N-dimR A dimR A ❚✉② ♥❤✐➟♥✱ ✈Ý ❞ô s❛✉ t r ó ữ trờ ợ r ♥❤á ❤➡♥ t❤ù❝ sù✳ ❱Ý ❞ô ✸✳✶✳✻✳ s❛♦ ❝❤♦ ❚å♥ t➵✐ ♠➠➤✉♥ ❆rt✐♥ N-dimR A < dimR A A tr➟♥ ✈➭♥❤ ◆♦❡t❤❡r✱ ➤Þ❛ ♣❤➢➡♥❣ (R, m) www.VNMATH.com ✷✾ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ (R, m) ❧➭ ♠✐Ị♥ ♥❣✉②➟♥✱ ❝❤✐Ị✉ ➤➢ỵ❝ ①➞② ❞ù♥❣ ❜ë✐ ❋❡rr❛♥❞ ✈➭ ❘❛②♥❛✉❞ ❬✷✵❪ s❛♦ ❝❤♦ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ➤➬② ➤đ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt q ❝❤✐Ị✉ R ❝ñ❛ R ❝ã ♠ét ✐➤➟❛♥ ✭①❡♠ t❤➟♠ ◆❛❣❛t❛ ❬✶✹✱ ❆✶✱ ❱Ý ❞ơ ✷❪✮✳ ❱× q ∈ Ass(R), dim R/q = ✈➭ ❝❤ó ý r➺♥❣ t❛ ❝ã ➤➻♥❣ ❝✃✉ ❣✐÷❛ ❝➳❝ R✲♠➠➤✉♥ Hm1 (R) ∼ = Hm1 (R) ♥➟♥ t❤❡♦ ❬✶✱ ➜Þ♥❤ ❧ý ✶✶✳✸✳✸❪ ❝đ❛ ❇r♦❞♠❛♥♥✲❙❤❛r♣✱ t❛ ❝ã q ∈ AttR (Hm1 (R)) ❚❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✶✳✺ s✉② r❛ q = q ∩ R ∈ AttR (Hm1 (R)) ❍➡♥ ♥÷❛✱ ❞♦ R ❧➭ ♠✐Ị♥ ♥❣✉②➟♥ ♥➟♥ Ass(R) = {0}✱ ✈× t❤Õ q = q ∩ R ∈ Ass(R) = {0} ❉♦ ➤ã s✉② r❛ AnnR (Hm1 (R)) = AnnR (Hm1 (R)) ∩ R ⊂ q ∩ R = ❱× t❤Õ✱ t❛ ❝ã dimR (Hm1 (R)) = dim R/ AnnR (Hm1 (R)) = dim R = ▼➷t ❦❤➳❝✱ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✸✳✶ ✈➭ ▼Ư♥❤ ➤Ị ✸✳✶✳✷✱✭✐✮ t❛ ❝ã N-dimR (Hm (R)) ❱❐②✱ t❛ ➤➲ ❝❤Ø r❛ ➤➢ỵ❝ sù tå♥ t➵✐ ❝ñ❛ ♠➠➤✉♥ ❆rt✐♥ A = Hm1 (R) = s❛♦ ❝❤♦ N-dimR A = < = dimR A ✸✳✷ ➜✐Ị✉ ❦✐Ư♥ AnnR (0 :A p) = p ❈➳❝ ❦Õt q✉➯ ❝đ❛ ♠ơ❝ t❤ø❝ 3.1 N-dimR A = dimR A ❝❤♦ t❤✃② ❦❤➠♥❣ ♣❤➯✐ ❦❤✐ ♥➭♦ t❛ ❝ò♥❣ ❝ã ➤➻♥❣ ▼ét ❝➞✉ ❤á✐ tù ♥❤✐➟♥ ➤➷t r❛ ❧➭ ❦❤✐ ♥➭♦ t❤× t❛ ❝ã ➤➻♥❣ t❤ø❝ tr➟♥✳ ➜Ĩ tr➯ ❧ê✐ ❝❤♦ ❝➞✉ ❤á✐ ♥➭②✱ tr➢í❝ ❤Õt t❛ ♥❤➽❝ ❧➵✐ ❦Õt q✉➯ s❛✉✳ www.VNMATH.com ✸✵ ❇ỉ ➤Ị ✸✳✷✳✶✳ ỗ R ữ s M t ❝ã ➤➻♥❣ t❤ø❝ AnnR (M/pM ) = p✱ ✈í✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè p ∈ V (AnnR M )✳ ❈❤ø♥❣ ♠✐♥❤✳ ❍✐Ĩ♥ ♥❤✐➟♥ t❛ ❝ã ❜❛♦ ❤➭♠ t❤ø❝ p ✈× p ∈ V (AnnR M ) = Supp M ⊆ AnnR (M/pM ) ◆❣➢ỵ❝ ❧➵✐✱ t❤❡♦ ❬✶✽✱ ❇ỉ ➤Ị ✾✳✷✵❪ ♥➟♥ Mp = 0✳ ❉♦ ➤ã (M/pM )p = Mp /pRp Mp = ì ế ợ tì ➤ã Mp = pRp Mp , s✉② r❛ Mp = t❤❡♦ ❇ỉ ➤Ị ◆❛❦❛②❛♠❛✱ ❞➱♥ ➤Õ♥ ♠➞✉ t❤✉➱♥✳ ❱❐②✱ p ∈ Supp(M/pM ) = V (AnnR (M/pM )) ❤❛② p ⊇ AnnR (M/pM ) ▼ét ❝➞✉ ❤á✐ tù ♥❤✐➟♥ ➤➷t r❛ ❧➭ ➤è✐ ♥❣➱✉ ❝ñ❛ ❦Õt q✉➯ tr➟♥ ❝ã ❧✉➠♥ ➤ó♥❣ ❝❤♦ ♠➠➤✉♥ ❆rt✐♥ ❦❤➠♥❣✳ ➜Ĩ t❤✉❐♥ t✐Ư♥ ❝❤♦ ✈✐Ö❝ tr➯ ❧ê✐ ❝➞✉ ❤á✐ ♥➭②✱ t❛ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✷✳ ❝❤ø❛ ❑ý ❤✐Ư✉ V (AnnR A) ❧➭ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè