Về đối đồng điều địa phương với giá cực đại và tính catenary của vành noether địa phương

VIỆN KHOA HỌC VÀ CƠNG NGHỆ VIỆT NAM VIỆN TỐN HỌC -oOo - Trần Nguyên An VÒ đối đồng điều địa phơng với giá cực đại V tính catenary vnh NOETHER địa phơng LUN N TIN SĨ TỐN HỌC HÀNỘI-2011 VIỆN KHOA HỌC VÀ CƠNG NGHỆ VIỆT NAM VIỆN TOÁN HỌC -oOo - Trn Nguyờn An Về đối đồng điều địa phơng với giá cực đại V tính catenary vnh NOETHER địa ph−¬ng Chuyên ngành: Đại số lý thuyết số Mã số: 62 46 05 01 LUẬN ÁN TIẾN SĨ TOÁN HỌC TẬP THỂ HƯỚNG DẪN KHOA HỌC: GS TSKH Nguyễn Tự Cường PGS TS Lê Thị Thanh Nhàn HÀNỘI-2011 ❚ã♠ t➽t ●✐➯ sö (R, m) ❧➭ ✈➭♥❤ ❣✐❛♦ ị tr M R ữ s A ❧➭ ♠ét R✲♠➠➤✉♥ ❆rt✐♥✳ ▼➠➤✉♥ A ➤➢ỵ❝ ❣ä✐ ❧➭ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ♥Õ✉ AnnR (0 :A p) = p ✈í✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè p ⊇ AnnR A ▲✉❐♥ ➳♥ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ✭∗✮ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ Hmi (M ) tr♦♥❣ ♠è✐ ❧✐➟♥ ❤Ư ✈í✐ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ tÝ♥❤ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ✈➭♥❤ ❝➡ së✳ ➜å♥❣ t❤ê✐ ❧✉❐♥ ➳♥ ❝ị♥❣ ➤Ị ❝❐♣ ➤Õ♥ ♠ét sè ✈✃♥ ➤Ị ✈Ị ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✱ ❝➠♥❣ t❤ø❝ ❜é✐ ❧✐➟♥ ❦Õt✱ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣✱ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ▲✉❐♥ ➳♥ ❜❛♦ ❣å♠ ❝❤➢➡♥❣✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✶✱ ❝❤ó♥❣ t➠✐ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ✈Ị ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ✈➭♥❤✱ ❝❤✐Ò✉✱ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ✈➭ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✳ ▼ét ➤➷❝ tr➢♥❣ ♠í✐ ✈Ị tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ t❤➠♥❣ q✉❛ ❤Ư t❤❛♠ số ợ trì tr tết ố ủ r ❈❤➢➡♥❣ ✷✱ ❝❤ó♥❣ t➠✐ ➤➷❝ tr➢♥❣ tÝ♥❤ ❝❤✃t ✭∗✮ ❝❤♦ Hmi (M ) q✉❛ t❐♣ ❣✐➯ ❣✐➳ t❤ø i ❝ñ❛ M ❀ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ✭∗✮ ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ i ➤Þ❛ ♣❤➢➡♥❣ Hm (M ) tr♦♥❣ ♠è✐ ❧✐➟♥ ❤Ư ✈í✐ tÝ♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ ✈➭♥❤ R/ AnnR M ✈➭ tÝ♥❤ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ❝➳❝ ✈➭♥❤ R/p ✈í✐ p ∈ Supp(M ) ❚r♦♥❣ ❈❤➢➡♥❣ ✸✱ ❝❤ó♥❣ t➠✐ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ✭∗✮ ❝ñ❛ ♠➠➤✉♥ ❆rt✐♥ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥✱ ➤➷❝ ❜✐Ưt ❧➭ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ tr♦♥❣ ♠è✐ ❧✐➟♥ ❤Ư ✈í✐ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ✈➭♥❤ ❝➡ së ✈➭ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✳ ❚r♦♥❣ ❈❤➢➡♥❣ ✹✱ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✿ ➤➢❛ r❛ ❝➠♥❣ t❤ø❝ ❜é✐ ❧✐➟♥ ❦Õt ❝đ❛ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ Hmi (M ) Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮❀ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣❀ ♥❣❤✐➟♥ ❝ø✉ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ Hmi (M ) ❧✐➟♥ ❤Ư ✈í✐ t❐♣ ❣✐➳ ✈➭ ❝➳❝ t❐♣ ❣✐➯ ❣✐➳ ❝ñ❛ M ✳ ❆❜str❛❝t ▲❡t (R, m) ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡ ◆♦❡t❤❡r✐❛♥ ❧♦❝❛❧ r✐♥❣ ❛♥❞ ❧❡t M ❜❡ ❛ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ R✲♠♦❞✉❧❡✳ ❆♥ ❆rt✐♥✐❛♥ R✲♠♦❞✉❧❡ A ✐s s❛✐❞ t♦ s❛t✐s❢② t❤❡ ♣r♦♣❡rt② ✭∗✮ ✐❢ AnnR (0 :A p) = p ❢♦r ❡✈❡r② ♣r✐♠❡ ✐❞❡❛❧ p ⊇ AnnR A ■♥ t❤✐s t❤❡s✐s✱ ✇❡ st✉❞② t❤❡ ♣r♦♣❡rt② ✭∗✮ ♦❢ t❤❡ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s Hmi (M ) ❝♦♥❝❡r♥✐♥❣ ❝❛t❡♥❛r✐t②✱ ✉♥✐✈❡rs❛❧ ❝❛t❡♥❛r✐t②✱ ❛♥❞ ✉♥♠✐①❡❞♥❡ss ♦❢ t❤❡ ❜❛s❡ r✐♥❣✳ ❲❡ ❛❧s♦ ❞❡❛❧ ✇✐t❤ t❤❡ ❞✐♠❡♥s✐♦♥✱ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ❢♦r♠✉❧❛❡✱ t❤❡ s❤✐❢t❡❞ ❧♦❝❛❧✐③❛t✐♦♥ ♣r✐♥❝✐♣❧❡✱ ❛♥❞ t❤❡ s❡t ♦❢ ❛tt❛❝❤❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✳ ❚❤❡ t❤❡s✐s ✐s ❞✐✈✐❞❡❞ ✐♥t♦ ❢♦✉r ❝❤❛♣t❡rs✳ ■♥ ❈❤❛♣t❡r ✶ ✇❡ r❡❝❛❧❧ s♦♠❡ ❜❛s✐❝ ❦♥♦✇❧❡❞❣❡ ♦♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❝❛t❡♥❛r✐t② ♦❢ r✐♥❣s✱ ❛tt❛❝❤❡❞ ♣r✐♠❡s✱ ❛♥❞ t❤❡ ♣r♦♣❡rt② ✭∗✮ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✳ ❆ ♥❡✇ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♣r♦♣❡rt② ✭∗✮ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ✐♥ t❡r♠s ♦❢ s②st❡♠s ♦❢ ♣❛r❛♠❡t❡rs ✐s ♣r♦✈❡❞ ✐♥ t❤❡ ❧❛st s❡❝t✐♦♥ ♦❢ t❤✐s ❝❤❛♣t❡r✳ ■♥ ❈❤❛♣t❡r ✷ ✇❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♣r♦♣❡rt② ✭∗✮ ♦❢ s✉♣♣♦rt ♦❢ Hmi (M ) ✈✐❛ it❤ ♣s❡✉❞♦✲ M ✳ ❲❡ st✉❞② t❤❡ ♣r♦♣❡rt② ✭∗✮ ❢♦r Hmi (M ) ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ✉♥✐✈❡rs❛❧ ❝❛t❡♥❛r✐t② ♦❢ t❤❡ r✐♥❣ R/ AnnR M ❛♥❞ ✉♥♠✐①❡❞♥❡ss ♦❢ t❤❡ r✐♥❣s R/p ❢♦r p ∈ Supp(M ) ■♥ ❈❤❛♣t❡r ✸ ✇❡ st✉❞② t❤❡ ♣r♦♣❡rt② ✭∗✮ ❢♦r q✉❛s✐ ✉♥♠✐①❡❞ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ st✉❞② t❤❡ ♣r♦♣❡rt② ✭∗✮ ❢♦r q✉❛s✐✲✉♥♠✐①❡❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ❝♦♥❝❡r♥✐♥❣ ❝❛t❡♥❛r✐t② ♦❢ t❤❡ ❜❛s❡ r✐♥❣✱ ❛♥❞ t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡s❡ ♠♦❞✉❧❡s✳ ■♥ ❈❤❛♣t❡r ✹ ✇❡ ❣✐✈❡ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣❡rt② ✭∗✮ ♦❢ ❧♦❝❛❧ ❝♦❤♦✲ ♠♦❧♦❣② ♠♦❞✉❧❡s✿ ❣✐✈❡ t❤❡ ❛ss♦❝✐❛t✐✈✐t② ❢♦r♠✉❧❛❡ ❢♦r ♠✉❧t✐♣❧✐❝✐t② ♦❢ Hmi (M ) i ✐♥ ❝❛s❡ Hm (M ) s❛t✐s❢✐❡s t❤❡ ♣r♦♣❡rt② ✭∗✮❀ st✉❞② t❤❡ s❤✐❢t❡❞ ❧♦❝❛❧✐③❛t✐♦♥ ♣r✐♥❝✐✲ i ♣❧❡ ❛♥❞ t❤❡ s❡t ♦❢ ❛tt❛❝❤❡❞ ♣r✐♠❡s ♦❢ Hm (M ) ❝♦♥❝❡r♥✐♥❣ s✉♣♣♦rt ❛♥❞ ♣s❡✉❞♦✲ s✉♣♣♦rts ♦❢ M✳ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ➤➞② ❧➭ ❝➠♥❣ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ r✐➟♥❣ t➠✐✳ ❈➳❝ ❦Õt q✉➯ ✈✐Õt ❝❤✉♥❣ ✈í✐ t➳❝ ❣✐➯ ❦❤➳❝ ➤➲ ➤➢ỵ❝ sù ♥❤✃t trÝ ❝đ❛ ➤å♥❣ t➳❝ ❣✐➯ ❦❤✐ ➤➢❛ ✈➭♦ ❧✉❐♥ ➳♥✳ ❈➳❝ ❦Õt q✉➯ ❝ñ❛ ❧✉❐♥ ➳♥ ❧➭ ♠í✐ ✈➭ ❝❤➢❛ tõ♥❣ ➤➢ỵ❝ ❛✐ ❝➠♥❣ ❜è tr♦♥❣ ❜✃t ❦× ❝➠♥❣ tr×♥❤ ♥➭♦ ❦❤➳❝✳ ❚➳❝ ❣✐➯ ❚r➬♥ ◆❣✉②➟♥ ❆♥ ▲ê✐ ❝➯♠ ➡♥ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ tí✐ t❤➬② t➠✐ ●❙✳ ❚❙❑❍✳ ◆❣✉②Ơ♥ ❚ù ❈➢ê♥❣✳ ❚❤➬② ➤➲ ❝❤Ø ❞➵② ❝❤♦ t➠✐ tõ ♥❤÷♥❣ ❜➭✐ ❤ä❝ ➤➬✉ t✐➟♥ ❝đ❛ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ➤Õ♥ ♥❤÷♥❣ ❝❤đ ➤Ị tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✳ ❚❤➬② ➤➲ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥ t➠✐ tõ ❦❤✐ t➠✐ ❧➭♠ ❧✉❐♥ ✈➝♥ t❤➵❝ sÜ ✈➭ ❣✐ê ➤➞② ❧➭ ❧✉❐♥ ➳♥ t✐Õ♥ sÜ✳ P❤➢➡♥❣ ♣❤➳♣ ➤ä❝ s➳❝❤✱ ❝➳❝❤ ♣❤➳t ❤✐Ư♥ ✈➭ ❣✐➯✐ q✉②Õt ✈✃♥ ➤Ị✱ ♥❤÷♥❣ ý t➢ë♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ♠➭ t❤➬② ❤➢í♥❣ ❞➱♥ ➤➲ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ➳♥ ♥➭② ✈➭ tr➢ë♥❣ t❤➭♥❤ ❤➡♥ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✳ ❚❤➬② ❧✉➠♥ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❝❤♦ t➠✐ ❝ã ❞Þ♣ ❣✐❛♦ ❧➢✉ q✉è❝ tÕ ➤Ĩ t➠✐ t❤➟♠ tù t✐♥✳ ◆❤➢ ♠ét ♥❣➢ê✐ ❝❤❛✱ t❤➬② ❧✉➠♥ ✉è♥ ♥➽♥ t➠✐ ➤Ĩ t➠✐ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✱ ❣✐ó♣ ➤ì t➠✐ ❝➯ ✈Ị ✈❐t ❝❤✃t ✈➭ t✐♥❤ t❤➬♥✳ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ tí✐ ❝➠ t➠✐ P●❙✳ ❚❙✳ ▲➟ ❚❤Þ ❚❤❛♥❤ ◆❤➭♥✳ ❈➠ ❧➭ ♥❣➢ê✐ t❤➬② ➤➬✉ t✐➟♥ ➤➲ ➤➢❛ t➠✐ ➤Õ♥ ✈í✐ t♦➳♥ ❤✐Ư♥ ➤➵✐✱ ➤Õ♥ ✈í✐ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ❈➠ ❧✉➠♥ t❐♥ t×♥❤✱ tØ ♠Ø ❤➢í♥❣ ➤➱♥ t➠✐ tõ ❦❤✐ t➠✐ ❧➭♠ ➤Ò t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ s✐♥❤ ✈✐➟♥✱ ❧✉❐♥ ✈➝♥ ➤➵✐ ❤ä❝ ✈➭ ➤➷❝ ❜✐Öt ❧➭ ❧✉❐♥ ➳♥ t✐Õ♥ sÜ✳ ❑❤➠♥❣ ❝❤Ø tr✉②Ò♥ ❝❤♦ t➠✐ sù s❛② ♠➟ tr♦♥❣ ❣✐➯♥❣ ❞➵② ✈➭ ♥❣❤✐➟♥ ❝ø✉✱ ❝➠ ❧✉➠♥ q✉❛♥ t➞♠✱ ➤é♥❣ ✈✐➟♥✱ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ ❝➠♥❣ ✈✐Ư❝✱ tr♦♥❣ ❝✉é❝ sè♥❣✳ ❙ù t❐♥ t➞♠ ✈í✐ ♥❣❤Ị✱ ✈í✐ ❤ä❝ trß ❝đ❛ ❝➠ sÏ ❧➭ ❝➳✐ ➤Ý❝❤ ❝❤♦ t➠✐ ❤ä❝ ❤á✐✱ ♥♦✐ t❤❡♦✳ ▲✉❐♥ ➳♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ ❤❛✐ ♥❣➢ê✐ t❤➬② ●❙✳ ❚❙❑❍✳ ◆❣✉②Ơ♥ ❚ù ❈➢ê♥❣ ✈➭ P●❙✳ ❚❙✳ ▲➟ ❚❤Þ ❚❤❛♥❤ ◆❤➭♥✳ ột ữ t tỏ ò ết s s➽❝ ➤Õ♥ ❤❛✐ ♥❣➢ê✐ t❤➬② ❝ñ❛ t➠✐ ✈➭ sÏ ♣❤✃♥ ➤✃✉ ❤➡♥ ♥÷❛ ➤Ĩ ①ø♥❣ ➤➳♥❣ ✈í✐ ❝➠♥❣ ❧❛♦ ❝đ❛ t❤➬②✱ ❝➠✱ ①ø♥❣ ➤➳♥❣ ✈í✐ ♥✐Ị♠ t✐♥ ❝đ❛ t❤➬②✱ ❝➠ ➤➲ ❞➭♥❤ ❝❤♦ t➠✐✳ ❚➠✐ ①✐♥ ❝❤➞♥ trä♥❣ ❝➯♠ ➡♥ ●❙✳ ❚❙❑❍✳ ▲➟ ❚✉✃♥ ❍♦❛ ✈➭ ❚❙✳ ❋r❡❞ ❘♦❤r❡r ➤➲ ➤ä❝ ✈➭ ❣ã♣ ♥❤÷♥❣ ý ❦✐Õ♥ q✉ý ❜➳✉ ❝❤♦ ❧✉❐♥ ➳♥✳ ❚➠✐ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ ❱✐Ö♥ ❚♦➳♥ ❤ä❝✱ ❝➳❝ ♣❤ß♥❣ ❝❤ø❝ ♥➝♥❣✱ ❚r✉♥❣ t➞♠ ➜➭♦ t➵♦ s❛✉ ➤➵✐ ❤ä❝ ❝đ❛ ❱✐Ư♥ ❚♦➳♥ ❤ä❝ ➤➲ ❝❤♦ t➠✐ ♠ét ♠➠✐ tr➢ê♥❣ ❤ä❝ t❐♣✱ ♥❣❤✐➟♥ ❝ø✉ ❧ý t➢ë♥❣ ➤Ó t➠✐ ❝ã t❤Ó ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ➳♥ ♥➭②✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❇❛♥ ❣✐➳♠ ❤✐Ö✉ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥ ➤➲ ❝❤♦ t➠✐ ❝➡ ❤é✐ ➤➢ỵ❝ ➤✐ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❇❛♥ ❝❤ñ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥ ✈➭ tỉ ➜➵✐ sè ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉ ①Õ♣ ❝➠♥❣ ✈✐Ư❝ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ t➠✐ ❧➭♠ ♥❣❤✐➟♥ ❝ø✉ s✐♥❤✳ ❚➠✐ ①✐♥ ữ ệ ị ✈➭ ➤❛♥❣ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ t➵✐ ♣❤ß♥❣ ➜➵✐ sè ✈➭ ♣❤ß♥❣ ▲ý t❤✉②Õt sè ❝đ❛ ❱✐Ư♥ ❚♦➳♥ ❤ä❝ ề ữ tr ổ ỗ trợ s tr ❦❤♦❛ ❤ä❝ ❝ò♥❣ ♥❤➢ tr♦♥❣ ❝✉é❝ sè♥❣✳ ❈✉è✐ ❝ï♥❣ t➠✐ ố tỏ ò ết s s tớ ữ t tr ì ủ ì ữ ➤➲ ➤é♥❣ ✈✐➟♥✱ ❝❤✐❛ s❰ ♠ä✐ ❦❤ã ❦❤➝♥ ❝ï♥❣ t➠✐ s✉èt ♥❤÷♥❣ ♥➝♠ t❤➳♥❣ q✉❛ ➤Ĩ t➠✐ ❝ã t❤Ĩ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ➳♥ ♥➭②✳ ✶ ▼ô❝ ❧ô❝ ▼ë ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✵ ✶✳✶ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷ ❚Ý♥❤ ❝❛t❡♥❛r② ❝ñ❛ ✈➭♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❇✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ❚Ý♥❤ ❝❤✃t ✭∗✮ ❝❤♦ ♠➠➤✉♥ ❆rt✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ❈❤➢➡♥❣ ✷✳ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✷✹ ✷✳✶ ●✐➯ ❣✐➳ ✈➭ ❣✐➯ ❝❤✐Ò✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❚Ý♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ ✈➭♥❤ ❝➡ së ❈❤➢➡♥❣ ✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✸✼ ✸✳✶ ▼➠➤✉♥ ❆rt✐♥ tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✷ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ❈❤➢➡♥❣ ✹✳ ø♥❣ ❞ơ♥❣ ❝đ❛ tÝ♥❤ ❝❤✃t ✭∗✮ ✺✸ ✹✳✶ ❇é✐ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✷ ❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✸ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt✱ t❐♣ ❣✐➳ ✈➭ t❐♣ ❣✐➯ ❣✐➳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ❑Õt ❧✉❐♥ ❝ñ❛ ❧✉❐♥ ➳♥ ✼✹ ❈➳❝ ❝➠♥❣ tr×♥❤ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❧✉❐♥ ➳♥ ✼✺ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✼✻ ✷ ▼ë ➤➬✉ ▲ý t❤✉②Õt ♠➠➤✉♥ ố ề ị ợ rt r ❧➬♥ ➤➬✉ t✐➟♥ tr♦♥❣ ❬✶✽❪ ✈➭ ♥❤❛♥❤ ❝❤ã♥❣ trë t❤➭♥❤ ụ ữ ệ ủ số ì ❤ä❝ ➤➵✐ sè✳ ❉♦ ➤ã ❧ý t❤✉②Õt ♥➭② ➤➲ t❤✉ ❤ót sù q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ✭①❡♠ ❝➳❝ ❝➠♥❣ tr×♥❤ ❬✺❪✱ ❬✻❪✱ ❬✶✽❪✱ ❬✷✵❪✱ ❬✷✾❪✱ ❬✸✵❪✱ ❬✸✷❪✱ ❬✸✻❪✱ ❬✹✽❪✱ ❬✺✺❪✱ ❬✺✻❪✮✳ ▼➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ❝❤♦ t❛ ♥❤✐Ị✉ t❤➠♥❣ t✐♥ ✈Ị ♠➠➤✉♥ ❜❛♥ ➤➬✉ ❝ị♥❣ ♥❤➢ ✈Ị ✈➭♥❤ ❝➡ së✳ ●✐➯ sư ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ✈➭ ❝❤✐Ị✉ ❑r✉❧❧ (R, m) M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ ❝ã dim M = d✳ ❈❤♦ I ❧➭ ♠ét ✐➤➟❛♥ ❝ñ❛ R✳ ❑❤✐ ➤ã✱ ➜Þ♥❤ ❧ý tr✐Ưt t✐➟✉ ✈➭ ❦❤➠♥❣ tr✐Ưt t✐➟✉ ❝đ❛ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✭①❡♠ ❬✻✱ ➜Þ♥❤ ❧ý ✻✳✶✳✷✱ ➜Þ♥❤ ❧ý ✻✳✶✳✹❪✮ ♥ã✐ r➺♥❣ ❝❤✐Ị✉ ❝đ❛ ♣❤➢➡♥❣ M ❧➭ sè i ❧í♥ ♥❤✃t ♠➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ Hmi (M ) ❦❤➠♥❣ tr✐Ưt t✐➟✉✱ ❝ß♥ ➤é s➞✉ ❝đ❛ M tr♦♥❣ ✐➤➟❛♥ I ❧➭ sè i ❜Ð ♥❤✃t s❛♦ ❝❤♦ HIi (M ) ❦❤➠♥❣ tr✐Ưt t✐➟✉✳ ❚Ý♥❤ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ tÝ♥❤ ❆rt✐♥ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝ị♥❣ ❧✐➟♥ ❤Ư ❝❤➷t ❝❤Ï ✈í✐ ❝➳❝ ❜✃t ❜✐Õ♥ ❝đ❛ M ✈➭ R✳ ❈ơ t❤Ĩ ●✳ ❋❛❧t✐♥❣s ➤➲ ➤➢❛ r❛ ❝➠♥❣ t❤ø❝ tÝ♥❤ ❝❤Ø sè ➤➬✉ t✐➟♥ ♠➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ s➞✉ ❝đ❛ ♠➠➤✉♥ tr♦♥❣ HIi (M ) ❦❤➠♥❣ ❤÷✉ ❤➵♥ s✐♥❤ t❤➠♥❣ q✉❛ ➤é Mp ✈➭ ➤é ❝❛♦ ❝đ❛ ✐➤➟❛♥ I + p/p✱ ✈í✐ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè Supp(M ) ✭❬✺✺❪✱ ❬✺✻❪✮❀ ♥➝♠ ✷✵✵✶✱ ❘✳ ▲u ¨ ✈➭ ❩✳ ❚❛♥❣ ➤➲ ❝❤Ø r❛ r➺♥❣ ➤é s➞✉ ❧ä❝ ❝ñ❛ M tr♦♥❣ ✐➤➟❛♥ I ✭tø❝ ❧➭ ❝❐♥ tr➟♥ ❝ñ❛ ❝➳❝ ➤é ❞➭✐ ❝ñ❛ ❝➳❝ ❞➲② ❧ä❝ ❝❤Ý♥❤ q✉② ❝ñ❛ ♣❤➢➡♥❣ M tr♦♥❣ I ✮ ❧➭ ❝❤Ø sè i ❜Ð ♥❤✃t ♠➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ HIi (M ) ❦❤➠♥❣ ❆rt✐♥ ✭❬✷✽❪✮✳ ➜➷❝ ❜✐Öt✱ ❣➬♥ ➤➞② ❝➳❝ t➳❝ ❣✐➯ ◆✳ ❚✳ ❈➢ê♥❣✱ ◆✳ ❚✳ ❉✉♥❣ ✈➭ ▲✳ ❚✳ ◆❤➭♥ ❬✶✸❪ ✸ ➤➲ ❝❤Ø r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ✈í✐ ❣✐➳ ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ✈➭ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ✈➭♥❤ ❝➡ së✳ ❚Ý♥❤ ❝❤✃t ✭∗✮ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❧➬♥ ➤➬✉ t✐➟♥ ❜ë✐ ◆✳ ❚✳ ❈➢ê♥❣ ✈➭ ▲✳ ❚✳ ◆❤➭♥ tr♦♥❣ ❬✶✷❪ ♥❤➺♠ ♥❣❤✐➟♥ ❝ø✉ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✳ ❚Ý♥❤ ❝❤✃t ♥➭② ♥❣➭② ❝➭♥❣ ➤➢ỵ❝ q✉❛♥ t➞♠ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ♠➠➤✉♥ ❆rt✐♥✱ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ ❝✃✉ tró❝ ❝đ❛ ✈➭♥❤ ❝➡ së ✭①❡♠ ❬✶✷❪✱ ❬✺✾❪✱ ❬✻✵❪✱ ❬✷✻❪✱ ❬✷✼❪✮✳ ❚r➢í❝ ❤Õt✱ ♥❤➽❝ ❧➵✐ r➺♥❣ ♠ét ♠➠➤✉♥ ❆rt✐♥ A ➤➢ỵ❝ ❣ä✐ ❧➭ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ế AnnR (0 :A p) = p ỗ ♥❣✉②➟♥ tè p ⊇ AnnR A ❑❤✐ ➤ã ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ✈➭ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ✈➭♥❤ tr♦♥❣ ❬✶✸❪ ➤➢ỵ❝ ♣❤➳t ❜✐Ĩ✉ ♥❤➢ s❛✉✿ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t Hmd (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈➭♥❤ R/ AnnR Hmd (M ) ❧➭ ❝❛t❡♥❛r②✳ ▼ơ❝ ➤Ý❝❤ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ➳♥ ❧➭ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ ❝❤✃t ✭∗✮ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ i ❜✃t ❦ú✱ ø♥❣ ✈í✐ ❣✐➳ ❧➭ ✐➤➟❛♥ ❝ù❝ ➤➵✐ tr♦♥❣ ♠è✐ ❧✐➟♥ ❤Ư ✈í✐ tÝ♥❤ ❝❛t❡♥❛r②✱ tÝ♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ tÝ♥❤ ❦❤➠♥❣ tré♥ ❧➱♥ ❝đ❛ ✈➭♥❤ ❝➡ së✳ ➜å♥❣ t❤ê✐ ❧✉❐♥ ➳♥ ➤➢❛ r❛ ø♥❣ ❞ô♥❣ ❝đ❛ tÝ♥❤ ❝❤✃t ✭∗✮ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❝❤✐Ị✉ ❝ñ❛ ♠➠➤✉♥ ❆rt✐♥✱ ❝➠♥❣ t❤ø❝ ❜é✐ ❧✐➟♥ ❦Õt✱ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣✱ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ▲✉❐♥ ➳♥ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ✹ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ♥❤➢ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣✱ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ✈➭♥❤✱ ❝❤✐Ị✉✱ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ✈➭ tÝ♥❤ ❝❤✃t ✭∗✮ ❝ñ❛ ♠➠➤✉♥ ❆rt✐♥✳ ▼ét ➤➷❝ tr➢♥❣ ♠í✐ ✈Ị tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ rt t q ệ t số ợ trì tr ♣❤➬♥ ❝✉è✐ ❝đ❛ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✷✱ ✸✱ ✹ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ t❤✉ ➤➢ỵ❝ ❝đ❛ ❧✉❐♥ ➳♥✱ ✈✐Õt ❞ù❛ tr➟♥ ❝➳❝ ❜➭✐ ❜➳♦ ❬✸✾❪✱ ❬✹✵❪ ✈➭ ❬✶❪✳ ❚r♦♥❣ s✉èt ❧✉❐♥ ➳♥✱ ❝❤ó♥❣ t➠✐ ❧✉➠♥ ❣✐➯ t❤✐Õt ♣❤➢➡♥❣ ◆♦❡t❤❡r✱ (R, m) ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ➤Þ❛ M ❧➭ ♠ét R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ị✉ ❑r✉❧❧✱ dim M = d ✻✻ ệ t ú t tì ợ ề ệ ✈➭ ➤đ ➤Ĩ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ♥❤á ❤➡♥ d t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ị ì ết q s ó tể ➤➢❛ r❛ ♠ét ❝➳❝❤ t✐Õ♣ ❝❐♥ ♠í✐✳ ➜Þ♥❤ ❧ý ✹✳✷✳✹✳ i ≥ ❧➭ ♠ét sè ♥❣✉②➟♥✳ ●✐➯ sö R/ AnnR M ❈❤♦ ❧➭ ❝❛t❡♥❛r②✳ ❑❤✐ ➤ã ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✳ i−dim R/p ✭✐✮ AttRp (HpRp (Mp )) = {qRp | q ∈ AttR (Hmi (M )), q ⊆ p}, ✈í✐ p ∈ Spec(R)✳ ✭✐✐✮ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮❀ ✭✐✐✐✮ i−dim R/p HpRp ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ (Mp ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ p ∈ Supp(M )✳ ⇒ ✭✐✐✮ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ✭✐✮ ⇒ ✭✐✐✐✮ ❝ñ❛ ➜✐♥❤ ❧ý ✹✳✷✳✸✳ ✭✐✐✮ i−dim R/p ⇒ ✭✐✮✳ ▲✃② qRp ∈ AttRp (HpRp tÝ♥❤ ❝❤✃t ✭∗✮ ♥➟♥ i−dim R/p HpRp (Mp )) ❱× Hmi (M ) t❤á❛ ♠➲♥ (Mp ) ❝ị♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ t❤❡♦ ▼Ư♥❤ ➤Ị ✹✳✷✳✷✳ ó t ị ý idim R/p PsuppRp ì ✈❐② i−dim R/p qRp ∈ PsuppRp ➜Þ♥❤ ❧ý ✶✳✸✳✹✱ t❛ ❝ã ♠➲♥ i−dim R/p (Mp ) = V(AnnRp HpRp (Mp )) (Mp ) t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✸✳✶✳ ▼➷t ❦❤➳❝ t❤❡♦ q ∈ AttR (Hmi (M )) ▲✃② q1 ∈ AttR Hmi (M ) t❤á❛ q ⊇ q1 ❱× Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✶✳✷✱ PsuppiR (M ) = V(AnnR Hmi (M )) ❉♦ ➤ã q1 ❱× i−dim R/p ∈ PsuppiR (M ) ❚❤❡♦ ❇ỉ ➤Ị ✹✳✷✳✶✱ t❛ ❝ã q1 Rp ∈ PsuppRp (Mp ) qRp ⊇ q1 Rp ✈➭ tÝ♥❤ ❝❤✃t tỉ✐ t✐Ĩ✉ ❝đ❛ qRp ✱ t❛ ❝ã q = q1 ì idim R/p AttRp (HpRp ợ ❧➵✐✱ ❧✃② q ➤Ò ✶✳✸✳✶✱ (Mp )) ⊆ {qRp | q ∈ AttR (Hmi (M )), q ⊆ p} ⊆ p t❤á❛ ♠➲♥ q ∈ AttR (Hmi (M )) ❑❤✐ ➤ã t❤❡♦ ▼Ö♥❤ q ∈ V(AnnR Hmi (M ))✳ ❉♦ ➤ã q ∈ PsuppiR (M ) t❤❡♦ ➜Þ♥❤ ✻✼ i−dim R/p ❧ý ✷✳✶✳✷✳ ❚❤❡♦ ❇ỉ ➤Ị ✹✳✷✳✶✱ t❛ ❝ã qRp ∈ PsuppRp i−dim R/p qRp ⊇ q1 Rp ∈ PsuppRp ❉♦ ✈❐② (Mp )✳ ●✐➯ sö (Mp ) q ⊇ q1 ✈➭ q1 ∈ PsuppiR (M ) t❤❡♦ ❇ỉ ➤Ị ✹✳✷✳✶✳ ❚Ý♥❤ tè✐ t✐Ĩ✉ ❝đ❛ q ❦Ð♦ t❤❡♦ q = q1 ❉♦ ➤ã i−dim R/p qRp = q1 Rp ∈ PsuppRp ❚❤❡♦ ▼Ö♥❤ ➤Ò ✹✳✷✳✷✱ i−dim R/p HpRp (Mp ) (Mp ) t❤á❛ tí t ì t ị ý ✈➭ ▼Ư♥❤ ➤Ị ✶✳✸✳✶✱ i−dim R/p qRp ∈ PsuppRp ✭✐✐✮ i−dim R/p (Mp ) = AttRp (HpRp (Mp )) ⇔ ✭✐✐✮ t❤❡♦ ▼Ư♥❤ ➤Ị ✹✳✷✳✷✳ ❍Ư q✉➯ s❛✉ ❝❤Ø r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝✃✉ tró❝ ❝đ❛ ✈➭♥❤ ✈➭ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐✱ tÝ♥❤ ❝❤✃t ✭∗✮ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐ ❝đ❛ ♠ét ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ tÝ♥❤ ❝❤✃t ✭∗✮ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐ ❝đ❛ ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳ ♠➠➤✉♥ ➤ã✳ ❍Ö q✉➯ ✹✳✷✳✺✳ ✭✐✮ ✈➭♥❤ ◆Õ✉ ❈➳❝ ♠Ö♥❤ ➤Ị s❛✉ ❧➭ ➤ó♥❣✿ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ t ị ể ị ọ i tì R/ AnnR M ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ R/p ❧➭ ❦❤➠♥❣ tré♥ ❧➱♥ ✈í✐ ♠ä✐ p ∈ AssR (M )✳ ✭✐✐✮ ●✐➯ sư (R, m) ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ❝ã ❝➳❝ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ❑❤✐ ➤ã i−dim R/p AttRp (HpRp ✈í✐ ♠ä✐ i ✈➭ ✈í✐ ♠ä✐ (Mp )) = {qRp | q ∈ AttR (Hmi (M )), q ⊆ p} p ∈ Spec(R)✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ AttR Hmi (M ) = AttR Hmi (M ) ✈í✐ ♠ä✐ i t❤× Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ♠ä✐ i✳ ✻✽ ✭✐✐✐✮ Hmi (M ) ∗✮ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ❱× ♠ä✐ i✱ ♥➟♥ t❤❡♦ ✭✐✮ ✈í✐ ♠ä✐ i ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ i (M ) HpR p ∗✮ ✈í✐ ♠ä✐ i ✈➭ ✈í✐ ♠ä✐ p ∈ Supp(M )✳ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ⇒ ✭✐✐✮ ❝đ❛ ➜Þ♥❤ ❧ý ✹✳✷✳✹✱ Hmi (M ) t❤á❛ ♠➲♥ tí t ọ i ì t ị ❧ý ✷✳✷✳✶✱ R/ AnnR M ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ R/p ❧➭ ❦❤➠♥❣ tré♥ ❧➱♥ ✈í✐ ♠ä✐ p ∈ AssR (M )✳ ✭✐✐✮ ❱× (R, m) ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ ❝ã ❝➳❝ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ♥➟♥ t❤❡♦ ❍Ö q✉➯ ✷✳✶✳✸✱ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ i✳ ❉♦ ✈❐② ❦❤➻♥❣ ➤Þ♥❤ ➤➬✉ t✐➟♥ ❝đ❛ ✭✐✐✮ ➤➢ỵ❝ s✉② r❛ t❤❡♦ ✭✐✐✮ ⇒ ✭✐✮ ❝đ❛ ➜Þ♥❤ ❧ý ✹✳✷✳✹✳ ❱í✐ ♠Ư♥❤ ➤Ị t❤ø ❤❛✐ ❝đ❛ ✭✐✐✮✱ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ i−dim R/p AttRp (HpRp ▲✃② i−dim R/p qRp ∈ AttRp (HpRp i−dim R/p (Mp )) = AttRp (HpRp (Mp ))✳ ❑❤✐ ➤ã i−dim R/p qRp ⊇ q Rp ∈ AttRp (HpRp ❙✉② r❛ (Mp )) q ⊇ q ∈ AttR Hmi (M ) t❤❡♦ ➜Þ♥❤ ❧ý ✶✳✸✳✹✳ ❚õ ❣✐➯ t❤✐Õt✱ t❛ s✉② r❛ i−dim R/p q = q ❱× ✈❐② qRp ∈ AttRp (HpRp ✭✐✐✐✮ ❱× ✈➭♥❤ (Mp )) (Mp ))✳ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ i ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✶ i R/ AnnR M ❧➭ ❝❛t❡♥❛r②✳ ❱× ✈❐② t❤❡♦ ▼Ư♥❤ ➤Ị ✹✳✷✳✷✱ t❛ ❝ã HpR (M ) p t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ i ✈➭ ✈í✐ ♠ä✐ p ∈ Supp(M )✳ ➜✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳ ✹✳✸ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt✱ t❐♣ ❣✐➳ ✈➭ t❐♣ ❣✐➯ ❣✐➳ ●✐➯ sö X ❧➭ ♠ét t❐♣ ❝♦♥ tr♦♥❣ Spec(R)✱ t❛ ❦ý ❤✐Ư✉ X ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ tè✐ t✐Ĩ✉ ❝đ❛ ❑❤✐ ◦ X ✱ X ❧➭ ❜❛♦ ➤ã♥❣ ❝ñ❛ X t❤❡♦ t➠♣➠ ❩❛r✐s❦✐ ✈➭ X ❧➭ X \ {m} R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥✱ ▼✳ ❇r♦❞♠❛♥♥ ✈➭ ❘✳ ❨✳ ❙❤❛r♣ ✭①❡♠ ❬✻✱ ➜Þ♥❤ ❧ý ✶✶✳✸✳✶✷❪✮ ➤➲ ➤➢❛ r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ t❐♣ ❝➳❝ ✻✾ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ✈➭ t❐♣ ❣✐➳ ♥❤➢ s❛✉ ◦ min{p ∈ Supp(M ) : depth Mp + dim R/p = i} i−1 ◦ = (min AttR (Hmi (M ))) AttR (Hmj (M )) \ j=0 ◆Õ✉ Hmi (M ) tỏ tí t tì t ị ❧ý ✷✳✶✳✷ PsuppiR (M ) = V(AnnR Hmi (M )) ❱× ✈❐② PsuppiR (M ) = AttR (Hmi (M ))✳ ➜Þ♥❤ ❧ý s❛✉ ❝❤ø♥❣ ♠✐♥❤ ❧➵✐ ❦Õt q✉➯ tr➟♥ ❝đ❛ ▼✳ ❇r♦❞♠❛♥♥ ✈➭ ❘✳ ❨✳ ❙❤❛r♣ ✈í✐ ❣✐➯ t❤✐Õt ②Õ✉ ❤➡♥✱ ➤å♥❣ t❤ê✐ ❝❤Ø r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt✱ ❝➳❝ t❐♣ ❣✐➯ ❣✐➳ ✈➭ t❐♣ ❣✐➳ ❝đ❛ ➜Þ♥❤ ❧ý ✹✳✸✳✶✳ ●✐➯ sư M✳ i≥0 ❧➭ ♠ét sè ♥❣✉②➟♥✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ ➤ó♥❣✳ ✭✐✮ {p ∈ Supp(M ) | depth Mp + dim R/p = i} = PsuppiR (M ) \ ✭✐✐✮ ◆Õ✉ i−1 j j=0 PsuppR (M )✳ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ t❤× {p ∈ AttR (Hmi (M )) | depth Mp + dim R/p = i} i−1 = AttR (Hmi (M )) PsuppjR (M ) \ j=0 ✭✐✐✐✮ ◆Õ✉ Hmj (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ j i t❤× min{p ∈ Supp(M )| depth Mp + dim R/p = i } i−1 = (min AttR (Hmi (M ))) V(AnnR Hmj (M )) \ j=0 ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ▲✃② p ∈ PsuppiR (M ) \ i−dim(R/p) PsuppiR (M ) ♥➟♥ HpRp i−1 j j=0 PsuppR (M )✳ ❱× p ∈ (Mp ) = ❉♦ ➤ã p ∈ Supp(M ) ✈➭ depth Mp ≤ i − dim(R/p)✳ ●✐➯ sö r➺♥❣ depth Mp + dim R/p = j < i ✼✵ ❱× j−dim(R/p) depth Mp = j − dim R/p ♥➟♥ HpRp (Mp ) = ❚õ ➤ã ❦Ð♦ t❤❡♦ p ∈ PsuppjR (M ) ➜✐Ị✉ ♥➭② tr➳✐ ✈í✐ ❣✐➯ t❤✐Õt✳ ❱× ✈❐② i−1 i PsuppR (M )\ PsuppjR (M ) j=0 ◆❣➢ỵ❝ ❧➵✐✱ ❧✃② i−dim(R/p) HpRp ♠➲♥ ⊆ {p ∈ Supp(M ) | depth Mp +dim R/p = i} p ∈ Supp(M ) t❤á❛ ♠➲♥ depth Mp + dim R/p = i✳ ❉♦ ➤ã (Mp ) = ❚õ ➤ã t❛ ❝ã p ∈ PsuppiR (M ) ●✐➯ sö tå♥ t➵✐ j t❤á❛ j−dim(R/p) ≤ j < i ✈➭ p ∈ PsuppjR (M ) t❤× HpRp (Mp ) = ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ depth Mp + dim R/p ≤ j < i ➜ã ❧➭ ➤✐Ị✉ ✈➠ ❧ý✳ ❱× ✈❐② {p ∈ Supp(M ) | depth Mp +dim R/p = i} ⊆ i−1 i PsuppR (M )\ PsuppjR (M ) j=0 i i ✭✐✐✮ ❱× Hm (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ♥➟♥ PsuppR (M ) = V(AnnR Hmi (M )) t ị ý ì ứ ❝ñ❛ ✭✐✐✮ ❧➭ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ❝ñ❛ ✭✐✮✳ ✭✐✐✐✮ ❱× ✷✳✶✳✷✱ Hmj (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮✱ ✈í✐ ♠ä✐ j ≤ i ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý PsuppjR (M ) = V(AnnR Hmj (M )) ✈í✐ ♠ä✐ j ≤ i✳ ●✐➯ sö i−1 p∈ (min AttR (Hmi (M ))) V(AnnR Hmi (M )) \ j=0 ❱× t❤Õ t❤❡♦ ✭✐✮✱ t❛ ❝ã p∈ i−1 i PsuppR (M )\ PsuppjR (M ) j=0 ●✐➯ sö r➺♥❣ = {p ∈ Supp(M ) | depth Mp +dim R/p = i} p ⊇ q, q ∈ min{p ∈ Supp(M ) | depth Mp + dim R/p = i}✳ ▲➵✐ t❤❡♦ ✭✐✮✱ t❛ ❝ã q ∈ PsuppiR (M ) ▼➷t ❦❤➳❝✱ p ∈ AttR (Hmi (M )) = PsuppiR (M ) ✼✶ ❱× ✈❐② p = q ∈ min{p ∈ Supp(M ) | depth Mp + dim R/p = i} ◆❣➢ỵ❝ ❧➵✐✱ ❧✃② p ∈ min{p ∈ Supp(M ) | depth Mp + dim R/p = i} ❉♦ ➤ã i−1 p∈ PsuppiR (M ) i−1 PsuppjR (M ) \ = PsuppiR (M ) j=0 V(AnnR Hmj (M )) \ j=0 t❤❡♦ ✭✐✮ ✈➭ t❤❡♦ ❣✐➯ t❤✉②Õt✳ ❑❤✐ ➤ã tå♥ t➵✐ q ∈ PsuppiR (M ) t❤á❛ ♠➲♥ p ⊇ q ❍✐Ó♥ ♥❤✐➟♥ t❛ ❝ã i−1 i−1 V(AnnR Hmj (M )) q∈ / j=0 PsuppjR (M ) = j=0 i (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ✈➭ ✭✐✐✮ ❝đ❛ ▼Ư♥❤ ▲➵✐ ✈× Hm ➤Ị ✶✳✸✳✶✱ t❛ ❝ã q ∈ PsuppiR (M ) = AttR Hmi (M ) ❉♦ ✈❐② t❤❡♦ ✭✐✐✮ ➤ã q ∈ {p ∈ AttR (Hmi (M )) | depth Mp + dim R/p = i}✳ ❉♦ q ❧➭ ♣❤➬♥ tư ❝đ❛ t❐♣ {p ∈ Supp(M ) | depth Mp + dim R/p = i} ❚õ tÝ♥❤ tè✐ t✐Ĩ✉ ❝đ❛ p✱ t❛ ❝ã p = q ❈❤ó ý r➺♥❣ ❣✐➯ t❤✐Õt Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ tr♦♥❣ ♠Ư♥❤ ➤Ị ✭✐✐✮ ✈➭ Hmj (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ j ≤ i tr♦♥❣ ✭✐✐✐✮ ❝đ❛ ị ý tể ỏ ợ í ụ ①Ðt (R, m) ❧➭ ♠✐Ị♥ ♥❣✉②➟♥✱ ➤Þ❛ ♣❤➢➡♥❣✱ ❝❤✐Ị✉ ➤➢ỵ❝ ①➞② ❞ù♥❣ ❜ë✐ ▼✳ ❋❡rr❛♥❞ ✈➭ ❉✳ ❘❛②♥❛✉❞ ❬✺✸❪ t❤á❛ ♠➲♥ R ❝ã ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè ♥❤ó♥❣ p ❝❤✐Ò✉ ❚❛ ❝ã Hm (R) ❦❤➠♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✭①❡♠ ❬✶✷✱ ❱Ý ❞ơ ✹✳✶❪✮✳ ❉Ơ tÝ♥❤ ợ ì Psupp0R (R) = V(AnnR Hm0 (R)) = ∅ t❤é❝ ✈Õ tr➳✐ ❝ñ❛ ❝➳❝ ❝➠♥❣ t❤ø❝ tr♦♥❣ ✭✐✐✮✱ ✭✐✐✐✮ ♥❤➢♥❣ ♥ã ❦❤➠♥❣ t❤✉é❝ ✈Õ ♣❤➯✐ ❝ñ❛ ❝➳❝ ❝➠♥❣ t❤ø❝ ➤ã✳ ❈➠♥❣ t❤ø❝ ✭✐✐✐✮ tr♦♥❣ ➜Þ♥❤ ❧ý ✹✳✸✳✶ ❝ã ♠è✐ ❧✐➟♥ ❤Ư ✈í✐ ➜Þ♥❤ ❧ý tr✐Ưt ❝đ❛ ❋❛❧t✐♥❣s✳ ●✐➯ sư b ⊆ a ❧➭ ❝➳❝ ✐➤➟❛♥ ❝đ❛ R✳ ❚❤❡♦ t❤✉❐t ♥❣÷ ❝đ❛ ▼✳ ❇r♦❞♠❛♥♥ ✼✷ ✈➭ ❘✳ ❨✳ ❙❤❛r♣ ❬✻✱ ❈❤➢➡♥❣ ✾❪✱ b ✲❝❤✐Ị✉ ❤÷✉ ❤➵♥ fab (M ) ủ M t ứ a ợ ị ĩ ❜ë✐ fab (M ) := inf{i ∈ N : b ✈➭ (AnnR Hai (M ))} b✲tè✐ t✐Ĩ✉ a✲➤é s➞✉ ➤✐Ị✉ ỉ ủ M ợ ị ĩ ba (M ) = inf{depth Mp + ht(a + p)/p : p ∈ Spec(R) \ V(b)} ◆Õ✉ R ❧➭ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ tt tớ ì tứ tì ị ❧ý tr✐Ưt ➤ó♥❣ tr➟♥ R✱ ♥❣❤Ü❛ ❧➭ fab (M ) = λba (M ) ✭①❡♠ ❬✻✱ ❇➭✐ t❐♣ ✾✳✻✳✻❪✮✳ ❱❐♥ ụ ị ý t ứ ợ ết q✉➯ tr➟♥ ♠ét ❝❤♦ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ a = m ✈í✐ ♠ét ❣✐➯ t❤✐Õt ❦❤➳❝ ❧➭ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ i (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮✳ ❈❤ó ý r➺♥❣ tr♦♥❣ ❈❤➢➡♥❣ ✷ ❝❤ó♥❣ t➠✐ ➤➲ ➤✐Ị✉ Hm ❝❤Ø r❛ ❧✉➠♥ tå♥ t➵✐ ♠ét ✈➭♥❤ R ❝❛t❡♥❛r② ♥❤➢♥❣ ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤å♥❣ t❤ê✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ R ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ✈➭ R ❝ã ❝➳❝ t❤í ❤×♥❤ t❤ø❝ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ tr➟♥ ➤ã tå♥ t➵✐ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ M ♠➭ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮✳ Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ i✳ ❍Ư q✉➯ ✹✳✸✳✷✳ ●✐➯ sư ♠ét ✐➤➟❛♥ ❝đ❛ R✳ ❑❤✐ ➤ã fmb (M ) = λbm (M )✳ ❈❤ø♥❣ ♠✐♥❤✳ ●ä✐ ●✐➯ sö b ❧➭ i ✈➭ j ❧➭ ❝➳❝ sè ♥❣✉②➟♥✳ ❚❤❡♦ ❬✻✱ ▼Ư♥❤ ➤Ị ✼✳✷✳✶✶❪✱ t❛ ❝ã AnnR Hmi (M ) = p i (M ) p∈min AttR Hm ❱× ✈❐② b⊆ ▼➷t ❦❤➳❝ b⊆ AnnR Hmi (M ) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ AttR Hmi (M ) \ V(b) = ∅ AnnR Hmj (M ) ❦Ð♦ t❤❡♦ V(b) ⊇ V( AnnR Hmj (M )) ❱× Hmj (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✶✳✷ V(b) ⊇ V( AnnR Hmj (M )) = V(AnnR Hmi (M )) = PsuppjR (M ) ữ ị tr ✈➭ ➜Þ♥❤ ❧ý ✹✳✸✳✶✱ t❛ ❝ã fmb (M ) = inf{i ∈ N : b = inf{i ∈ N : b (AnnR Hmi (M ))} (AnnR Hmi (M )) ❛♥❞ b ⊆ (AnnR Hmj (M )), ∀j < i} i−1 = inf{i ∈ N : AttR Hmi (M ) PsuppjR (M ) ∪ V(b)] = ∅} \[ j=0 i−1 = inf{i ∈ N : [min AttR Hmi (M ) PsuppjR (M )] \ V(b) = ∅} \ j=0 = inf{i ∈ N : i = depth Mp + dim R/p, p ∈ Supp(M ) \ V(b)} = inf{depth Mp + dim R/p, p ∈ Spec(R) \ V(b)} = λbm (M ) b (M ) ❱❐② fm = λbm (M )✳ ❑Õt ❧✉❐♥ ❈❤➢➡♥❣ ✹ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ t ợ ữ ết q s r ❝➠♥❣ t❤ø❝ ❜é✐ ❧✐➟♥ ❦Õt ❝❤♦ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ ♥❣✉②➟♥ ❦❤➠♥❣ ➞♠ ❝❤♦ tr➢í❝ ❞➢í✐ ❣✐➯ t❤✐Õt tr♦♥❣ ➤ã Hmi (M ) ✈í✐ i ❧➭ sè Hmi (M ) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮✱ q ❧➭ ✐➤➟❛♥ m ♥❣✉②➟♥ s➡✳ ✲ ➜➢❛ r❛ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ❝❤♦ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ✈➭♥❤ Hmd (M ) t❤➠♥❣ q✉❛ tÝ♥❤ ❝❛t❡♥❛r② ❝ñ❛ R/ AnnR Hmd (M )✳ ✲ ➜➢❛ r❛ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ĩ i−dim R/p AttRp (HpRp ✈í✐ (Mp )) = {qRp | q ∈ AttR (Hmi (M )), q ⊆ p}, p ∈ Spec(R) ✈➭ i ❧➭ ♠ét sè ♥❣✉②➟♥ ❦❤➠♥❣ ➞♠ ❝❤♦ tr➢í❝ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ✈➭♥❤ R/ AnnR M ❧➭ ❝❛t❡♥❛r②✳ ✲ ➜➢❛ r❛ ♠ét ❧✐➟♥ ❤Ư ❣✐÷❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ t❐♣ ❣✐➳ ✈➭ ❝➳❝ t❐♣ ❣✐➯ ❣✐➳ ❝ñ❛ M✳ ✼✹ ❑Õt ❧✉❐♥ ❝ñ❛ ❧✉❐♥ ➳♥ ✶✳ ➜➷❝ tr➢♥❣ tÝ♥❤ ❝❤✃t ❣ä✐ ❧➭ tÝ♥❤ ❝❤✃t ✭∗✮ ❝❤♦ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ Hmi (M ) t❤➠♥❣ q✉❛ t❐♣ ❣✐➯ ❣✐➳ Psuppi (M )✳ i ✷✳ ❈❤Ø r❛ ❝➳❝ q✉❛♥ ❤Ư ❣✐÷❛ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ Hm (M ) ✈➭ tÝ♥❤ ❝❛t❡♥❛r② ♣❤ỉ ❞ơ♥❣ ❝đ❛ ✈➭♥❤ R/ AnnR M ✱ tÝ♥❤ ❦❤➠♥❣ tré♥ ❧➱♥ ❝ñ❛ ✈➭♥❤ R/p ✈í✐ p ∈ Ass(M ) ✸✳ ➜➷❝ tr➢♥❣ tÝ♥❤ ❝❤✃t ✭∗✮ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ Hmi (M ) tù❛ ❦❤➠♥❣ tré♥ ❧➱♥ t❤➠♥❣ q✉❛ tÝ♥❤ ❝❛t❡♥❛r② ❝đ❛ ✈➭♥❤ ❝➡ së ✈➭ ❝❤✐Ị✉ ◆♦❡t❤❡r ❝đ❛ ✹✳ Hmi (M ) ✳ ø♥❣ ❞ơ♥❣ ❝đ❛ tÝ♥❤ ❝❤✃t ✭∗✮ ➤Ĩ ➤➢❛ r❛ ❝➠♥❣ t❤ø❝ ❜é✐ ❧✐➟♥ ❦Õt ❝❤♦ Hmi (M ) ✈➭ ❝❤Ø r❛ ❝➳❝ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ❝➳❝ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt✱ ❣✐➯ ❣✐➳ ✈➭ ❣✐➳ q✉❛ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛✳ ✺✳ ▼ét sè ❜➭✐ t♦➳♥ ♣❤➳t tr✐Ĩ♥ trù❝ t✐Õ♣ ❝đ❛ ❧✉❐♥ ➳♥✿ ◆❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❧í♣ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ♣❤➢➡♥❣ (R, m) ♠➭ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ Hmi (R) t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭∗✮ ✈í✐ ♠ä✐ i❀ ➜➷❝ tr➢♥❣ ❧í♣ ✈➭♥❤ ➤Þ❛ i ♣❤➢➡♥❣ ◆♦❡t❤❡r (R, m) ♠➭ SuppR (M ) ❧➭ ➤ã♥❣ ✈í✐ ♠ä✐ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ M ✈➭ ✈í✐ ♠ä✐ i✳ ✼✺ ❈➳❝ ❝➠♥❣ tr×♥❤ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❧✉❐♥ ➳♥ ✶✳ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ❚✳ ◆✳ ❆♥ ✭✷✵✵✾✮✱ ✧❖♥ t❤❡ ✉♥♠✐①❡❞♥❡ss ❛♥❞ t❤❡ ✉♥✐✈❡rs❛❧ ❝❛t❡♥❛r✐❝✐t② ♦❢ ❧♦❝❛❧ r✐♥❣s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ✸✷✶✱ ✱ ❏✳ ❆❧❣❡❜r❛ ♣♣✳ ✸✵✸✲✸✶✶✳ ✷✳ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ❚✳ ◆✳ ❆♥ ✭✷✵✶✵✮✱ ✧❖♥ t❤❡ ❝❛t❡♥❛r✐❝✐t② ♦❢ ◆♦❡t❤❡r✐❛♥ ❧♦❝❛❧ r✐♥❣s ❛♥❞ q✉❛s✐ ✉♥♠✐①❡❞ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ❈♦♠♠✳ ✱ ✸✽✱ ♣♣✳ ❆❧❣❡❜r❛ ✸✼✷✽✲✸✼✸✻✳ ✸✳ ❚✳ ◆✳ ❆♥ ✭✷✵✶✶✮✱ ✧❖♥ t❤❡ ❛tt❛❝❤❡❞ ♣r✐♠❡s ❛♥❞ ❙❤✐❢t❡❞ ▲♦❝❛❧✐③❛t✐♦♥ Pr✐♥❝✐✲ ♣❧❡ ❢♦r ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ❆❧❣❡❜r❛ ❈♦❧❧♦q✉✐✉♠ ✭t♦ ❛♣♣❡❛r✮✳ ❈➳❝ ❦Õt q✉➯ tr♦♥❣ ❧✉❐♥ ➳♥ ➤➲ ➤➢ỵ❝ ❜➳♦ ❝➳♦ ✈➭ t❤➯♦ ❧✉❐♥ t➵✐ ✲ ❳❡♠✐♥❛ ➜➵✐ sè ✈➭ ▲ý t❤✉②Õt sè ✲ ❱✐Ö♥ ❚♦➳♥ ❤ä❝✳ ✲ ❳❡♠✐♥❛ ➜➵✐ sè ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✲ ➜➵✐ ❤ä❝ ❚❤➳✐ ◆❣✉②➟♥✳ ✲ ➜➵✐ ❤é✐ ❚♦➳♥ ❤ä❝ t♦➭♥ q✉è❝✱ ◗✉② ◆❤➡♥✱ ✵✽✴✷✵✵✽✳ ✲ ❍é✐ ♥❣❤Þ ♥❣❤✐➟♥ ❝ø✉ s✐♥❤ ❝đ❛ ❱✐Ư♥ ❚♦➳♥ ❤ä❝✱ ✶✵✴✷✵✵✽✱ ộ ị số ì ọ ✲ ❚➠ ♣➠✱ ❍✉Õ✱ ✵✾✴✷✵✵✾✳ ✲ ❍é✐ t❤➯♦ ❧✐➟♥ ❦Õt ◆❤❐t ❇➯♥ ✲ ❱✐Ưt ◆❛♠ ❧➬♥ t❤ø ♥➝♠ ✈Ị ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✱ ❍➭ ◆é✐✱ ✵✶✴✷✵✶✵✳ ✲ ❳❡♠✐♥❛ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✱ ❚r➢ê♥❣ ➤➵✐ ❤ä❝ tỉ♥❣ ❤ỵ♣ ▼❡✐❥✐✱ ◆❤❐t ❇➯♥✱ ✵✸✴✷✵✶✵✳ ✼✻ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❚✐Õ♥❣ ❆♥❤ ❬✶❪ ❚✳ ◆✳ ❆♥ ✭✷✵✶✶✮✱ ✧❖♥ t❤❡ ❛tt❛❝❤❡❞ ♣r✐♠❡s ❛♥❞ ❙❤✐❢t❡❞ ▲♦❝❛❧✐③❛t✐♦♥ Pr✐♥✲ ❝✐♣❧❡ ❢♦r ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ❆❧❣❡❜r❛ ❈♦❧❧♦q✉✐✉♠ ❬✷❪ ▼✳ ❋✳ ❆t✐②❛❤ ❛♥❞ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✭✶✾✻✾✮✱ ✭t♦ ❛♣♣❡❛r✮✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❝♦♠♠✉t❛t✐✈❡ ✱ ❘❡❛❞✐♥❣ ▼❛ss✿ ❆❞❞✐s♦♥✲❲❡s❧❡②✳ ❛❧❣❡❜r❛ ❬✸❪ ▼✳ ❇r♦❞♠❛♥♥ ✭✶✾✼✽✮✱ ✧❆ ♣❛rt✐❝✉❧❛r ❝❧❛ss ♦❢ r❡❣✉❧❛r ❞♦♠❛✐♥s✧✱ ❏✳ ❆❧❣❡✲ ✱ ✺✹✱ ♣♣✳ ✸✻✻✲✸✼✸✳ ❜r❛ ❬✹❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❈✳ ❘♦tt❤❛✉s ✭✶✾✽✸✮✱ ✧❆ ♣❡❝✉❧✐❛r ✉♥♠✐①❡❞ ❞♦♠❛✐♥✧✱ ✱ ✭✹✮✽✼✱ ♣♣✳ ✺✾✻✲✻✵✵✳ Pr♦❝✳ ❆▼❙✳ ❬✺❪ ▼✳ ❇r♦❞♠❛♥♥✱ ❈✳ ❘♦tt❤❛✉s ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✷✵✵✵✮✱ ✧❖♥ ❛♥♥✐❤✐❧❛t♦rs ❛♥❞ ❛ss♦❝✐❛t❡❞ ♣r✐♠❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ❏✳ P✉r❡ ❆♣♣❧✳ ✱ ✶✺✸✱ ♣♣✳ ✶✾✼✲✷✷✼✳ ❆❧❣❡❜r❛ ❬✻❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✾✽✮✱ ▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡❜r❛✐❝ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ●❡♦♠❡tr✐❝ ❆♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✼❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✷✵✵✷✮✱ ✧❖♥ t❤❡ ❞✐♠❡♥s✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝✲ ✐t② ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ❬✽❪ ❲✳ ❇r✉♥s ❛♥❞ ❏✳ ❍❡r③♦❣ ✭✶✾✾✽✮✱ ◆❛❣♦②❛ ▼❛t❤✳ ❏✳ ✱ ✶✻✼✱ ♣♣✳ ✷✶✼✲✷✸✸✳ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✱ ❈❛♠❜r✐❞❣❡ r✐♥❣s ❯♥✐✈❡rs✐t② Pr❡ss ✭❘❡✈✐s❡❞ ❡❞✐t✐♦♥✮✳ ❬✾❪ ■✳ ❙✳ ❈♦❤❡♥ ✭✶✾✹✻✮✱ ✧❖♥ t❤❡ str✉❝t✉r❡ ❛♥❞ ✐❞❡❛❧ t❤❡♦r② ♦❢ ❝♦♠♣❧❡t❡ ❧♦❝❛❧ r✐♥❣s✧✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✺✾✱ ♣♣✳ ✺✹✲✶✵✻✳ ❬✶✵❪ ■✳ ❙✳ ❈♦❤❡♥ ✭✶✾✺✹✮✱ ✧▲❡♥❣t❤ ♦❢ ♣r✐♠❡ ✐❞❡❛❧ ❝❤❛✐♥s✧✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✼✻✱ ♣♣✳ ✻✺✹✲✻✻✽✳ ❬✶✶❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✶✾✾✾✮✱ ✧❉✐♠❡♥s✐♦♥✱ ♠✉❧t✐♣❧✐❝✐t② ❛♥❞ ❍✐❧❜❡rt ❢✉♥❝t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ✶✱ ♣♣✳ ✶✼✾✲✶✾✻✳ ❊❛st✲❲❡st ❏✳ ♦❢ ▼❛t❤❡♠❛t✐❝s ✱ ✼✼ ❬✶✷❪ ◆✳ ❚✳ ❈✉♦♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✷✵✵✷✮✱ ✧❖♥ t❤❡ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥s✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ✱ ✭✷✮✸✵✱ ♣♣✳ ✶✷✶✲✶✸✵✳ ❬✶✸❪ ◆✳ ❚✳ ❈✉♦♥❣✱ ◆✳ ❚✳ ❉✉♥❣ ❛♥❞ ▲✳ ❚✳ ◆❤❛♥ ✭✷✵✵✼✮✱ ✧❚♦♣ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ❛♥❞ t❤❡ ❝❛t❡♥❛r✐❝✐t② ♦❢ t❤❡ ✉♥♠✐①❡❞ s✉♣♣♦rt ♦❢ ❛ ❢✐♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠♦❞✉❧❡✧✱ ✱ ✭✺✮✸✺✱ ♣♣✳ ✶✻✾✶✲✶✼✵✶✳ ❈♦♠♠✳ ❆❧❣❡❜r❛ ❬✶✹❪ ❉✳ ❉❡❧❢✐♥♦ ❛♥❞ ❚✳ ▼❛r❧❡② ✭✶✾✾✼✮✱ ✧❈♦❢✐♥✐t❡ ♠♦❞✉❧❡s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧✲ ♦❣②✧✱ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛✱ ✶✶✺✱ ♣♣✳ ✶✵✼✲✶✶✶✳ ❬✶✺❪ ▼✳ ❚✳ ❉✐❜❛❡✐ ❛♥❞ ❙✳ ❨❛ss❡♠✐ ✭✷✵✵✺✮✱ ✧❆tt❛❝❤❡❞ ♣r✐♠❡s ♦❢ t❤❡ t♦♣ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s ✇✐t❤ r❡s♣❡❝t t♦ ❛♥ ✐❞❡❛❧✧✱ ❆r❝❤✳ ▼❛t❤✳ ✱ ✽✹✱ ♣♣✳ ✷✾✷✲✷✾✼✳ ❬✶✻❪ ❉✳ ❊✐s❡♥❜✉❞ ✭✶✾✾✺✮✱ ❈♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ✇✐t❤ ❛ ✈✐❡✇ t♦✇❛r❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣ ❇❡r❧✐♥ ✲ ❍❡✐❞❡❧❜❡r❣ ✲ ◆❡✇ ❨♦r❦✳ ❬✶✼❪ ❑✳ ❋✉❥✐t❛ ✭✶✾✼✺✮✱ ✧❙♦♠❡ ❝♦✉♥t❡r❡①❛♠♣❧❡s r❡❧❛t❡❞ t♦ ♣r✐♠❡ ❝❤❛✐♥s ✐♥ ✐♥t❡❣r❛❧ ❞♦♠❛✐♥s✧✱ ❍✐r♦s❤✐♠❛ ▼❛t❤✳ ❏✳ ❬✶✽❪ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✭✶✾✻✼✮✱ ✱ ✺✱ ♣♣✳ ✹✼✸✲✹✽✺✳ ✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✷✵✱ ▲♦❝❛❧ ❤♦♠♦❧♦❣② ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥ ✲ ❍❡✐❞❡❧❜❡r❣ ✲ ◆❡✇ ❨♦r❦✳ ❬✶✾❪ ❘✳ ❍❛rts❤♦r♥❡ ✭✶✾✼✵✮✱ ✧❆❢❢✐♥❡ ❞✉❛❧✐t② ❛♥❞ ❝♦❢✐♥✐t❡♥❡ss✧✱ ■♥✈✳ ▼❛t❤✳✱ ✾✱ ♣♣✳ ✶✹✺✲✶✻✹✳ ❬✷✵❪ ❈✳ ❍✉♥❡❦❡ ✭✶✾✾✷✮✱ ✧Pr♦❜❧❡♠s ♦♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✱ ❋r❡❡ r❡s♦❧✉t✐♦♥s ✐♥ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr②✧✱ ✭❙✉♥❞❛♥❝❡✱ ❯t❛❤✱ ✶✾✾✵✮ ❘❡s✳ ◆♦t❡s ▼❛t❤✳✱ ✷✱ ♣♣✳ ✾✸✲✶✵✽✳ ❬✷✶❪ ❈✳ ❍✉♥❡❦❡ ❛♥❞ ❆✳ ❚❛②❧♦r ✭✷✵✵✼✮✱ ▲❡❝t✉r❡s ♦♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ✱ ❈♦♥✲ t❡♠♣♦r❛r② ▼❛t❤❡♠❛t✐❝s✱ ✹✸✻✱ ❆▼❙✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✺✶✾✾✳ ❬✷✷❪ ❍✳ ❈✳ ❍✉t❝❤✐♥s ✭✶✾✽✶✮✱ ❊①❛♠♣❧❡s ♦❢ ❝♦♠♠✉t❛t✐✈❡ ✱ P♦❧②❣♦♥❛❧✱ r✐♥❣s P❛ss❛✐❝✱ ◆❡✇ ❏❡rs❡②✳ ❬✷✸❪ ■✳ ❑❛♣❧❛♥s❦② ✭✶✾✼✹✮✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ ✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦ Pr❡ss ✭❘❡✈✐s❡❞ ❡❞✐t✐♦♥✮✳ ❬✷✹❪ ❉✳ ❑✐r❜② ✭✶✾✼✸✮✱ ✧❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ❛♥❞ ❍✐❧❜❡rt ♣♦❧②♥♦♠✐❛❧s✧✱ ✱ ✷✹✱ ♣♣✳ ✹✼✲✺✼✳ ❏✳ ▼❛t❤✳ ◗✉❛rt ✳ ✼✽ ❬✷✺❪ ❉✳ ❑✐r❜② ✭✶✾✾✵✮✱ ✧❉✐♠❡♥s✐♦♥ ❛♥❞ ❧❡♥❣t❤ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ❏✳ ▼❛t❤✳ ❖①❢♦r❞✳ ◗✉❛rt✳ ✱ ✭✷✮✹✶✱ ♣♣✳ ✹✶✾✲✹✷✾✳ ❬✷✻❪ ▲✳ ●✳ ▲✐ ✭✷✵✶✵✮✱ ✧❱❛♥✐s❤✐♥❣ ❚❤❡♦r❡♠ ♦❢ ❉✉❛❧ ❇❛ss ◆✉♠❜❡rs✧✱ ✳ ❛r❳✐✈✿♠❛t❤✳❆❈✴✶✵✵✾✳✷✷✷✾ ❬✷✼❪ ▲✳ ●✳ ▲✐ ✭✷✵✶✵✮✱ ✧❉✉❛❧ ❇❛ss ◆✉♠❜❡rs ❛♥❞ ❈♦✲❈♦❤❡♥ ▼❛❝❛✉❧❛② s rt u ă ❚❛♥❣ ✭✷✵✵✶✮✱ ✧❚❤❡ ❢✲❞❡♣t❤ ♦❢ ❛♥ ✐❞❡❛❧ ♦♥ ❛ ♠♦❞✉❧❡✧✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ Pr♦❝✳ ✱ ✭✼✮ ✶✸✵✱ ♣♣✳ ✶✾✵✺✲✶✾✶✷✳ ❬✷✾❪ ●✳ ▲✉②❜❡③♥✐❦ ✭✶✾✾✸✮✱ ✧❋✐♥✐t❡♥❡ss ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧② ♠♦❞✉❧❡s ✭❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ D✲♠♦❞✉❧❡s t♦ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✮✧✱ ■♥✈❡♥t✳ ▼❛t❤✳✱ ✭✶✮ ✶✶✸✱ ♣♣✳ ✹✶✲✺✺✳ ❬✸✵❪ ●✳ ▲②✉❜❡③♥✐❦ ✭✷✵✵✵✮✱ ✧❋✐♥✐t❡♥❡ss ♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✲ ✉❧❡s ❢♦r r❡❣✉❧❛r ❧♦❝❛❧ r✐♥❣s ♦❢ ♠✐①❡❞ ❝❤❛r❛❝t❡r✐st✐❝✿ ❚❤❡ ✉♥r❛♠✐①❡❞ ❝❛s❡✧✱ ❈♦♠♠✳ ❆❧❣✳✱ ✷✽✱ ♣♣✳ ✺✽✻✼✲✺✽✽✷✳ ❬✸✶❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞✱ ✧❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠♠✉✲ t❛t✐✈❡ r✐♥❣✧✱ ❙②♠♣♦s✐❛ ▼❛t❤❡♠❛t✐❝❛ ✱ ✶✶✱ ♣♣✳ ✷✸✲✹✸✳ ❬✸✷❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✼✷✮✱ ✧❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♦❢ ❝❡rt❛✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ✭✷✮ ✷✸✱ ♣♣✳ ✶✾✼✲✷✵✹✳ ❖①❢♦r❞✱ ❬✸✸❪ ❍✳ ▼❛ts✉♠✉r❛ ✭✶✾✽✻✮✱ ✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r② Pr❡ss✳ ❬✸✹❪ ❙✳ ▼❝❆❞❛♠ ❛♥❞ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✻✮✱ ✧▼❛①✐♠❛❧ ❝❤❛✐♥s ♦❢ ♣r✐♠❡ ✐❞❡❛❧s ✐♥ ✐♥t❡❣r❛❧ ❡①t❡♥s✐♦♥ ❞♦♠❛✐♥s✧✱ ✱ ✷✷✹✱ ♣♣✳ ✶✵✸✲✶✶✻✳ ❚r❛♥s✳ ❆▼❙✳ ❬✸✺❪ ❙✳ ▼❝❆❞❛♠ ❛♥❞ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✼✮✱ ✧❙❡♠✐✲❧♦❝❛❧ t❛✉t r✐♥❣s✧✱ ❯♥✐✈✳ ▼❛t❤✳ ❏✳ ■♥❞✐❛♥❛ ✱ ✷✻✱ ♣♣✳ ✼✸✲✼✾✳ ❬✸✻❪ ▲✳ ▼❡❧❦❡rss♦♥ ✭✶✾✾✺✮✱ ✧❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❛ ❝r✐t❡r✐♦♥ ❢♦r ❆rt✐♥✐❛♥♥❡ss ♦❢ ❛ ♠♦❞✉❧❡✧✱ ✱ ✶✵✶✱ ♣♣✳ ✷✾✶✲✸✵✸✳ ❏✳ P✉r❡ ❆♣♣❧✳ ❆❧❣❡❜r❛ ❬✸✼❪ ▼✳ ◆❛❣❛t❛ ✭✶✾✻✷✮✱ ▲♦❝❛❧ r✐♥❣s ✱ ■♥t❡rs❝✐❡♥❝❡✱ ◆❡✇ ❨♦r❦✳ ❬✸✽❪ ▼✳ ◆❛❣❛t❛ ✭✶✾✽✵✮✱ ✧❖♥ t❤❡ ❝❤❛✐♥ ♣r♦❜❧❡♠ ♦❢ ♣r✐♠❡ ✐❞❡❛❧✧✱ ✱ ✽✵✱ ♣♣✳ ✶✵✼✲✶✶✻✳ ▼❛t❤✳ ❏✳ ◆❛❣♦②❛ ✼✾ ❬✸✾❪ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ❚✳ ◆✳ ❆♥ ✭✷✵✵✾✮✱ ✧❖♥ t❤❡ ✉♥♠✐①❡❞♥❡ss ❛♥❞ t❤❡ ✉♥✐✈❡rs❛❧ ❝❛t❡♥❛r✐❝✐t② ♦❢ ❧♦❝❛❧ r✐♥❣s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ✸✷✶✱ ✱ ❏✳ ❆❧❣❡❜r❛ ♣♣✳ ✸✵✸✲✸✶✶✳ ❬✹✵❪ ▲✳ ❚✳ ◆❤❛♥ ❛♥❞ ❚✳ ◆✳ ❆♥ ✭✷✵✶✵✮✱ ✧❖♥ t❤❡ ❝❛t❡♥❛r✐❝✐t② ♦❢ ◆♦❡t❤❡r✐❛♥ ❧♦❝❛❧ r✐♥❣s ❛♥❞ q✉❛s✐ ✉♥♠✐①❡❞ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ✱ ✸✽✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ♣♣✳ ✸✼✷✽✲✸✼✸✻✳ ❬✹✶❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✻✾✮✱ ✧❖♥ q✉❛s✐✲✉♥♠✐①❡❞ ❧♦❝❛❧ ❞♦♠❛✐♥s✱ t❤❡ ❛❧t✐t✉❞❡ ❢♦r♠✉❧❛✱ ❛♥❞ t❤❡ ❝❤❛✐♥ ❝♦♥❞✐t✐♦♥ ❢♦r ♣r✐♠❡ ✐❞❡❛❧s ✭■✮✧✱ ✾✶✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ♣♣✳ ✺✵✽✲✺✷✽✳ ❬✹✷❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✶✮✱ ✧❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❝❛t❡♥❛r② r✐♥❣s✧✱ ▼❛t❤✳ ✾✸✱ ✾✹✱ ❏✳ ♣♣✳ ✶✵✼✵✲✶✶✵✽✳ ❬✹✸❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✷✮✱ ✧❈❛t❡♥❛r② r✐♥❣s ❛♥❞ t❤❡ ❛❧t✐t✉❞❡ ❢♦r♠✉❧❛✧✱ ▼❛t❤✳ ❆♠❡r✳ ❆♠❡r✳ ❏✳ ♣♣✳ ✹✺✽✲✹✻✻✳ ❬✹✹❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✼✽✮✱ ✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈❤❛✐♥ ❝♦♥❥❡❝t✉r❡s ✐♥ r✐♥❣ t❤❡♦r② ▼❛t❤❡♠❛t✐❝s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✹✺❪ ▲✳ ❏✳ ❘❛t❧✐❢❢ ✭✶✾✽✹✮✱ ✧❖♥ ❛s②♠♣t♦t✐❝ ♣r✐♠❡ ❞✐✈✐s♦rs✧✱ ✶✶✶✱ P❛❝✐❢✐❝ ❏✳ ▼❛t❤✳ ✱ ♣♣✳ ✸✾✺✲✹✶✸✳ ❬✹✻❪ ❘✳ ◆✳ ❘♦❜❡rts ✭✶✾✼✺✮✱ ✧❑r✉❧❧ ❞✐♠❡♥s✐♦♥ ❢♦r ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s ♦✈❡r q✉❛s✐ ❧♦❝❛❧ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s✧✱ ✱ ✭✷✮✷✻✱ ♣♣✳ ✷✻✾✲✷✼✸✳ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞ ❬✹✼❪ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✼✺✮✱ ✧❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r ✐♥❥❡❝t✐✈❡ ♠♦❞✉❧❡s ♦✈❡r ❝♦♠♠✉t❛t✐✈❡ ◆♦❡t❤❡r✐❛♥✱ ❧♦❝❛❧ r✐♥❣s ✧✱ ❊❞✐♥❜✉r❣❤ ▼❛t❤✳ ❙♦❝✳✱ ✷✵✱ ♣♣✳ ✶✹✸✲✶✺✶✳ ❬✹✽❪ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✼✺✮✱ ✧❙♦♠❡ r❡s✉❧ts ♦♥ t❤❡ ✈❛♥✐s❤✐♥❣ ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ✱ ✸✵✱ ♣♣✳ ✶✼✼✲✶✾✺✳ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝ ❬✹✾❪ ❘✳ ❨✳ ❙❤❛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♣❡ ❛♥❞ P✳ ❱❛s♠♦s ✭✶✾✼✷✮✱ ❯♥✐✈❡rs✐t② Pr❡ss✳ ■♥❥❡❝t✐✈❡ ♠♦❞✉❧❡s✱ ❈❛♠❜r✐❞❣❡ ✽✵ ❬✺✶❪ ❩✳ ❚❛♥❣ ❛♥❞ ❍✳ ❩❛❦❡r✐ ✭✶✾✾✹✮✱ ✧❈♦✲❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❛♥❞ ♠♦❞✉❧❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢r❛❝t✐♦♥s✧✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✱ ✭✻✮✷✷✱ ♣♣✳ ✷✶✼✸✲ ✷✷✵✹✳ ❬✺✷❪ ❖✳ ❩❛r✐s❦✐ ❛♥❞ P✳ ❙❛♠✉❡❧ ✭✶✾✻✵✮✱ ✱ ❱♦❧✳ ✷✱ ❱❛♥ ❈♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ ◆♦str❛♥❞✱ ◆❡✇ ❨♦r❦✳ ❚✐Õ♥❣ P❤➳♣ ❬✺✸❪ ❉✳ ❋❡rr❛♥❞ ❛♥❞ ▼✳ ❘❛②♥❛✉❞ ✭✶✾✼✵✮✱ ✧❋✐❜r❡s ❢♦r♠❡❧❧❡s ❞✬✉♥ ❛♥♥❡❛✉ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥✧✱ ❆♥♥✳ ❙❝✐✳ ❊✬❝♦❧❡ ◆♦r♠✳ ❙✉♣✳ ✱ ✭✹✮✸✱ ♣♣✳ ✷✾✺✲✸✶✶✳ ❚✐Õ♥❣ ➜ø❝ ❬✺✹❪ ◆✳ ❚✳ ❈✉♦♥❣✱ P✳ ❙❝❤❡♥③❡❧ ❛♥❞ ◆✳ ❱✳ ❚r✉♥❣ ✭✶✾✼✽✮✱ ✧❱❡r❛❧❧❣❡♠❡✐♥❡rt❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧♥✧✱ ✱ ✽✺✱ ♣♣✳ ✺✼✲✼✸✳ t r ă r tr r r ❋❛❧t✐♥❣s ✭✶✾✼✽✮✱ ✧U ♣❡♥✧✱ ✸✵✱ ❆r❝❤✳ ▼❛t❤✳ ♣♣✳ ✹✼✸✲✹✼✻✳ ❬✺✻❪ ●✳ ❋❛❧t✐♥❣s ✭✶✾✽✶✮✱ ✧❉❡r ❊♥❞❧✐❝❤❦❡✐tss❛t③ ✐♥ ❞❡r ❧♦❦❛❧❡♥ ❑♦❤♦♠♦❧♦❣✐❡✧✱ ▼❛t❤✳ ❆♥♥✳ ✷✺✺✱ ♣♣✳ ✹✺✲✺✻✳ ❬✺✼❪ ❲✳ ❑r✉❧❧ ✭✶✾✸✼✮✱ ✧❩✉♠ ❉✐♠❡♥s✐♦♥s❜❡❣r✐❢❢ ❞❡r ✐❞❡❛❧t❤❡♦r✐❡✧✱ ✹✷✱ ▼❛t❤✳ ❩✳✱ ♣♣✳ ✼✹✺✲✼✻✻✳ ❬✺✽❪ P✳ ❙❝❤❡♥③❡❧ ✭✶✾✽✷✮✱ ❇✉❝❤s❜❛✉♠ ✲ r✐♥❣❡ ❉✉❛❧✐s✐❡r❡♥❞❡ ❦♦♠♣❧❡①❡ ✐♥ ❞❡r ❧♦❦❛❧❡♥ ❛❧❣❡❜r❛ ✉♥❞ ✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✾✵✼✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ❇❡r❧✐♥ ✲ r r ă r ssrt r s o ăsr U r ă r u o ăsr U ăr R t ✷✷✺✲✷✸✶✳ ⊂ R✧✱ ❆r❝❤✳ ...VIỆN KHOA HỌC VÀ CƠNG NGHỆ VIỆT NAM VIỆN TỐN HỌC -oOo - Trần Nguyên An VÒ đối đồng điều địa phơng với giá cực đại V tính catenary vnh NOETHER địa phơng Chuyờn ngnh: i số lý