ĐẠI HỌC THÁI NGUYÊN ĐẠI HỌC SƯ PHẠM NGUYỄN QUANG BẠO TẬP IĐÊAN NGUYÊN TỐ GẮN KẾT VÀ TÍNH CHẤT DỊCH CHUYỂN ĐỊA PHƯƠNG 2012 ▼ô❝ ❧ô❝ ▼ë ➤➬✉ ✶ ❈❤➢➡♥❣ ✶✳ ❑✐Õ♥ t❤ø❝ ❝❤✉➬♥ ❜Þ ✸ ✶✳✶✳ ❱➭♥❤ ✈➭ ♠➠➤✉♥ ❆rt✐♥ ✸ ✶✳✷✳ ▼➠➤✉♥ ❊①t ✈➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✻ ❈❤➢➡♥❣ ✷✳ ❇✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ✶✸ ✷✳✶✳ ❇✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✶✸ ✷✳✷✳ ❙ù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✶✻ ✷✳✸✳ ❚❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ✷✼ ✷✳✹✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt q✉❛ ➤å♥❣ ❝✃✉ ♣❤➻♥❣ ✈➭ ➤è✐ ♥❣➱✉ ▼❛t❧✐s ✸✹ ❈❤➢➡♥❣ ✸✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ✸✽ ✸✳✶✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ✸✳✷✳ ❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ✸✽ ✹✷ ✹✽ ▼ë ➤➬✉ ❚r♦♥❣ s✉èt ❧✉❐♥ ✈➝♥ ♥➭②✱ t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt ❝➳❝ ✈➭♥❤ ❧➭ ❣✐❛♦ ❤♦➳♥✱ ◆♦❡t❤❡r✳ P❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ✈➭ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ➤ã♥❣ ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ♠➠➤✉♥ tr➟♥ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ◆♦❡t❤❡r✳ ▲ý t❤✉②Õt ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❝❤♦ ♠ét ✐➤➟❛♥ ❤❛② ❝❤♦ ♠ét ♠➠➤✉♥ ➤➢ỵ❝ ①❡♠ ♥❤➢ ♠ë ré♥❣ ❝đ❛ ➜Þ♥❤ ❧ý ❝➡ ❜➯♥ ❝❤♦ sè ❤ä❝✿ ♠ét sè tù ♥❤✐➟♥ ❧í♥ ❤➡♥ ➤➢ỵ❝ ♣❤➞♥ tÝ❝❤ t❤➭♥❤ tÝ❝❤ ❝ñ❛ ❝➳❝ t❤õ❛ sè ♥❣✉②➟♥ tè ✈➭ sù ♣❤➞♥ tÝ❝❤ ➤ã ❧➭ ❞✉② ♥❤✃t ♥Õ✉ ❦❤➠♥❣ ❦Ó ➤Õ♥ t❤ø tù ❝➳❝ ♥❤➞♥ tư✳ ▼ét ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ ❜✐Ĩ✉ ❞✐Ơ♥ N= ❝ñ❛ ♠➠➤✉♥ ❝♦♥ r i=1 Qi , tr♦♥❣ ➤ã ỗ N ủ R M ột Qi pi −♥❣✉②➟♥ s➡✳ ▲Ý t❤✉②Õt ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ■✳ ●✳ ▼❛❝✲ ❞♦♥❛❧❞ ❬▼❛❝❪ ♥➝♠ ✶✾✼✸ t❤❡♦ ♠ét ♥❣❤Ü❛ ♥➭♦ ➤ã ❧➭ ➤è✐ ♥❣➱✉ ✈í✐ ❧Ý t❤✉②Õt ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡✿ ▼ét R−♠➠➤✉♥ M ❧➭ t❤ø ❝✃♣ ♥Õ✉ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ x tr➟♥ M ❧➭ t♦➭♥ ❝✃✉ ❤♦➷❝ ❧ị② ❧✐♥❤ ✈í✐ ♠ä✐ x ∈ R ▼➠➤✉♥ M ❧➭ ❞✐Ơ♥ ➤➢ỵ❝ ❜✐Ĩ✉ ♥Õ✉ ♥ã ❧➭ tỉ♥❣ ❝đ❛ ♥❤÷♥❣ ♠➠➤✉♥ t❤ø ❝✃♣ M = S1 + S2 + · · · + Sn , tr♦♥❣ ➤ã ❝➳❝ ✈➭ ❝➳❝ Si ❧➭ ♠➠➤✉♥ pi t❤ø ❝✃♣ i = 1, , n ◆Õ✉ ❝➳❝ pi ❧➭ ♣❤➞♥ ❜✐Öt Si ❧➭ ❦❤➠♥❣ ❜á ➤✐ ➤➢ỵ❝ tr♦♥❣ sù ♣❤➞♥ tÝ❝❤ tr➟♥ ủ M tì tí ó ợ ọ tÝ❝❤ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉ ❝đ❛ t❐♣ M ✳ ❍➡♥ ♥÷❛ ❦❤✐ ➤ã {p1 , , pn } ❝❤Ø ♣❤ô t❤✉é❝ M ♠➭ ❦❤➠♥❣ ♣❤ô t❤✉é❝ ✈➭♦ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉ ❝đ❛ ❤✐Ư✉ ❧➭ M ✱ t❛ ❣ä✐ ♥ã ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ❣➽♥ ❦Õt ❝ñ❛ M, ✈➭ ❦Ý Att M P❤➞♥ tÝ❝❤ t❤ø ❝✃♣ ✈➭ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ➤ã♥❣ ✈❛✐ trß q✉❛♥ tr♦♥❣ tr♦♥❣ ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ♠➠➤✉♥ ❆rt✐♥✳ ✶ ➜è✐ ề ị ợ tệ rt ữ ố ề ị ♣❤➢➡♥❣ ➤➲ trë t❤➭♥❤ ❝➠♥❣ ❝ơ ❦❤➠♥❣ t❤Ĩ t❤✐Õ✉ tr♦♥❣ ❍×♥❤ ❤ä❝ ➤➵✐ sè✱ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ❚❤❡♦ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐ ❧➭ ❝➳❝ ♠➠➤✉♥ ❆rt✐♥✳ ❈❤Ý♥❤ ✈× ✈❐② ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ♥➭② ❧➭ ❝➬♥ t❤✐Õt✳ ◆é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ❧➵✐ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✈➭ ❘✳ ❨✳ ❙❤❛r♣ tr♦♥❣ ❜➭✐ ❜➳♦ ✧❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♦❢ ❝❡rt❛✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ◗✉❛rt✳ ❏✳ ▼❛t❤✳ ❖①❢♦r❞✱ ✭✷✮ ✷✸✱ ♣♣✳ ✶✾✼✲✷✵✹ ✭✶✾✼✷✮ ✈➭ ❝ñ❛ ❘✳ ❨✳ ❙❤❛r♣ tr♦♥❣ ❜➭✐ ❜➳♦ ✧❙♦♠❡ r❡s✉❧ts ♦♥ t❤❡ ✈❛♥✐s❤✐♥❣ ♦❢ ❧♦❝❛❧ ❝♦✲ ❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝ ✱ ✸✵✱ ♣♣✳ ✶✼✼✲✶✾✺ ✭✶✾✼✺✮✳ ❇➟♥ ❝➵♥❤ ➤ã ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❤Ư t❤è♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ ✈Ị ♣❤➞♥ tÝ❝❤ t❤ø ❝✃♣ ✈➭ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝ñ❛ ♠➠➤✉♥ t❤❡♦ ❜➭✐ ❜➳♦ ✧❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣✧✱ ❙②♠♣♦s✐❛ ▼❛t❤❡♠❛t✐❝❛ ✱ ✶✶✱ ♣♣✳ ✷✸✲✹✸ ✭✶✾✼✸✮ ❝đ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ✈í✐ sù ❝❤Ø ❞➵② ❤➢í♥❣ ❞➱♥ ♥❤✐Ưt t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ ❚r➬♥ ◆❣✉②➟♥ ❆♥✱ ♥❤➞♥ ❞Þ♣ ♥➭② ❡♠ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬②✳ ❊♠ ❝ị♥❣ ①✐♥ ➤➢ỵ❝ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ ❝❤➞♥ t❤➭♥❤ ➤Õ♥ ❑❤♦❛ ❚♦➳♥✱ ❑❤♦❛ s❛✉ ➜➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ P❤➵♠ ❚❤➳✐ ◆❣✉②➟♥ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ ❡♠ tr♦♥❣ t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ t➵✐ tr➢ê♥❣✳ ❳✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ư♣✱ ❜➵♥ ❜❒ tr♦♥❣ ❧í♣ ❝❛♦ ❤ä❝ ❚♦➳♥ ❑✶✽ ➤➲ q✉❛♥ t➞♠✱ ➤é♥❣ ✈✐➟♥✱ ❣✐ó♣ ➤ì ❡♠ tr♦♥❣ t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ❧✉❐♥ ✈➝♥✳ ✷ ❈❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✳✶ ❱➭♥❤ ✈➭ ♠➠➤✉♥ ❆rt✐♥ ❚❛ ❧✉➠♥ ❣✐➯ t❤✐Õt ❝➳❝ ✈➭♥❤ ❧➭ ✈➭♥❤ ❣✐❛♦ ♦➳♥ ◆♦❡t❤❡r ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ✈➭ A ❧➭ R✲♠➠➤✉♥✳ ❑❤✐ ➤ã A ➤➢ỵ❝ ❣ä✐ rt ế ỗ ❝đ❛ A ➤Ị✉ ❞õ♥❣ ♥❣❤Ü❛ ❧➭ ♥Õ✉ ♠➠➤✉♥ ❝♦♥ ❝đ❛ ❱➭♥❤ A0 ⊇ A1 ⊇ ⊇ An ⊇ ❧➭ ♠ét ❞➲② ❣✐➯♠ ❞➬♥ ❝➳❝ A t❤× tå♥ t➵✐ k ∈ N s❛♦ ❝❤♦ Ak = An ✈í✐ ♠ä✐ n ≥ k ✳ R ➤➢ỵ❝ ❣ä✐ ❧➭ ✈➭♥❤ ❆rt✐♥ ♠ä✐ ❞➲② ❣✐➯♠ ❝➳❝ ✐➤➟❛♥ ❝ñ❛ ♥Õ✉ ♥ã ❧➭ ♠ét R✲♠➠➤✉♥ ❆rt✐♥✱ tø❝ ❧➭ R ➤Ị✉ ❞õ♥❣✳ ▼Ư♥❤ ➤Ị s❛✉ ❝❤♦ t❛ ♠ét ➤✐Ị✉ ❦✐Ư♥ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ➤Þ♥❤ ♥❣❤Ü❛ ♠➠➤✉♥ ❆rt✐♥✳ ✶✳✶✳✷ ▼Ư♥❤ ➤Ị✳ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ✈➭ A ❧➭ ♠ét R✲♠➠➤✉♥✳ ❑❤✐ ➤ã ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿ ✭✐✮ A ❧➭ ♠➠➤✉♥ ❆rt✐♥❀ ỗ t rỗ ủ A ➤Ị✉ ❝ã ♣❤➬♥ tư ❝ù❝ t✐Ĩ✉✳ ➜Ĩ ➤Ị ❝❐♣ ➤Õ♥ ♠ét ✈➭✐ tÝ♥❤ ❝❤✃t ❝ñ❛ ♠➠➤✉♥ ❆rt✐♥✱ s❛✉ ➤➞② t❛ sÏ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ➤é ❞➭✐ ❝đ❛ ♠➠➤✉♥✳ ✶✳✶✳✸ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❦❤➳❝ ❦❤➠♥❣ ✈➭ M ❧➭ ♠ét R✲♠➠➤✉♥✳ ✭✐✮ ▼ét ❞➲② M0 ⊆ M1 ⊆ · · · ⊆ Mn = M ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝đ❛ M ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ✭✐✐✮ ❳Ý❝❤ ✳ ①Ý❝❤ = M0 ⊂ M1 ⊂ ⊂ Mn = M ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❞➲② ❤ỵ♣ t❤➭♥❤ ❝đ❛ M ♥Õ✉ Mi+1 /Mi ❧➭ ❝➳❝ ♠➠➤✉♥ ➤➡♥ ✈í✐ ♠ä✐ i = 0, 1, , n − 1✱ tø❝ ❧➭ Mi+1 /Mi ❝ã ➤ó♥❣ ❤❛✐ ♠➠➤✉♥ ❝♦♥ ❧➭ ✈➭ ❝❤Ý♥❤ ♥ã✳ ✭✐✐✐✮ ➜é ❞➭✐ ❝đ❛ M ✱ ❦Ý ❤✐Ư✉ ❧➭ ❞➭✐ ❝đ❛ ❝➳❝ ①Ý❝❤ ❝ã ❞➵♥❣ R (M )✱ ❧➭ ❝❐♥ tr➟♥ ➤ó♥❣ ❝đ❛ ❝➳❝ ➤é = M0 ⊂ M1 ⊂ ⊂ Mn = M, tr♦♥❣ ➤ã Mi = Mi+1 ✈í✐ ♠ä✐ i = 0, 1, , n − ▼ét R✲♠➠➤✉♥ M ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ♥Õ✉ M ❝ã Ýt ♥❤✃t ♠ét ❞➲② ❤ỵ♣ t❤➭♥❤✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭② ❝➳❝ ❞➲② ❤ỵ♣ t❤➭♥❤ ❝đ❛ ❝ï♥❣ ➤é ❞➭✐ ✈➭ ❦❤✐ ➤ã ➤é ❞➭✐ ❝đ❛ M ✱ ❦Ý ❤✐Ư✉ ❧➭ ❝đ❛ ♠ét ❞➲② ❤ỵ♣ t❤➭♥❤ ♥➭♦ ➤ã ❝đ❛ ❣✐➯♠ t❤ù❝ sù ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ M ❝ã R (M )✱ ❝❤Ý♥❤ ❧➭ ộ M tế ữ ỗ t ❤♦➷❝ M ➤Ị✉ ❝ã ➤é ❞➭✐ ❦❤➠♥❣ ✈➢ỵt q✉➳ ➤é ủ ợ t ị ý ế ✭✐✐✮ ◆Õ✉ ❚❛ ❝ã ❝➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ❧➭ ➤ó♥❣✳ R ❧➭ ✈➭♥❤ ❆rt✐♥ t❤× ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ R ➤Ị✉ tè✐ ➤➵✐✳ R ❧➭ ✈➭♥❤ ❆rt✐♥ t❤× R ❝ã ❤÷✉ ❤➵♥ ✐➤➟❛♥ tè✐ ➤➵✐✳ ❇➞② ❣✐ê t❛ sÏ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ❝đ❛ ✈➭♥❤ ✈➭ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥✳ ✹ ✶✳✶✳✺ ➜Þ♥❤ ♥❣❤Ü❛✳ p0 R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❦❤➳❝ ❦❤➠♥❣✳ ▼ét ❞➲② pn tr♦♥❣ ➤ã p0 , p1 , , pn ❧➭ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ p1 ✭✐✮ ❈❤♦ R✱ ❣ä✐ ❧➭ ♠ét ❞➲② ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ R ➤é ❞➭✐ ♥✳ ❈❐♥ tr➟♥ ❝ñ❛ ❝➳❝ ➤é ❞➭✐ ❝ù❝ ➤➵✐ ❝ñ❛ ❞➲② ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ R ❝ã ❞➵♥❣ p = p0 ➤➢ỵ❝ ❣ä✐ ❧➭ ✭✐✐✮ ➤é ❝❛♦ p1 ❝đ❛ p✱ ❦ý ❤✐Ư✉ ❧➭ ❈❤✐Ị✉ ❝đ❛ ✈➭♥❤ pn ht p R í ệ dim R ợ ị ♥❣❤Ü❛ ❧➭ ❝❐♥ tr➟♥ ❝ñ❛ ➤é ❝❛♦ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣ R dim(R) = Sup{ht p| p ∈ Spec(R)} ❈❤✐Ị✉ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ R✳ ◆Õ✉ dim R ữ tì ó ộ ❝ñ❛ ❞➲② ✐➤➟❛♥ ♥❣✉②➟♥ tè ❞➭✐ ♥❤✃t tr♦♥❣ ✭✐✐✐✮ ❈❤♦ M = ❧➭ ♠ét R✲♠➠➤✉♥✳ ❑❤✐ ➤ã R✳ ❝❤✐Ò✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥ M ✱ ❦ý ❤✐Ư✉ ❧➭ dim M ✱ ❧➭ dim R/ Ann M ✳ ◆Õ✉ M ❧➭ ♠➠➤✉♥ ❦❤➠♥❣ t❤× t❛ q✉② ➢í❝ dim M = −1✳ ✶✳✶✳✻ ▼Ư♥❤ ➤Ị✳ ✈➭ R = ❧➭ ✈➭♥❤ ❆rt✐♥ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ R ❧➭ ✈➭♥❤ ◆♦❡t❤❡r dim R = ✶✳✶✳✼ ❇ỉ ➤Ị✳ ❈❤♦ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✳ ❈❤♦ A ❧➭ R✲♠➠➤✉♥✳ ❈➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ❧➭ ➤ó♥❣ ✭✐✮ (A) < ∞ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ A ✈õ❛ ❧➭ ◆♦❡t❤❡r ✈õ❛ ❧➭ ❆rt✐♥✳ ✭✐✐✮ ❈❤♦ (A) = n < ∞ ❧➭ ♠➠➤✉♥ ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥✳ mn A = ✺ ❑❤✐ ➤ã ✶✳✷ ▼➠➤✉♥ ❊①t ✈➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✶✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ▼ét ❣✐➯✐ ①➵ ➯♥❤ ❝ñ❛ M ❧➭ ♠ét ❞➲② ❦❤í♣ −→ P2 −→ P1 −→ P0 M tr ó ỗ Pi ①➵ ➯♥❤✳ ✶✳✷✳✷ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ M, N ❧➭ ❝➳❝ R−♠➠➤✉♥ ✈➭ n ≥ ❧➭ ♠ét sè tù ♥❤✐➟♥✳ ▼➠➤✉♥ ❞➱♥ s✉✃t ♣❤➯✐ t❤ø n ❝đ❛ ❤➭♠ tư Hom(−, N ) ø♥❣ ✈í✐ M ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➠➤✉♥ n ❝đ❛ M ✈➭ N ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ ♠ë ré♥❣ t❤ø u u ExtnR (M, N ) ❈ơ t❤Ĩ✱ ♥Õ✉ −→ P2 −→ P1 −→ P0 −→ M −→ ❧➭ ♠ét ❣✐➯✐ ①➵ ➯♥❤ ❝đ❛ M, t➳❝ ➤é♥❣ ❤➭♠ tư Hom(−, N ) ✈➭♦ ❞➲② ❦❤í♣ tr➟♥ t❛ ❝ã ♣❤ø❝ u∗ u∗ −→ Hom(P0 , N ) −→ Hom(P1 , N ) −→ Hom(P2 , N ) −→ ❑❤✐ ➤ã ExtnR (M, N ) = Ker u∗n+1 / Im u∗n ❧➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ t❤ø n ❝đ❛ ♣❤ø❝ tr➟♥ ✭♠➠➤✉♥ ♥➭② ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ✈✐Ư❝ ❝❤ä♥ ❣✐➯✐ ①➵ ➯♥❤ ❝đ❛ M ✮✳ ✶✳✷✳✸ ▼Ư♥❤ ➤Ị✳ s✐♥❤ ✈í✐ ♠ä✐ ◆Õ✉ M, N ❧➭ ❤÷✉ ❤➵♥ s✐♥❤ tì ExtnR (M, N ) ữ n ết q✉➯ ❞➢í✐ ➤➞② ❝❤♦ t❛ tÝ♥❤ ❝❤✃t ❣✐❛♦ ❤♦➳♥ ❣✐÷❛ Ext ✈➭ ❤➭♠ tư ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳✳ ✶✳✷✳✹ ▼Ư♥❤ ➤Ị✳ ◆Õ✉ S ❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ ❝đ❛ R t❤× S −1 (ExtnR (M, N )) ∼ = ExtnS −1 R (S −1 M, S −1 N ), ✻ tr♦♥❣ ➤ã S −1 ❧➭ ❤➭♠ tư ➤Þ❛ ♣❤➢➡♥❣ ❤♦➳✳ ➜➷❝ ❜✐Ưt✱ (ExtnR (M, N ))p ∼ = ExtnRp (Mp , Np ) ✈í✐ ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ✶✳✷✳✺ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư ●✐➯ sư p ❝đ❛ R f : (R , m ) −→ (R, m) ❧➭ ♠ét t♦➭♥ ❝✃✉ ✈➭♥❤✳ p ∈ Spec(R), p = f −1 (p) ∈ Spec(R ) ❑❤✐ ➤ã f ❝➯♠ s✐♥❤ t♦➭♥ ❝✃✉ f : Rp −→ Rp , t❤á❛ ♠➲♥ ●✐➯ sö ➤ã tå♥ t➵✐ f (r /s ) = f (r )/f (s ) ✈í✐ ♠ä✐ r ∈ R M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ j ✈➭ s ∈R \p ❧➭ sè ♥❣✉②➟♥ ❦❤➠♥❣ ➞♠✳ ❑❤✐ Rp ✲♠➠➤✉♥ ExtjR (Mp , Rp ) ✈➭ p j ExtjR (Mp , Rp ) ∼ = (ExtR (M, R ))p p ♥❤➢ ❝➳❝ Rp ✲♠➠➤✉♥✳ ➜è✐ ề ị ợ tệ rt ữ ố ề ị ♣❤➢➡♥❣ ➤➲ trë t❤➭♥❤ ❝➠♥❣ ❝ơ ❦❤➠♥❣ t❤Ĩ t❤✐Õ✉ tr♦♥❣ ❍×♥❤ ❤ä❝ ➤➵✐ sè✱ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ❚r➢í❝ t✐➟♥ t❛ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❤➭♠ tư I ✲①♦➽♥✳ ✶✳✷✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ♥❣❤Ü❛ ΓI (N ) = ❈❤♦ I ❧➭ ủ R ỗ R N t ị (0 :N I n ) ◆Õ✉ f : N −→ N ❧➭ ➤å♥❣ ❝✃✉ ❝➳❝ n≥0 R−♠➠❞✉♥ t❤× t❛ ❝ã ➤å♥❣ ❝✃✉ f ∗ : ΓI (N ) −→ ΓI (N ) ❝❤♦ ❜ë✐ f ∗ (x) = f (x) ❑❤✐ ➤ã ΓI (−) ❧➭ ❤➭♠ tư ❦❤í♣ tr➳✐ tõ ♣❤➵♠ trï ❝➳❝ R−♠➠➤✉♥ ➤Õ♥ ♣❤➵♠ trï ❝➳❝ R−♠➠➤✉♥✳ ❍➭♠ tö ΓI (−) ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ tư I−①♦➽♥✳ ✼ ▼ét ❣✐➯✐ ♥é✐ ①➵ ❝đ❛ N ❧➭ ♠ét ❞➲② ❦❤í♣ −→ N −→ E0 −→ E1 −→ E2 −→ tr ó ỗ Ei ộ ú ý ỗ ề ó ộ ➜Þ♥❤ ♥❣❤Ü❛✳ ❞➱♥ s✉✃t ♣❤➯✐ t❤ø ♠➠➤✉♥ ❈❤♦ N ❧➭ R−♠➠➤✉♥ ✈➭ I ❧➭ ✐➤➟❛♥ ❝ñ❛ R ▼➠➤✉♥ n ❝ñ❛ ❤➭♠ tư I−①♦➽♥ ΓI (−) ø♥❣ ✈í✐ N ➤➢ỵ❝ ❣ä✐ ❧➭ ➤è✐ ➤å♥❣ ➤✐Ị✉ t❤ø n ❝đ❛ N ✱ ❦Ý ❤✐Ư✉ ❧➭ HIn (N ) ❈ơ t❤Ĩ✱ ♥Õ✉ u u E1 −→ E2 −→ N −→ E0 −→ ❧➭ ❣✐➯✐ ♥é✐ ①➵ ❝ñ❛ N, t➳❝ ➤é♥❣ ❤➭♠ tö ΓI (−) t❛ ❝ã ♣❤ø❝ u∗ u∗ −→ Γ(E0 ) −→ Γ(E1 ) −→ Γ(E2 ) −→ ❑❤✐ ➤ã HIn (N ) = Ker u∗n / Im u∗n−1 ❧➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ t❤ø n ❝đ❛ ♣❤ø❝ tr➟♥ ✭♥ã ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ✈✐Ư❝ ❝❤ä♥ ❣✐➯✐ ♥é✐ ①➵ ❝đ❛ N ✮✳ ❙❛✉ ➤➞② ❧➭ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ✶✳✷✳✽ ▼Ư♥❤ ➤Ị✳ ✭✐✮ N ❧➭ ♠ét R−♠➠➤✉♥✳ HI0 (N ) ∼ = ΓI (N ) ✭✐✐✮ ❱í✐ N = N/ΓI (N ) t❛ ❝ã HIn (N ) ∼ = HIn (N ) ✈í✐ n ế ỗ N N −→ N −→ ❧➭ ❞➲② ❦❤í♣ ♥❣➽♥ t❤× ✈í✐ n ❝ã ➤å♥❣ ❝✃✉ ♥è✐ HIn (N ) −→ HIn+1 (N ) s❛♦ ❝❤♦ t❛ ❝ã ❞➲② ❦❤í♣ ❞➭✐ −→ ΓI (N ) −→ ΓI (N ) −→ ΓI (N ) −→ HI1 (N ) −→ HI1 (N ) −→ HI1 (N ) −→ HI2 (N ) −→ ✭✐✈✮ ◆Õ✉ S ❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ ❝đ❛ R t❤× S −1 (HIn (N )) ∼ = HSn−1 I (S −1 N )✳ ✽ ❈❤♦ ϕ : (R, m) −→ (S, n) ♣❤➢➡♥❣ ✈➭ M ❧➭ ♠ét s✐♥❤✳ ❑❤✐ ➤ã ♥Õ✉ a N ❧➭ ➤å♥❣ ❝✃✉ ❣✐÷❛ tr ị R ữ s N ♣❤➻♥❣ tr➟♥ ❧➭ S ✲♠➠➤✉♥ ❤÷✉ ❤➵♥ R✱ p ∈ Spec(R) ✈➭ N/pN = t❤× ϕ(AssS (N/pN )) = AssR (N/pN ) = p, AssS (M ⊗R N ) = AssS (N/pN ), p∈AssR M a ✈í✐ ϕ : Spec(S) −→ Spec(R) ✷✳✹✳✷ ❇ỉ ➤Ị✳ ●✐➯ sư (R, m) ♠➠➤✉♥ ◆♦❡t❤❡r✳ ❑❤✐ ➤ã ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ D(N ) ❧➭ ♠ét R✲♠➠➤✉♥ ❆rt✐♥ ✈➭ N ❧➭ ♠ét R✲ Att D(N ) = Ass(N ) ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö p ∈ Ass(N )✳ ❑❤✐ ➤ã N ❝ã ♠➠➤✉♥ ❝♦♥ B ♠➭ f p = Ann(B) ❚õ ❞➲② ❦❤í♣ −→ B −→ N, t❛ ❝ã ❞➲② ❦❤í♣ D(N ) −→ D(B) −→ ❙✉② r❛ D(B) ∼ = D(N )/ ker f ❚❛ ❝ã p = Ann(B) = Ann(D(B)) = Ann(D(N )/ ker f ) ❚❤❡♦ ❍Ö q✉➯ ✷✳✷✳✻✱ t❛ ❝ã ◆❣➢ỵ❝ ❧➵✐ ❣✐➯ sư p ∈ Att(D(N )) p ∈ Att(D(N )) ❚❤❡♦ ❍Ö q✉➯ ✷✳✷✳✻✱ t❛ ❝ã p = Ann(D(N )/L) ✈í✐ L ❧➭ ♠➠➤✉♥ ❝♦♥ ❝đ❛ D(N )✳ ❚õ ❞➲② ❦❤í♣ D(N ) −→ D(N )/L −→ t❛ ❝ã ❞➲② ❦❤í♣ −→ D(D(N )/L) −→ D(D(N ))✳ ❉♦ ✈❐② D(D(N )) ❝ã ♠➠➤✉♥ ❝♦♥ ♠➭ ❧✐♥❤ ❤ã❛ tư ❧➭ p✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ➤è✐ ♥❣➱✉ ▼❛t❧✐s✱ t❛ ❝ã ➤➻♥❣ ❝✃✉ ❝♦♥ D(D(N )) ∼ = N ⊗R R✳ ❉♦ ➤ã tå♥ t➵✐ R✲♠➠➤✉♥ C ❝ñ❛ N ⊗R R ♠➭ AnnR (C) = p✳ ●ä✐ C R ❧➭ R✲♠➠➤✉♥ ❝♦♥ ❝ñ❛ N ⊗R R s✐♥❤ ❜ë✐ C ✳ ❑❤✐ ➤ã C R ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ AnnR (C R) ∩ R = AnnR (C R) = AnnR (C) = p ✸✹ ❚õ ➤ã s✉② r❛ tå♥ t➵✐ p ∈ AssR (C R) s❛♦ ❝❤♦ p ∩ R = p ❱× p ∈ AssR (N ⊗R R) ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✹✳✶✱ t❛ ❝ã p ∈ AssR N ❙❛✉ ➤➞② t❛ sÏ ①❡♠ ①Ðt t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝ñ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ❣✐➳ ❝ù❝ ➤➵✐ ❧✐➟♥ ❤Ư ✈í✐ ➤å♥❣ ❝✃✉ ♣❤➻♥❣✳ ❚r➢í❝ ❤Õt t❛ ❝ã ❜ỉ ➤Ị s❛✉✳ ✷✳✹✳✸ ❇ỉ ➤Ị✳ ✈➭♥❤✳ ▲✃② ●✐➯ sư R, S ❧➭ ❝➳❝ ✈➭♥❤ ✈➭ f : R −→ S q ∈ Spec(S) ✈➭ ➤➷t qc = f −1 (q)✳ ●✐➯ sö N ❧➭ ♠ét ➤å♥❣ ❝✃✉ ❧➭ S ✲♠➠➤✉♥ ❦❤➳❝ ❦❤➠♥❣ ❝ã ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉ N = K1 + K2 + · · · + Kn , (∗) tr♦♥❣ ➤ã Ki ♠➠➤✉♥ t❤❡♦ ❧➭ S ✲♠➠➤✉♥ qi ✲t❤ø f ✳ ❑❤✐ ➤ã Ki ❧➭ ❝✃♣ i = 1, , n qci ✲t❤ø ❝✃♣✱ N ❳❡♠ N ✈➭ Ki ❧➭ R✲ ❝ã ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ AttR (N ) = {qc |q ∈ AttS (N )} ❈❤ø♥❣ ♠✐♥❤✳ ●ä✐ r i ∈ {1, , n} ❳Ðt tù ➤å♥❣ ❝✃✉ Ki −→ Ki ✳ ❚❛ ❝ã r.x = f (r).x ✈➭ f (r) ∈ qi ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ r ∈ qci ❱× ✈❐② ♣❤Ð♣ ♥❤➞♥ ❧➭ t♦➭♥ ❝✃✉ ❤❛② ❧ò② ❧✐♥❤ tï② t❤❡♦ r t❤✉é❝ ❤❛② ❦❤➠♥❣ t❤✉é❝ qci ✳ ❉♦ ➤ã R✲♠➠➤✉♥ Ki ❧➭ qci ✲t❤ø ❝✃♣ ✈➭ (∗) ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ❝ñ❛ R✲♠➠➤✉♥ N ✳ ❚❤❡♦ ◆❤❐♥ ①Ðt ✷✳✶✳✻ t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❈❤ó ý ❦❤➠♥❣ ♠➠➤✉♥ ♥➭♦ t❤õ❛ tr♦♥❣ ♣❤➞♥ tÝ❝❤ ë ✷✳✹✳✹ ▼Ư♥❤ ➤Ị✳ ●✐➯ sư ♣❤➢➡♥❣ t❤á❛ ♠➲♥ s✐♥❤ ❦❤➳❝ ❦❤➠♥❣✱ mS i (∗)✳ f : (R, m) −→ (S, n) ❧➭ n✲♥❣✉②➟♥ s➡✳ ●✐➯ sö ❧➭ ➤å♥❣ ❝✃✉ ♣❤➻♥❣ ➤Þ❛ M ❧➭ ❧➭ sè ♥❣✉②➟♥ ❦❤➠♥❣ ➞♠ s❛♦ ❝❤♦ R✲♠➠➤✉♥ Hmi (M ) = 0✳ qc = f −1 (q) ✈í✐ q ∈ Spec(S) ❑❤✐ ➤ã Hni (M ⊗R S) = ✈➭ AttR (Hmi (M )) = {qc : q ∈ AttS (Hni (M ⊗R S))} ✸✺ ❤÷✉ ❤➵♥ ➜➷t ❈❤ø♥❣ ♠✐♥❤✳ ❣✐➯ t❤✐Õt t❛ ❝ã ❚❤❡♦ ➜Þ♥❤ ❧ý ❝❤✉②Ĩ♥ ❝➡ së ♣❤➻♥❣ ✭➜Þ♥❤ ❧ý ✶✳✷✳✶✶✮ ✈➭ t❤❡♦ Hni (M ⊗R S) ∼ = Hmi (M ) ⊗R S ●✐➯ sö Hmi (M ) = S1 + S2 + · · · + Sn ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ t❤✉ ❣ä♥ ❝đ❛ ❝✃♣✱ R✲♠➠➤✉♥ Hmi (M ) ✈í✐ Si ❧➭ pi ✲t❤ø i = 1, , n ỗ j {1, , n} ❣ä✐ uj : Sj −→ Hmi (M ) ❧➭ ♣❤Ð♣ ♥❤ó♥❣✳ ❱× S ❧➭ R✲♣❤➻♥❣ ♥➟♥ ➤å♥❣ ❝✃✉ uj ⊗ IdS : Sj ⊗R S −→ Hmi (M ) ⊗R S ❧➭ ➤➡♥ ❝✃✉✳ ❉♦ ➤ã Tj = (uj ⊗ IdS )[Sj ⊗R S] ∼ = Sj ⊗R S ✈➭ Hmi (M ) ⊗R S = T1 + T2 + Ã Ã Ã + Tn () ỗ Tj S ✲♠➠➤✉♥ ❆rt✐♥ ❦❤➳❝ ❦❤➠♥❣ ✈➭ ✈× ✈❐② ♥ã ❝ã ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣✳ ❱× Tj ∼ = Sj ⊗R S ♥➟♥ Tj ❧➭ R✲♠➠➤✉♥ t❤ø ❝✃♣✳ ❚❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✸ t❛ ❝ã {f −1 (q)|q ∈ AttS (Tj )} = {pj } ❉♦ ✈❐② ✈í✐ ≤ j < ≤ n t❤× AttS (Tj ) ∩ AttS (T ) = ∅ ❚❤✉ ❣ä♥ ♣❤➞♥ tÝ❝❤ t❤ø ❝✃♣ ❝ñ❛ ❝➳❝ Tj ①Ðt ♥❤➢ S ♠➠➤✉♥✳ ▲➢✉ ý ✈× S ❧➭ ❤♦➭♥ t♦➭♥ ♣❤➻♥❣ tr➟♥ R ♥➟♥ ❦❤➠♥❣ t❤Ó ❜á ❜✃t ỳ Tj tr tứ () ỗ Tj sÏ ❣✐÷ ❧➵✐ Ýt ♥❤✃t ♠ét ❤➵♥❣ tư s❛✉ ❦❤✐ t❤✉ ❣ä♥✳ ❚❤❛② ✈➭♦ (∗) t❛ sÏ ❝ã ♣❤➞♥ tÝ❝❤ t❤✉ ❣ä♥ ❝ñ❛ Hmi (M ) ⊗R S ①Ðt ♥❤➢ S ♠➠➤✉♥✳ ❚õ ➤ã t❛ ❝ã ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✸✻ ❈❤➢➡♥❣ ✸ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ✸✳✶ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ▼ô❝ ♥➭② ♠➠ t➯ t❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t✳ ✸✳✶✳✶ ➜Þ♥❤ ❧ý✳ ●✐➯ sư (R, m) ❧➭ ✈➭♥❤ ị tr M R ữ ❤➵♥ s✐♥❤✱ ❝❤✐Ò✉ d✳ ❑❤✐ ➤ã Hmd (M ) = ✈➭ Att(Hmd (M )) = {p ∈ Ass M : dim R/p = d} ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ✶✳✷✳✶✸ t❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✶✾ q✉② ♥➵♣ t❤❡♦ ❝❤✐Ị✉ ●✐➯ sö Hmi (M ) ❧➭ ♠➠➤✉♥ ❆rt✐♥ ✈➭ ❞♦ ➤ã Hmi (M ) ❧➭ ♠➠➤✉♥ ❜✐Ĩ✉ ❞✐Ơ♥ ➤➢ỵ❝✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ d ❝ñ❛ M ✳ d = 0✳ ❑❤✐ ➤ã M ❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ❤❛② tå♥ t➵✐ n ∈ N s❛♦ ✸✼ ❝❤♦ mn M = ❱× ✈❐② Hm0 (M ) ∼ = Γm (M ) = M = ❉♦ ➤ã t❤❡♦ ❍Ö q✉➯ ✷✳✸✳✼ t❛ ❝ã Att(Hmd (M )) = Att(M ) = {m} = Ass(M ) = {p ∈ Ass(M ) : dim R/p = 0} ●✐➯ sư d > ✈➭ ❦❤➻♥❣ ➤Þ♥❤ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ ❝➳❝ ♠➠➤✉♥ ❝ã ❝❤✐Ị✉ ♥❤á t❤ù❝ d rờ ợ M R ữ ❤➵♥ s✐♥❤ ❝❤✐Ò✉ d ♠➭ M ❦❤➠♥❣ ❝ã ♠➠➤✉♥ ❝♦♥ ❦❤➳❝ ❦❤➠♥❣ ❝ã ❝❤✐Ò✉ t❤ù❝ sù ♥❤á ❤➡♥ d✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ Hmd (M ) = ✈➭ Att(Hmd (M ) = Ass(M ) ❱× d > ♥➟♥ m ∈ / Ass(M )✳ ❉♦ ➤ã tå♥ t➵✐ r ∈ m ❧➭ ♣❤➬♥ tư M ✲❝❤Ý♥❤ q✉②✳ ❱× ✈❐② ◆Õ✉ depth M ≥ ●✐➯ sö Hmd (M ) = 0✳ d = 1✱ t❛ ❝ã ≤ depth M ≤ dim M = ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ depth M = ✈➭ ❧➭ ➤✐Ò✉ ✈➠ ❧ý t❤❡♦ tÝ♥❤ ❝❤✃t ✈Ị ➤é s➞✉ ❝đ❛ ♠➠➤✉♥✳ ●✐➯ sư d > ❱× r ❧➭ M ✲❝❤Ý♥❤ q✉② ♥➟♥ dim M/rM = d − 1✳ ❚❛ ❝ã ❞➲② ❦❤í♣ r −→ M −→ M −→ M/rM −→ ❱× Hmd (M ) = ♥➟♥ ❞➲② ❦❤í♣ ♥➭② ❝➯♠ s✐♥❤ ❞➲② ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ò✉ r Hmd−1 (M ) −→ Hmd−1 (M ) −→ Hmd−1 (M/rM ) −→ ✸✽ ❚õ ➤ã s✉② r❛ Hmd−1 (M )/rHmd−1 (M ) ∼ = Hmd−1 (M/rM ) d−1 ❚❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ Hm (M/rM ) ♠✐♥❤ = ♥➟♥ Hmd−1 (M ) = ❚❛ ❝❤ø♥❣ m ∈ Att(Hmd−1 (M )) ❚❤❐t ✈❐② ❣✐➯ sö m ∈ / Att(Hmd−1 (M )) ❑❤✐ ➤ã t❤❡♦ ➜Þ♥❤ ❧ý ❚r➳♥❤ ♥❣✉②➟♥ tè✱ t❛ ❝ã m        q d−1 p∈Att(Hm (M )) ❱× ✈❐② tå♥ t➵✐ r1 q∈Ass(M ) ∈ m ❧➭ M ✲❝❤Ý♥❤ q✉② ✈➭ t❤❡♦ ▼Ö♥❤ ➤Ò ✷✳✷✳✾✱ t❛ ❝ã Hmd−1 (M ) = r1 Hmd−1 (M ) ➜✐Ị✉ ♥➭② ❧➭ ✈➠ ❧ý ✈× Hmd−1 (M )/rHmd−1 (M ) = ❑Ð♦ t❤❡♦ m ∈ Att(Hmd−1 (M )) ●✐➯ sö Att(Hmd−1 (M )) \ {m} = {p1 , · · · , pt } ▲➵✐ t❤❡♦ ➜Þ♥❤ ❧ý ❚r➳♥❤ ♥❣✉②➟♥ tè tå♥ t➵✐ t r2 ∈ m \ pi i=1 ❉♦ ➤ã r2 ❧➭    q q∈Ass(M ) M ✲❝❤Ý♥❤ q✉②✱ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥✱ t❛ ❝ã Hmd−1 (M )/r2 Hmd−1 (M ) ∼ = Hmd−1 (M/r2 M ) ❚❤❡♦ ❣✐➯ t❤✐Õt q✉② ♥➵♣ Hmd−1 (M/r2 M ) = ✈➭ Att(Hmd−1 (M/r2 M )) = {p ∈ Ass(M/r2 M ) : dim R/p = d − 1} ▼➷t ❦❤➳❝ t❤❡♦ ✷✳✷✳✷ Att(Hmd−1 (M )/r2 Hmd−1 (M )) ⊆ {p ∈ Att(Hmd−1 (M )) : r2 ∈ p} = {m} ì d > ị tr ♠➞✉ t❤✉➱♥ ✈í✐ ♥❤❛✉✳ ➜✐Ị✉ ♥➭② ❝❤ø♥❣ tá Hmd (M ) = ❚❛ ❝❤ø♥❣ ♠✐♥❤ t➵✐ Att(Hmd (M )) = Ass(M ) ❱× depth M ≥ ♥➟♥ tå♥ r ∈ m ❧➭ M ✲❝❤Ý♥❤ q✉②✳ ❱× ✈❐② dim M/rM = d − ❚õ ❞➲② ❦❤í♣ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝➯♠ s✐♥❤ tõ ❞➲② ❦❤í♣ r −→ M −→ M −→ M/rM −→ ✈➭ t❤❡♦ ➜Þ♥❤ ❧ý ❚r✐Ưt t✐➟✉ ❝đ❛ ●r♦t❤❡♥❞✐❡❝❦ ✭➜Þ♥❤ ❧ý ✶✳✷✳✶✷✮✱ t❛ ❝ã Hmd (M ) = rHmd (M ) ❉♦ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✾✱ t❛ ❝ã m\ p⊆m\ d (M )) q∈Att(Hm p∈Ass(M ) ▲✃② ❜✃t ❦ú q q ∈ Att(Hmd (M ))✳ ❚õ ❜❛♦ ❤➭♠ t❤ø❝ tr➟♥ ✈➭ t❤❡♦ ➜Þ♥❤ ❧ý ❚r➳♥❤ ♥❣✉②➟♥ tè✱ t❛ ❝ã q ⊆ p ✈í✐ p ∈ Ass(M ) ❚❛ ❝ã Ann M ⊆ Ann Hmd (M ) ⊆ q ⊆ p ❱× d = dim R/ Ann M = dim R/p ♥➟♥ q = p ❉♦ ➤ã Att(Hmd (M )) ⊆ Ass(M ) ◆❣➢ỵ❝ ❧➵✐ ❣✐➯ sư p ∈ Ass(M ) ❚❤❡♦ ❣✐➯ t❤✐Õt✱ t❛ ❝ã dim R/p = d ❚❤❡♦ ❧ý t❤✉②Õt ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ tå♥ t➵✐ ♠➠➤✉♥ p✲♥❣✉②➟♥ s➡ ❝ã Q ❝ñ❛ M ✳ ❚❛ M/Q = ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ Ass(M/Q) = {p} ❚❛ ❝ã M/Q ❦❤➠♥❣ ❝ã ♠➠➤✉♥ ❝♦♥ ❦❤➳❝ ❦❤➠♥❣ ❝❤✐Ò✉ t❤ù❝ sù ♥❤á ❤➡♥ d ✈× tr➳✐ ❧➵✐ t❤× Ass(M/Q) ❝ã ✐➤➟❛♥ ♥❣✉②➟♥ tè ❦❤➳❝ p✳ ❚❤❡♦ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥ ➳♣ ❞ô♥❣ ❝❤♦ ♠➠➤✉♥ M/Q t❤❛② ❝❤♦ ♠➠➤✉♥ M ✱ t❛ ❝ã Hmd (M ) = ✈➭ ∅ = Att(Hmd (M/Q)) ⊆ Ass(M/Q) = {p} ✹✵ ❉♦ ➤ã Att(Hmd (M/Q)) = {p} ❱× dim Q < d + ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ❚r✐Ưt t✐➟✉ ❝đ❛ ●r♦t❤❡♥❞✐❡❝❦ ✭➜Þ♥❤ ❧ý ✶✳✷✳✶✷✮✱ t❛ ❝ã Hmd+1 (Q) = ❚õ ❞➲② ❦❤í♣ −→ Q −→ M −→ M/Q −→ ✈➭ ị tr t ó t ữ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ Hmd (M ) −→ Hmd (M/Q)✳ ❉♦ ➤ã t❛ ❝ã t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✷ {p} = Att(Hmd (M/Q)) ⊆ Att(Hmd (M )) ❱× ✈❐② Ass(M ) ⊆ Att(Hmd (M )) ❚r➢ê♥❣ ❤ỵ♣ ✷✿ ●✐➯ sư M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ ❦❤➳❝ ❦❤➠♥❣✱ ❝❤✐Ị✉ d✳ ❚❤❡♦ ❇ỉ ➤Ị ✶✳✷✳✾✱ t❛ ❝ã Att(Hmd (M )) = Att(Hmd (G)) = Ass(G) = {p ∈ Ass(M ) : dim R/p = d} ✸✳✶✳✷ ❍Ư q✉➯✳ ●✐➯ sư ❤➵♥ s✐♥❤ ❝ã ❝❤✐Ị✉ ✸✳✷ (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ M ❧➭ R✲♠➠➤✉♥ ❤÷✉ d > 0✳ ❑❤✐ ➤ã Hmd (M ) ữ s í t ị ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ➜Þ❛ ♣❤➢➡♥❣ ❤ã❛ ❧➭ ♠ét ❝➠♥❣ ❝ơ ❤÷✉ ❤✐Ư✉ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ◆❤➽❝ ❧➵✐ ♠ét tÝ♥❤ ❝❤✃t q✉❡♥ t❤✉é❝ ❝❤Ø r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❧✐➟♥ ❦Õt ❝đ❛ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ♣❤➢➡♥❣ ❤♦➳ ❝đ❛ ♥ã t➵✐ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè p AssRp (Mp ) = {qRp | q ∈ AssR M, q ⊆ p} ✹✶ M ✈➭ ➤Þ❛ ✈í✐ ♠ä✐ p ∈ Spec(R) ➜è✐ ✈í✐ ❝➳❝ ♠➠➤✉♥ ❆rt✐♥ t❛ ❝ị♥❣ ♠✉è♥ t×♠ ♠ét ❝➠♥❣ t❤ø❝ t➢➡♥❣ tù ♥❤➢ ✈❐② ❝❤♦ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt✳ ❈➳❝ ❦Õt q✉➯ ë ♠ơ❝ ♥➭② ➤➢ỵ❝ ❧✃② tr♦♥❣ ❜➭✐ ❜➳♦ ➤➝♥❣ ♥➝♠ ✶✾✼✺ ❝ñ❛ ❘✳❨✳ ❙❤❛r♣ ❬✻❪ ❣✐➯✐ q✉②Õt ♠ét ♣❤➬♥ ✈✃♥ ➤Ị tr➟♥✳ ✸✳✷✳✶ ➜Þ♥❤ ❧ý (R, m) ✭❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ tỉ♥❣ q✉➳t✮✳ ●✐➯ sư ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ủ ị rst M R ữ s✐♥❤✳ ●✐➯ sö p ∈ Spec(R)✱ dim R/p = t ✈➭ i ❧➭ ♠ét sè ♥❣✉②➟♥✳ ▲✃② q ∈ Spec(R) t❤á❛ ♠➲♥ q ⊆ p✳ ❑❤✐ ➤ã qRp ∈ AttRp (HRi p (Mp )) ⇔ q ∈ AttR (Hmi+t (M )) ❈❤ø♥❣ ♠✐♥❤✳ i ❚r➢í❝ ❤Õt ❝❤ó ý r➺♥❣ ❝➳❝ ♠➠➤✉♥ HR (Mp ) ✈➭ Hmi+t (M ) ❧➭ p ♠➠➤✉♥ ❆rt✐♥ t❤❡♦ ➜Þ♥❤ ❧ý ✶✳✷✳✶✸ ♥➟♥ tå♥ t➵✐ ❝➳❝ t❐♣ ✈➭ AttRp (HRi p (Mp )) AttR (Hmi+t (M ))✳ ●✐➯ sö (R , m ) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥ ❝❤✐Ị✉ d ✈➭ f : R −→ R ❧➭ ♠ét t♦➭♥ ❝✃✉ ✈➭♥❤✳ ➜➷t E = E(R/m)✳ ➜➷t p = f −1 (p) ❑❤✐ ➤ã Rp ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥ ✈➭ dim R /p = t ❱× R ❧➭ ♠ét ✈➭♥❤ ●♦r❡♥st❡✐♥ ♥➟♥ t❛ ❝ã dim Rp = dim R − dim R /p = d − t ●ä✐ f : Rp −→ Rp ❧➭ t♦➭♥ ❝✃✉ ✈➭♥❤ ①➳❝ ➤Þ♥❤ ❜ë✐ f (r /s ) = f (r )/f (s ), ∀r ∈ R , ∀s ∈ R \ p ❚❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✷✳✺✱ ỗ j Z t ó Rp ✲♠➠➤✉♥ j ExtjR (Mp , Rp ) ∼ = (ExtR (M, R ))p p ✹✷ ❚❤❡♦ ➜Þ♥❤ ❧ý ➤è✐ ♥❣➱✉ ➤Þ❛ ♣❤➢➡♥❣ ✭➜Þ♥❤ ❧ý ✶✳✷✳✶✻✮✱ t❛ ❝ã ➤➻♥❣ ❝✃✉ i HpR (Mp ) ∼ = HomRp (ExtnR −t−i (Mp , Rp ), ERp (Rp /pRp )) p p ❚❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✷✱ t❛ ❝ã i qRp ∈ AttRp (HpR (Mp )) ⇔ qRp ∈ AssRp (ExtnR −t−i (Mp , Rp )) p p ⇔ qRp ∈ AssRp ((ExtdR−t−i (M, R ))p ) ⇔ q ∈ AssR (ExtdR−t−i (M, R )) ▲➵✐ t❤❡♦ ➜Þ♥❤ ❧ý ➜è✐ ♥❣➱✉ ➤Þ❛ ♣❤➢➡♥❣ t❛ ❝ã Hmi+t (M ) ∼ = HomR (ExtdR−t−i (M, R ), E) ◆➟♥ t❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✷ t❛ ❝ã ➤✐Ị✉ tr➟♥ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ q ✸✳✷✳✷ ❍Ư q✉➯✳ ●✐➯ sư (R, m) ❧➭ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥✳ ●ä✐ s✐♥❤ ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❧✃② Hmj (M ) = ✈➭ ∈ AttR (Hmi+t (M )) p ∈ Ass(M ) p ∈ Att(Hmj (M )) ●✐➯ sö M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ dim R/p = j ✳ ❍➡♥ ♥÷❛ ♥Õ✉ j > t❤× ❑❤✐ ➤ã Hmj (M ) ❦❤➠♥❣ ữ s ứ ì pRp Ass(Mp ) ♥➟♥ HpR (Mp ) ❧➭ Rp ✲♠➠➤✉♥ ❝ã ➤é ❞➭✐ p ❤÷✉ ❤➵♥✳ ❚❤❡♦ ❇ỉ ➤Ị ✷✳✸✳✼✱ t❛ ❝ã AttRp (HpR (Mp ) = {pRp } p ❚❤❡♦ ❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ tỉ♥❣ q✉➳t t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✹✸ ✸✳✷✳✸ ❍Ư q✉➯✳ ●✐➯ sư (R, m) ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ♠➭ ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥✱ ❑❤✐ ➤ã M ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝❤✐Ị✉ d✳ Hmd (M ) = ✈➭ {p ∈ Ass M : dim R/p = d} ⊆ Att(Hmd (M )) ❚❤❡♦ ❍Ö q✉➯ ✸✳✷✳✷✳ ❈❤ø♥❣ ♠✐♥❤✳ ✸✳✷✳✹ ➜Þ♥❤ ❧ý ✭❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ②Õ✉✮✳ ❧➭ ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣✱ ✈➭ ➤➷t M dim R/p = t ❧➭ ▲✃② R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ i ∈ Z✱ q ∈ Spec(R) ▲✃② ●✐➯ sö (R, m) p ∈ Spec(R) t❤á❛ ♠➲♥ q ⊆ p ✈➭ i qRp ∈ AttRp (HpR (Mp )) ❑❤✐ ➤ã q ∈ AttR (Hmi+t (M )) p ❈❤ø♥❣ ♠✐♥❤✳ ●ä✐ R ❧➭ ➤➬② ➤ñ ❤ã❛ ❝ñ❛ R t❤❡♦ t➠♣➠ m✲❛❞✐❝✱ f ❧➭ ➤å♥❣ ❝✃✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❚❛ ❝ã : R −→ R f ị ì R/pR = R/p ♥➟♥ dim R/pR = t ▲✃② p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ❝đ❛ pR t❤á❛ ♠➲♥ dim R/p = t ❑❤✐ ➤ã p ∩ R = p ✈➭ f ❝➯♠ s✐♥❤ ➤å♥❣ ❝✃✉ f : Rp −→ Rp , t❤á❛ ♠➲♥ f (a/s) = f (a)/f (s), a ∈ R, s ∈ R \ p ❑❤✐ ➤ã f ❧➭ ➤å♥❣ ❝✃✉ ♣❤➻♥❣ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ f (pRp ) ❧➭ pRp ✲♥❣✉②➟♥ s➡✳ ❉♦ ➤ã t❤❡♦ ➜Þ♥❤ ❧ý ❈❤✉②Ĩ♥ s➡ së ♣❤➻♥❣ ✭➜Þ♥❤ ❧ý ✶✳✷✳✶✶✮ i Hpi R (Mp ⊗Rp Rp ) ∼ (Mp ) ⊗Rp Rp = = HpR p p ❱× ✈❐② t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✹✳✹ tå♥ t➵✐ q ⊆ p, qRp ∈ AttRp (Hpi R (Mp ⊗Rp p Rp )), t❤á❛ ♠➲♥ f −1 (qRp ) = qRp ✹✹ ❚❛ ❧➵✐ ❝ã Mp ⊗Rp Rp ∼ = (M ⊗R R)p ♥❤➢ Rp ✲♠➠➤✉♥✳ ❉♦ ➤ã Hpi R ((M ⊗R R)p ) = ✈➭ qRp ∈ AttRp (Hpi R ((M p ❝ñ❛ ❝ã p ⊗R R)p ))✳ ●ä✐ m ❧➭ ✐➤➟❛♥ tè✐ ➤➵✐ R✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ❈✃✉ tró❝ ❝đ❛ ❈♦❤❡♥ ✭①❡♠ ❬✷✱ ❚❤❡♦r❡♠ ❆✳✷✷❪✮ t❛ R ❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ ♠ét ✈➭♥❤ rst ì dim R/p = t t ị ý ✸✳✷✳✶✱ t❛ ❝ã R✲♠➠➤✉♥ Hmi+t (M ⊗R R) ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❝ã ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❧➭ ❱× f ❧➭ ➤å♥❣ ❝✃✉ ➤Þ❛ ♣❤➢➡♥❣ ♣❤➻♥❣ ✈➭ mR = m ♥➟♥ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✹✳✹ t❛ ❝ã ❦Õt ❧➭ q Hmi+t (M ) ❧➭ R✲♠➠➤✉♥ ❦❤➳❝ ❦❤➠♥❣ ❝ã ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ q ∩ R ❱× f −1 (qRp ) = qRp ♥➟♥ t❛ ❝ã q ∩ R = q ✸✳✷✳✺ ❍Ư q✉➯✳ ●✐➯ sư (R, m) ❧➭ ♠ét ✈➭♥❤ ị ữ s j✳ ❑❤✐ ➤ã Hmj (M ) = ✈➭ ●ä✐ M ❧➭ R✲ p ∈ Ass(M ) ●✐➯ sö dim R/p = p ∈ Att(Hmj (M )) ❍➡♥ ♥÷❛ ♥Õ✉ j >0 tì Hmj (M ) ữ s ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã pRp ∈ AssRp (Mp )✳ ❉♦ ➤ã Mp = ✈➭ depth(Mp ) = ❱× ✈❐② HpR (Mp ) ∼ = ΓpRp (Mp ) = p ▼➠➤✉♥ ΓpRp (Mp ) ❧➭ ♠➠➤✉♥ ❝♦♥ ❝đ❛ ♠➠➤✉♥ Mp ♥➟♥ ❧➭ Rp ✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳ ❉♦ ➤ã ✷✳✸✳✼ t❛ ❝ã ΓpRp (Mp ) ❧✐♥❤ ❤ã❛ tư ❜ë✐ ❧ị② t❤õ❛ ❝đ❛ pRp ✳ ❚❤❡♦ ❇ỉ ➤Ị AttRp (HpR (Mp )) = {pRp }✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ✸✳✷✳✹ t❛ ❝ã ➤✐Ò✉ p ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛ ❦❤➠♥❣ ➤ó♥❣ tr➟♥ ✈➭♥❤ ✭❘✱ ♠✮ ❜✃t ❦ú✳ ✸✳✷✳✻ ❱Ý ❞ô✳ ❳Ðt (R, m) ề ị tr ề ợ ①➞② ❞ù♥❣ ❜ë✐ ▼✳ ❋❡rr❛♥❞ ✈➭ ❉✳ ❘❛②♥❛✉❞ ❬✼❪ t❤á❛ ♠➲♥ R ❝ã ✐➤➟❛♥ ✹✺ ♥❣✉②➟♥ tè ♥❤ó♥❣ q✉➯ ✸✳✷✳✺✱ p ❝❤✐Ò✉ ❜➺♥❣ 1✳ ❘â r➭♥❣ p ∩ R = 0✳ ▲➵✐ ❝ã t❤❡♦ ❍Ö p ∈ AttR Hm1 (R)✳ ❱× ✈❐② t❤❡♦ ❇ỉ ➤Ị ✷✳✹✳✹✱ t❛ ❝ã = p ∩ R ∈ AttR Hm1 (R) ▲✃② p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ã ➤é ❝❛♦ ❜➺♥❣ ❝ñ❛ R✱ t❛ ❝ã 0Rp ∈ / AttRp HpR (Rp ) p ❱× ♥Õ✉ tr➳✐ ❧➵✐ t❛ ❝ã = ht(p) = dim Rp /0Rp ≤ dim Rp / AnnRp HpR (Rp ) ≤ 0, p ✈➠ ❧ý✳ ❱❐② Hm1 (R) ❦❤➠♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ❤ã❛✳ ✹✻ ❈❤➢➡♥❣ ✹ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ ➤➵t ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ ✲ ▼ơ❝ ✷✳✶ ❧✉❐♥ ✈➝♥ ❣✐í✐ t❤✐Ư✉ ✈Ị ♠➠➤✉♥ t❤ø ❝✃♣ ✈➭ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✲ ▼ơ❝ ✷✳✷ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ✈Ị sù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✲ ▼ô❝ ✷✳✸ ❧✉❐♥ ✈➝♥ ❝❤♦ t❤✃② ✈➭✐ tÝ♥❤ ❝❤✃t ❝ñ❛ t❐♣ ❝➳❝ ✐➤❡❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ✲ ▼ơ❝ ✷✳✹ ❧✉❐♥ ✈➝♥ tr×♥❤ ❜➭② ✈Ị t❐♣ ✐➤❡❡❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt q✉❛ ➤å♥❣ ❝✃✉ ♣❤➻♥❣ ✈➭ q✉❛ ➤è✐ ♥❣➱✉ ▼❛t❧✐s ✲ ▼ô❝ ✸✳✶ ❧✉❐♥ ✈➝♥ ❝❤♦ t❤✃② ❦Õt q✉➯ ❝ñ❛ t❐♣ ✐➤➟ ❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt q✉❛ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t ✲ trì ề tí t ị ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣ ✹✼ ❚➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣✱ ❵❵▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✱ ✶✾✾✽✳ ❬✷❪ ❲✳ ❇r✉♥s ❛♥❞ ❏✳ ❍❡r③♦❣ ✭✶✾✾✽✮✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✱ ❈❛♠✲ r✐♥❣s ❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss ✭❘❡✈✐s❡❞ ❡❞✐t✐♦♥✮✳ ❬✸❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞✱ ❙❡❝♦♥❞❛r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❛ ❝♦♠✲ ✱ ❙②♠♣♦s✐❛ ▼❛t❤❡♠❛t✐❝❛✱ ✶✶ ✭✶✾✼✸✮✱ ✷✸✲✹✸✳ ♠✉t❛t✐✈❡ r✐♥❣ ❬✹❪ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✼✷✮✱ ✧❆♥ ❡❧❡♠❡♥t❛r② ♣r♦♦❢ ♦❢ t❤❡ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♦❢ ❝❡rt❛✐♥ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ❏✳ ▼❛t❤✳ ❖①❢♦r❞✱ ◗✉❛rt✳ ✭✷✮ ✷✸✱ ♣♣✳ ✶✾✼✲✷✵✹✳ ❬✺❪ ❍✳ ▼❛ts✉♠✉r❛ ✭✶✾✽✻✮✱ ✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✲ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r② ✈❡rs✐t② Pr❡ss✳ ❬✻❪ ❘✳ ❨✳ ❙❤❛r♣ ✭✶✾✼✺✮✱ ✧❙♦♠❡ r❡s✉❧ts ♦♥ t❤❡ ✈❛♥✐s❤✐♥❣ ♦❢ ❧♦❝❛❧ ❝♦❤♦✲ ♠♦❧♦❣② ♠♦❞✉❧❡s✧✱ ✱ ✸✵✱ ♣♣✳ ✶✼✼✲✶✾✺✳ Pr♦❝✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝ ❬✼❪ ❉✳ ❋❡rr❛♥❞ ❛♥❞ ▼✳ ❘❛②♥❛✉❞ ✭✶✾✼✵✮✱ ✧❋✐❜r❡s ❢♦r♠❡❧❧❡s ❞✬✉♥ ❛♥♥❡❛✉ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥✧✱ ✱ ✭✹✮✸✱ ♣♣✳ ✷✾✺✲✸✶✶✳ ❆♥♥✳ ❙❝✐✳ ❊✬❝♦❧❡ ◆♦r♠✳ ❙✉♣✳ ✹✽ ... ❜✐Ĩ✉ s❛✉ ❧➭ ➤ó♥❣✳ R ❧➭ ✈➭♥❤ ❆rt✐♥ t❤× ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ R ➤Ị✉ tè✐ ➤➵✐✳ R ❧➭ rt tì R ó ữ tố ❇➞② ❣✐ê t❛ sÏ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❝❤✐Ị✉ ❝đ❛ ✈➭♥❤ ✈➭ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥✳ ✹ ✶✳✶✳✺ ➜Þ♥❤ ♥❣❤Ü❛✳ p0 R... ❝➳❝ ♠➠➤✉♥ ❝♦♥ A = A1 + + An ✱ Ai ❧➭ pi − t❤ø ❝✃♣✱ ∀i = 1, n ❇✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ♥➭② ➤➢ỵ❝ ❣ä✐ tố tể ế ỗ Ai từ ❝➳❝ pi ❧➭ ➤➠✐ ♠ét ❦❤➳❝ ♥❤❛✉✳ ✷✳✶✳✻ ◆❤❐♥ ①Ðt✳ ●✐➯ sư A ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ ➤➢ỵ❝ ✈➭ A... ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ t❤ø ❝✃♣ ø♥❣ ✈í✐ ❝ï♥❣ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè✱ t❛ ❝ã t❤Ĩ rót ❣ä♥ ỗ ể ễ tứ t ột ể ễ tố t❤✐Ĩ✉✳ ✷✳✷ ❙ù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ❚r➢í❝ ❤Õt ❣✐➯ t❤✐Õt t❤✐Ĩ✉ A ❧➭ ♠➠➤✉♥