AnnR A✳ ❚❛ ♥ã✐ r➺♥❣ A t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ ♥Õ✉ AnnR (0 :A p) = p, ✈í✐ ♠ä✐ p ∈ V (AnnR A) ❱Ý ❞ô s❛✉ ❝❤♦ t❤✃② r➺♥❣✱ ❝➞✉ tr➯ ❧ê✐ ❝ñ❛ ❝➞✉ ❤á✐ tr➟♥ ❧➭ ♣❤ñ ➤Þ♥❤ ♥❣❛② ❝➯ ❦❤✐ ✈➭♥❤ R ❧➭ ➤Þ❛ ♣❤➢➡♥❣✱ ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ ♠➠➤✉♥ ❆rt✐♥ tr➟♥ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ❦❤➠♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❱Ý ❞ơ ✸✳✷✳✸✳ s❛♦ ❝❤♦ ❚å♥ t➵✐ ♠➠➤✉♥ ❆rt✐♥ A tr➟♥ ✈➭♥❤ ◆♦❡t❤❡r✱ ➤Þ❛ ♣❤➢➡♥❣ (R, m) A ❦❤➠♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ R ✈➭ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè p A = Hm1 (R) s❛♦ ❝❤♦ p = ♥❤➢ tr♦♥❣ ❱Ý ❞ô ✸✳✶✳✻✳ ▲✃② t✉ú ý ✈➭ p = m✳ ❱× AnnR A = p ∈ V (AnnR A) ▲✃② ♠ét ♣❤➬♥ tö = x ∈ p✳ ❚❛ ❝ã ❞➲② ❦❤í♣ s❛✉ x −→ R −→ R −→ R/xR −→ ♥➟♥ www.VNMATH.com ✸✶ ❚❤❡♦ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ tư ➤è✐ ➤å♥❣ ➤✐Ị✉ t❛ ➤➢ỵ❝ ❞➲② ❦❤í♣ ❞➭✐ x −→ Hm0 R −→ Hm0 (R) −→ Hm0 (R/xR) −→ x −→ Hm1 (R) −→ Hm1 (R) −→ Hm1 (R/xR) −→ ❈❤ó ý r➺♥❣ R ❧➭ ♠✐Ò♥ ♥❣✉②➟♥ ♥➟♥ Hm0 (R) = 0✱ ❞♦ ➤ã t❛ t❤✉ ➤➢ỵ❝ ❞➲② ❦❤í♣ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ x −→ Hm0 (R/xR) −→ Hm1 (R) −→ Hm1 (R) ❱× ✈❐②✱ Hm0 (R/xR) ∼ = (0 :Hm1 (R) xR) = (0 :A xR) ❱× Hm0 (R/xR) ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ♥➟♥ (0 :A xR) ❝ị♥❣ ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥✳ ❉♦ x∈p ♥➟♥ :A x ⊇ :A p✱ AnnR (0 :A p) ❧➭ m✲♥❣✉②➟♥ s✉② r❛ ➤é ❞➭✐ ❝đ❛ s➡✳ ❱× t❤Õ t❛ ❝ã :A p ❝ị♥❣ ❤÷✉ ❤➵♥✳ ❉♦ ➤ã AnnR (0 :A p) = p✱ ♥❣❤Ü❛ ❧➭ A ❦❤➠♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❚✉② ♥❤✐➟♥✱ ❧í♣ ♠➠➤✉♥ ❆rt✐♥ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ ✈➱♥ ❝ß♥ ❦❤➳ ré♥❣ ✈➭ ➤✐Ị✉ ♥➭② ❝❤ø♥❣ tá ✈✐Ư❝ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ ❧➭ ❤÷✉ Ý❝❤✳ ➜Ĩ ❝❤Ø r❛ ❝➳❝ ❧í♣ ♠➠➤✉♥ ♥➭②✱ tr➢í❝ ❤Õt✱ ❝❤ó♥❣ t❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ➤è✐ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ ▼❡❧❦❡rss♦♥ ✈➭ ❙❝❤❡♥③❡❧ ❬✶✸❪ ♥❤➢ s❛✉✿ ➜è✐ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ R✲♠➠➤✉♥ HomR (S −1 R; A) ❧➭ ❦❤í♣ ✈➭ ♥❣✉②➟♥ tè ❆rt✐♥ A ø♥❣ ✈í✐ t❐♣ ♥❤➞♥ ➤ã♥❣ S ❍ä ❝ị♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ❤➭♠ tö coSupp A = V (AnnR A)✱ tr♦♥❣ ➤ã coSupp A ❧➭ S −1 R✲♠➠➤✉♥ Hom(S −1 R; −) ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ p s❛♦ ❝❤♦ HomR (Rp ; A) = ❑Õt q✉➯ ♥➭② ❝❤♦ ♣❤Ð♣ t❛ ➤➢❛ r❛ ♠ét sè ❧í♣ ♠➠➤✉♥ ❆rt✐♥ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ ♥❤➢ s❛✉✳ ❇ỉ ➤Ị ✸✳✷✳✹✳ ◆Õ✉ R ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ➤➬② ➤đ ❤♦➷❝ A ❝❤ø❛ ♠➠➤✉♥ ❝♦♥ ➤➻♥❣ ❝✃✉ ✈í✐ ❜❛♦ ♥é✐ ①➵ ❝đ❛ R/m t❤× A t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư R ❧➭ ✈➭♥❤ ➤➬② ➤ñ✳ ❑❤✐ ➤ã✱ ➤è✐ ♥❣➱✉ ▼❛t❧✐s HomR (A; E) ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ▲✃② p ∈ V (AnnR A)✱ s✉② r❛ p ∈ SuppR (HomR (A; E))✳ www.VNMATH.com ✸✷ ❱× ✈❐②✱ ➳♣ ❞ơ♥❣ ❝➳❝ ❦Õt ❝đ❛ ❝đ❛ ➤è✐ ♥❣➱✉ ▼❛t❧✐s tr♦♥❣ ▼Ư♥❤ ➤Ị ✷✳✸✳✸ ✈➭ ❇ỉ ➤Ị ✸✳✷✳✶✱ t❛ ❝ã AnnR (0 :A p) = AnnR HomR ((0 :A p); E) = AnnR (HomR (A; E)/p HomR (A; E)) = p ❱× ✈❐②✱ A t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❳Ðt tr➢ê♥❣ ❤ỵ♣ A ❝❤ø❛ ♠➠➤✉♥ ❝♦♥ ➤➻♥❣ ❝✃✉ ✈í✐ ❜❛♦ ♥é✐ ①➵ ❝đ❛ E✳ ▲✃② p ∈ V (AnnR A) ❚❤❡♦ ❬✶✸✱ ❇ỉ ➤Ị ✹✳✶❪✱ t❛ ❝ã AssRp (HomR (Rp ; A)) ⊇ AssRp (HomR (Rp ; E(R/m))) = {qRp : q ⊆ p} ❱× t❤Õ✱ t❛ ❝ã ✐➤➟❛♥ ❝ù❝ ➤➵✐ pRp tr♦♥❣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ Rp ♣❤➯✐ t❤✉é❝ t❐♣ AssRp (HomR (Rp ; A))✳ ➜✐Ò✉ ♥➭② s✉② r❛ (0 :HomR (Rp ;A) pRp ) = ❙✉② r❛ HomR (Rp ; (0 :A p)) = ❉♦ ➤ã✱ t❤❡♦ ❬✶✸✱ ♣✳✶✸✵❪ t❛ ♥❤❐♥ ➤➢ỵ❝ p ⊇ AnnR (0 :A p) tõ ➤ã p = AnnR (0 :A p) ➜Þ♥❤ ❧ý s❛✉ ➤➞② ❧➭ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ t✐Õt ♥➭②✱ ❝❤♦ t❛ ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤Ĩ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠ét ♠➠➤✉♥ ❆rt✐♥ ➜Þ♥❤ ❧ý ✸✳✷✳✺✳ ❆rt✐♥✳ ◆Õ✉ ❈❤♦ A ❜➺♥❣ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♥ã✳ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ◆♦❡t❤❡r ✈➭ A ❧➭ R✲♠➠➤✉♥ A t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ t❤× N-dimR A = dimR A✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✶✳✷✱ ✭✐✐✮ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ❜✃t ➤➻♥❣ t❤ø❝ dimR A N-dimR A✳ ❈❤♦ a ❧➭ ✐➤➟❛♥ ❜✃t ❦ú ❝đ❛ R✳ ❱× rad AnnR (0 :A a) = ∩ p p∈V (AnnR (0:A a)) ✈➭ rad a + AnnR A = ∩ p, p∈V (a+AnnR A) www.VNMATH.com ✸✸ ❤➡♥ ♥÷❛✱ ♥Õ✉ p ⊇ AnnR (0 :A a) t❤× p ⊇ (a + AnnR A) ♥➟♥ râ r➭♥❣ t❛ ❧✉➠♥ ❝ã rad AnnR (0 :A a) ⊇ rad a + AnnR A ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜❛♦ ❤➭♠ t❤ø❝ ♥❣➢ỵ❝ ❧➵✐✳ ❱í✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè p ❝❤ø❛ a + AnnR A, ✈× p ⊇ a ♥➟♥ (0 :A p) ⊆ (0 :A a)✳ ❉♦ ➤ã✱ t❤❡♦ ❣✐➯ t❤✐Õt✱ t❛ ❝ã AnnR (0 :A a) ⊆ AnnR (0 :A p) = p ❱× t❤Õ✱ rad AnnR (0 :A a) ⊆ rad a + AnnR A ❑Õt ❤ỵ♣ ✈í✐ tr➟♥ t❛ ❝ã ➤➻♥❣ t❤ø❝✳ ❇➞② ❣✐ê✱ ❣✐➯ sö N-dimR A = d ❑❤✐ ➤ã✱ t❤❡♦ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✻ ✈Ị ❤Ư t❤❛♠ sè ë ❈❤➢➡♥❣ ✶✱ tå♥ t➵✐ ❝➳❝ ♣❤➬♥ tö R (0 :A s❛♦ ❝❤♦ x1 , , x d ∈ m (x1 , , xd )R) < ∞ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✷✳✼ ✈➭ ➳♣ ❞ơ♥❣ ➤➻♥❣ t❤ø❝ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥ ✈➭♦ ✐➤➟❛♥ a = (x1 , , xd )✱ t❛ ♥❤❐♥ ➤➢ỵ❝ = dimR (0 :A (x1 , , xd )R) dimR A − d = dimR R/((x1 , , xd )R + AnnR A) ❱× ✈❐②✱ t❛ ❝ã dimR A d✳ ❑Õt ❤ỵ♣ ✈í✐ ▼Ư♥❤ ➤Ị ✸✳✶✳✷✱ ✭✐✐✮ t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤ó ý r ề ợ ủ ị ý ❦❤➠♥❣ ➤ó♥❣✳ ➜Ĩ ❝❤Ø r❛ ✈Ý ❞ơ ❧➭♠ s➳♥❣ tá ♥❤❐♥ ①Ðt tr➟♥✱ t❛ ❝➬♥ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ö♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ♠➠➤✉♥ ❝➳❝ ➤❛ t❤ø❝ ♥❣➢ỵ❝ ➤➲ ➤➢ỵ❝ ➤➢❛ r❛ ❜ë✐ ▼❛❝❛✉❧❛② ❬✾❪ ✈➭ ➤➲ ➤➢ỵ❝ ➤Ị ❝❐♣ ➤Õ♥ tr♦♥❣ ❬✼❪ ✈➭ ❬✶✻❪ ♥❤➢ s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛ ✸✳✷✳✻✳ R✲♠➠➤✉♥✳ ♥❣➢ỵ❝ ❬✾❪ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❝ã ➤➡♥ ✈Þ ✈➭ ❜ë✐ ❝➳❝ ♣❤➬♥ tư ❝ã ❞➵♥❣ ❝ñ❛ t ❜✐Õ♥ tr➟♥ ✈➭♥❤ m = axi11 xitt ❧➭ t✱ ♠➠➤✉♥ ❝➳❝ ➤❛ t❤ø❝ ❑❤✐ ó ỗ số M [x1 , , xt ] M R[x1 , , xt ] ✈í✐ a ∈ M ✈➭ ➤➢ỵ❝ s✐♥❤ i1 , , i t www.VNMATH.com ✸✹ ❧➭ ❝➳❝ sè ♥❣✉②➟♥ ❦❤➠♥❣ ❞➢➡♥❣✳ P❤Ð♣ ❝é♥❣ tr♦♥❣ −1 M [x−1 , , xt ] ợ ị ĩ t tự tí ợ ị s m axi11 xitt = R[x1 , , xt ]✱ tr♦♥❣ ➤ã raxi11 +j1 xitt +jt ✈➭ ❜➺♥❣ r∈R ✈➭ a ∈ M✱ x = rxj11 xjt t ∈ ✈➭ t❛ ➤Þ♥❤ ♥❣❤Ü❛ tÝ❝❤ ♥Õ✉ t✃t ❝➯ ik + jk ➤Ị✉ ❦❤➠♥❣ ❞➢➡♥❣ ✈í✐ ♠ä✐ xm ❧➭ ♣❤➬♥ tư k = 1, 2, , t tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥❣➢ỵ❝ ❧➵✐✳ ▼Ư♥❤ ➤Ị ✸✳✷✳✼✳ ♥❣➢ỵ❝ −1 M [x−1 , , xt ] t❤✉é❝ ❬✼❪✱ ❬✶✻❪ ✭✐✮ ◆Õ✉ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥ t❤× ♠➠➤✉♥ ❝➳❝ ➤❛ t❤ø❝ −1 A[x−1 , , xt ] ❧➭ R[x1 , , xt ]✲♠➠➤✉♥ ❆rt✐♥✳ ✭✐✐✮ ❈❤♦ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥ ✈➭ ➤➷t S −1 = R[x1 , , xt ], K = A[x−1 , , xt ]✳ ❑❤✐ ➤ã N-dimS K = N-dimR A + t ❱Ý ❞ơ s❛✉ ❝❤Ø r❛ r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ ❝❤Ø ❧➭ ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤Ĩ ♠ét ♠➠➤✉♥ ❆rt✐♥ ❝ã ❝❤✐Ị✉ ◆♦❡t❤❡r ❜➺♥❣ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♥ã✳ ❱Ý ❞ô ✸✳✷✳✽✳ s❛♦ ❝❤♦ ❚å♥ t➵✐ ♠➠➤✉♥ ❆rt✐♥ A tr➟♥ ✈➭♥❤ ◆♦❡t❤❡r✱ ➤Þ❛ ♣❤➢➡♥❣ (R, m) N-dimR A = dimR A✱ ♥❤➢♥❣ A ❦❤➠♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư r➺♥❣ tå♥ t➵✐ ♥❤÷♥❣ ♠➠➤✉♥ ❆rt✐♥ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✭✐✮ tr➟♥ ✈➭♥❤ ➤Þ❛ R s❛♦ ❝❤♦ ❝➳❝ ➤✐Ị✉ ❦✐Ö♥ s❛✉ t❤♦➯ ♠➲♥ N-dimR A = dimR A > dimR A > N-dimR A ✭✐✐✮ ❚å♥ t➵✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤♦ A ,A p ∈ V (AnnR A ) ✈➭ p ∈ V (AnnR A ) s❛♦ AnnR (0 :A p) = p ➜➷t A = A ⊕ A ❑❤✐ ➤ã✱ t❛ ❝ã ❞➲② ❦❤í♣ −→ A −→ A −→ A −→ ❙ư ❞ơ♥❣ tÝ♥❤ ❝❤✃t ✈Ị ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ❝đ❛ ♠ét ❞➲② ❦❤í♣ tr♦♥❣ ▼Ư♥❤ ➤Ị ✶✳✶✳✸ ✈í✐ ❝❤ó ý r➺♥❣ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ò✉ s❛✉✿ AnnR A ⊆ AnnR A , t❛ ❝ã A www.VNMATH.com ✸✺ ✶✳ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥✳ ✷✳ N-dimR A = N-dimR A = dimR A = dimR A ✸✳ p ∈ V (AnnR A)✳ ❚✉② ♥❤✐➟♥✱ t❤❡♦ ❣✐➯ t❤✐Õt t❛ ❝ã AnnR (0 :A p) = AnnR (0 :A p) ∩ AnnR (0 :A p) = p ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá A ❦❤➠♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮✳ ❇➞② ❣✐ê✱ ❝❤ó♥❣ t❛ sÏ ❝❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ♠➠➤✉♥ ❧➭ ♠✐Ị♥ ♥❣✉②➟♥ ❝❤✐Ò✉ t A ✈➭ A ë tr➟♥✳ ❈❤♦ R ♥❤➢ tr♦♥❣ ❱Ý ❞ô ✸✳✶✳✻✳ ❈❤♦ S = R[[x1 , , xt ]]✱ ✈í✐ ❧➭ ✈➭♥❤ ỗ ỹ từ ì tứ t ế x1 , , xt tr➟♥ ✈➭♥❤ R✳ ▲✃② −1 A = k[[x−1 , , xt ]] ❧➭ ♠➠➤✉♥ ❝➳❝ ➤❛ t❤ø❝ ♥❣➢ỵ❝ tr➟♥ tr➢ê♥❣ ❑❤✐ ➤ã✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✸✳✷✳✼✱ AnnS A = mS A ❧➭ S ✲♠➠➤✉♥ ❆rt✐♥ ✈➭ k = R/m✳ N-dimS A = t✳ ❱× ♥➟♥ dimS A = dim(k[[x1 , , xt ]]) = t ❈❤♦ A = Hm1 (R) ❧➭ S ✲♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ s❛♦ ❝❤♦ xi A = ✈í✐ ♠ä✐ i = 1, , t ó ỗ t ❝♦♥ ❝ñ❛ A ✈➭ ❝❤Ø ❦❤✐ ♥ã ❧➭ S ✲♠➠➤✉♥ ❝♦♥ ❝đ❛ A ✳ ❱× ✈❐② ❧➭ A R✲♠➠➤✉♥ ❝♦♥ ❝đ❛ A ❝ị♥❣ ❧➭ ❦❤✐ S ✲♠➠➤✉♥ ❆rt✐♥ ✈➭ dimS A = 2; N-dimS A = ❘â r➭♥❣ r➺♥❣ ❝❤♦ AnnS A = (x1 , , xt )S ❈❤♦ p = m ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ S p ⊇ AnnS A ❑❤✐ ➤ã p ∈ V (AnnS A ) s❛♦ ❇➺♥❣ ❝➳❝❤ tÝ♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❱Ý ❞ơ ✸✳✷✳✸✱ t❛ ❝ã ➤✐Ị✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ AnnS (0 :A p) = p ❍Ö q✉➯ ✸✳✷✳✾✳ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ◆♦❡t❤❡r ✈➭ A ❧➭ R✲♠➠➤✉♥ ❆rt✐♥✳ ❑ý ❤✐Ư✉ R ❧➭ ➤➬② ➤đ t❤❡♦ t➠♣➠ m − adic ❝ñ❛ R✳ ❑❤✐ ➤ã t❛ ❝ã N-dimR A = dimR A www.VNMATH.com ✸✻ ❈❤ø♥❣ ♠✐♥❤✳ ❱× ✶✳✶✳✼✱ ✭✐✐✮✱ t❛ ❝ã ❦✐Ư♥ ✭✯✮ tr➟♥ A ❝ã ❝✃✉ tró❝ tù ♥❤✐➟♥ ❧➭ N-dimR A = N-dimR A✳ R✲♠➠➤✉♥ ▼➷t ❦❤➳❝✱ ❆rt✐♥ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị A ❝ị♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ R t❤❡♦ ❇ỉ ➤Ị ✸✳✷✳✹✳ ❱× t❤Õ✱ tõ ➜Þ♥❤ ❧ý ✸✳✷✳✺✱ t❛ ❝ã ♥❣❛② N-dimR A = N-dimR A = dimR A www.VNMATH.com ✸✼ ❑Õt ❧✉❐♥ ❚ã♠ ❧➵✐✱ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭② ❝❤ó♥❣ t➠✐ ➤➲ tr×♥❤ ❜➭② ❧➵✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❜➭✐ ❜➳♦✿ ✧❖♥ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥s✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧ ❝ñ❛ ◆✳ ❚✳ ❈➢ê♥❣ ✲ ▲✳ ❚✳ ◆❤➭♥ ✭✷✵✵✷✮ ✈➭ ♠ét ♣❤➬♥ ❦Õt q✉➯ ❝ñ❛ ❝➳❝ ❜➭✐ ❜➳♦✿ ✧❑r✉❧❧ ❞✐♠❡♥s✐♦♥ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦✈❡r q✉❛s✐✲❧♦❝❛❧ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s✧ ❝ñ❛ ❘✳ ◆✳ ❘♦❜❡rts ✭✶✾✼✺✮❀ ✧❉✐♠❡♥s✐♦♥ ❛♥❞ ❧❡♥❣t❤ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧ ❝ñ❛ ❉✳ ❑✐r❜② ✭✶✾✾✵✮ ✈➭ ✧❉✐♠❡♥s✐♦♥✱ ♠✉❧t✐♣❧✐❝✐t② ❛♥❞ ❍✐❧❜❡rt ❢✉♥❝t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧ ❝ñ❛ ◆✳ ❚✳ ❈➢ê♥❣ ✲ ▲✳ ❚✳ ◆❤➭♥ ✭✶✾✾✾✮✳ ❑Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ❣å♠ ❝➳❝ ♥é✐ ❞✉♥❣ s❛✉✳ ✶✳ ❍Ư t❤è♥❣ ❧➵✐ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ♥é✐ ❞✉♥❣ ❝ñ❛ ❧✉❐♥ ✈➝♥✳ ✷✳ ●✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ❦Õt q✉➯ ✈Ị ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✳ ➜➷❝ ❜✐Ưt ❧➭ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❤÷✉ ❤➵♥ ❝đ❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈í✐ ❜❐❝ ❝ñ❛ ➤❛ t❤ø❝ ❍✐❧❜❡rt ❝ñ❛ ♠ét ♠➠➤✉♥ ❆rt✐♥✳ ✸✳ ◆❣❤✐➟♥ ❝ø✉ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❦❤✐ ❝❤ó♥❣ ❧➭ ❆rt✐♥✿ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ t❤ø ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ❝❤✐Ị✉ i ✈í✐ ❝❤Ø sè i ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ✈í✐ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s rì ố q ệ ữ ❝❤✐Ị✉ ◆♦❡t❤❡r ✈➭ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ tr♦♥❣ tr➢ê♥❣ ợ tổ qt N-dimR A dimR A ỉ r ữ tr➢ê♥❣ ❤ỵ♣ ①➯② r❛ ❞✃✉ ♥❤á ❤➡♥ t❤ù❝ sù ✈➭ ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤Ĩ ❦❤✐ ♥➭♦ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ♠ét ♠➠➤✉♥ ❆rt✐♥ ❜➺♥❣ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♥ã✳ www.VNMATH.com ✸✽ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❇r♦❞♠❛♥♥✱ ▼✳ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✾✽✮✱ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡❜r❛✐❝ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ●❡♦♠❡tr✐❝ ❆♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❬✷❪ ◆✳ ❚✳ ❈✉♦♥❣✱ ◆✳ ❚✳ ❉✉♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✷✵✵✼✮✱ ✧❚♦♣ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ❛♥❞ t❤❡ ❝❛t❡♥❛r② ♦❢ t❤❡ ✉♥♠✐①❡❞ s✉♣♣♦rt ♦❢ ❛ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠♦❞✉❧❡✧✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✺✭✸✺✮✱ ♣♣✳ ✶✻✾✶✲✶✼✵✶✳ ❬✸❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✶✾✾✾✮✱ ✧❉✐♠❡♥s✐♦♥✱ ♠✉❧t✐♣❧✐❝✐t② ❛♥❞ ❍✐❧❜❡rt ❢✉♥❝t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ❊❛st✲❲❡st ❏✳ ▼❛t❤✳✱ ✶ ✭✷✮✱ ♣♣✳ ✶✼✾✲✶✾✻✳ ❬✹❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✷✵✵✷✮✱ ✧❖♥ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥s✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳✱ ✸✵✱ ♣♣✳ ✶✷✶✲✶✸✵✳ ❬✺❪ ❉❡♥✐③❧❡r✱ ■✳ ❍✳ ❛♥❞ ❘✳ ❨✳ ❙❤❛♣ ✭✶✾✾✻✮✱ ✧❈♦✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ♦✈❡r ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s✧✱ ●❧❛s❣♦✇ ▼❛t❤✳ ❏✳ ✸✽✱ ♣♣✳ ✸✺✾✲✸✻✻✳ ❬✻❪ ●r♦t❤❡♥❞✐❡❝❦✱ ❆✳ ✭✶✾✻✻✮✱ ✧▲♦❝❛❧ ❤♦♠♦❧♦❣②✧✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳ ✷✵✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✲❍❡✐❞❡❧❜❡r❣✲◆❡✇ ❨♦r❦ ✳ ❬✼❪ ❑✐r❜② ❉✳ ✭✶✾✼✸✮✱ ✧❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ❛♥❞ ❍✐❧❜❡rt ♣♦❧②♥♦♠✐❛❧s✧✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ✭❙❡r✳ ✷✮ ✷✹ ✭✷✮✱ ♣♣✳ ✹✼✲✺✼✳ ❬✽❪ ❑✐r❜②✱ ❉✳ ✭✶✾✾✵✮✱ ✧❉✐♠❡♥s✐♦♥ ❛♥❞ ❧❡♥❣t❤ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞✱ ✭❙❡r✳ ✷✮ ✹✶ ✭✷✮✱ ♣♣✳ ✹✶✾✲✹✷✾✳ www.VNMATH.com ✸✾ ❬✾❪ ▼❛❝❛✉❧❛②✱ ❋✳ ❙✳ ✭✶✾✶✻✮✱ ✧❆❧❣❡❜r❛✐❝ ❚❤❡♦r② ♦❢ ▼♦❞✉❧❛r s②st❡♠✧✱ ❈❛♠✲ ❜r✐❞❣❡ ❚r❛❝ts ✶✾✳ ❬✶✵❪ ▼❛❝❞♦♥❛❧❞✱ ■✳ ●✳ ✭✶✾✼✸✮✱ ✧❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣✧✱ ❙②♠♣♦s✐❛ ▼❛t❤❡♠❛t✐❝❛✳ ✶✶✱ ♣♣✳ ✷✸✲✹✸✳ ❬✶✶❪ ▼❛t❧✐s✱ ❊✳ ✭✶✾✺✽✮✱ ✧■♥❥❡❝t✐✈❡ ♠♦❞✉❧❡s ♦✈❡r ◆♦❡t❤❡r✐❛♥ r✐♥❣s✧✱ P❛❝✐❢✐❝ ❏✳ ▼❛t❤✳ ✽✱ ♣♣✳ ✺✶✶✲✺✷✽✳ ❬✶✷❪ ▼❛ts✉♠✉r❛✱ ❍✳ ✭✶✾✼✵✮✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❇❡♥❥❛♠✐♥✳ ❬✶✸❪ ▼❡❧❦❡rss♦♥✱ ▲✳ ❛♥❞ P✳ ❙❝❤❡♥③❡❧ ✭✶✾✾✺✮✱ ✧❚❤❡ ❝♦✲❧♦❝❛❧✐③❛t✐♦♥ ♦❢ ❛♥ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡✧✱ Pr♦❝✳ ❊❞✐♥❜✉r❣❤ ▼❛t❤✳ ❙♦❝✳ ✸✽✱ ♣♣✳ ✶✷✶✲✶✸✶✳ ❬✶✹❪ ◆❛❣❛t❛ ▼✳✱ ✭✶✾✻✷✮✱ ▲♦❝❛❧ r✐♥❣✱ ■♥t❡rs❝✐❡♥❝❡✱ ◆❡✇ ❨♦r❦✳ ❬✶✺❪ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ❚✳ ◆✳ ❆♥✳✱ ✭✷✵✵✽✮✱ ❖♥ t❤❡ ✉♥♠✐①❡❞♥❡ss ❛♥❞ ✉♥✐✈❡rs❛❧ ❝❛t❡♥❛r✐❝✐t② ♦❢ r✐♥❣ ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ Pr❡♣r✐♥t✳ ❬✶✻❪ ❘♦❜❡rts✱ ❘✳ ◆✳ ✭✶✾✼✺✮✱ ✧❑r✉❧❧ ❞✐♠❡♥s✐♦♥ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦✈❡r q✉❛s✐✲❧♦❝❛❧ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s✧✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞✱ ✭❙❡r✳ ✷✮ ✷✻✱ ♣♣✳ ✷✻✾✲✷✼✸✳ ❬✶✼❪ ❙❤❛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♣✱ ❘✳ ❨✳ ✭✶✾✾✵✮ ❙t❡♣s ✐♥ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✶✾❪ ❚❛♥❣✱ ❩✳ ❛♥❞ ❍✳ ❩❛❦❡r✐ ✭✶✾✾✹✮✱ ✧❈♦✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❛♥❞ ♠♦❞✉❧❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✧✱ ❈♦♠♠✳ ❆❧❣❡❜r❛✳✱ ✷✷ ✭✻✮✱ ♣♣✳ ✷✶✼✸✲ ✷✷✵✹✳ www.VNMATH.com ✹✵ ❬✷✵❪ ❋❡rr❛♥❞ ❉✳ ❛♥❞ ▼✳ ❘❛②♥❛✉❞ ✭✶✾✼✵✮✱ ✧❋✐❜r❡s ❢♦r♠❡❧❧❡s ❞✬✉♥ ❛♥♥❡❛✉ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥✱✧ ❆♥♥✳ ❙❝✐✳ ❊✬❝♦❧❡ ◆♦r♠✳ ❙✉♣✳✱ ✸ ✭✹✮✱ ♣♣✳ ✷✾✺✲✸✶✶✳ ...www.VNMATH.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM - TRẦN THỊ HƯỜNG CHIỀU NOETHER CỦA MÔĐUN ARTIN Chuyên ngành: Đại số lý thuyết số Mã số: 60.46.05 LUẬN VĂN THẠC SĨ KHOA HỌC TOÁN... www.VNMATH.com ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC SƯ PHẠM - TRẦN THỊ HƯỜNG CHIỀU NOETHER CỦA MÔĐUN ARTIN Chuyên ngành: Đại số lý thuyết số Mã số: 60.46.05 TÓM TẮT LUẬN VĂN THẠC SĨ KHOA