Tập iđêan nguyên tố gắn kết và tính chất dịch chuyển địa phương

..

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


▼ơ❝ ❧ơ❝
▼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 ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣
➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ tÝ♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣

✸✽

✸✳✶✳ ❚❐♣ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝ñ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉
➤Þ❛ ♣❤➢➡♥❣ ❝✃♣ ❝❛♦ ♥❤✃t
✸✳✷✳ ❚Ý♥❤ ❝❤✃t ❞Þ❝❤ ❝❤✉②Ĩ♥ ➤Þ❛ ♣❤➢➡♥❣
❚➭✐ ❧✐Ö✉ 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 ❝❤♦ ột
ợ ở rộ ủ ị ý ❜➯♥ ❝❤♦ sè ❤ä❝✿ ♠ét sè
tù ♥❤✐➟♥ ❧í♥ ❤➡♥

1 ➤➢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=

❝đ❛

r
i=1 Qi , tr ó ỗ

N ủ R M ❧➭ ♠ét

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
ễ ợ

ể

ế ó tổ ủ ữ tứ ❝✃♣

M = S1 + S2 + · · · + Sn ,
tr♦♥❣ ➤ã ❝➳❝
✈➭ ❝➳❝

Si ❧➭ ♠➠➤✉♥ pi t❤ø ❝✃♣ i = 1, ..., n. ◆Õ✉ ❝➳❝ pi ❧➭ ♣❤➞♥ ❜✐Ưt

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 ♠➭ ❦❤➠♥❣ ♣❤ơ 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♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ♠➠➤✉♥ ❆rt✐♥✳

✶


ố ồ ề ị ợ ớ tệ ở rt
ữ ố ồ 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♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥ tr×♥❤
❜➭② ❧➵✐ ❝➳❝ ❦Õt q✉➯ ❝ñ❛ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ✈➭ ❘✳ ❨✳ ❙❤❛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❝ ✈Ò
♣❤➞♥ 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 ò ết s
s ế t
ũ ợ ử ❧ê✐ ❝➯♠ ➡♥ ❝❤➞♥ t❤➭♥❤ ➤Õ♥ ❑❤♦❛ ❚♦➳♥✱ ❑❤♦❛
s❛✉ ➜➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ P❤➵♠ ❚❤➳✐ ◆❣✉②➟♥ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥
t❤✉❐♥ ❧ỵ✐ ❝❤♦ ❡♠ tr♦♥❣ t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ t trờ ợ
ì ồ ệ ❜❒ 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 ợ ọ 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 ❣✐➯♠ ❝➳❝ ✐➤➟❛♥ ❝đ❛

♥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 ❝➳❝ ➤✐Ị✉ ệ s t


A rt

ỗ t rỗ ủ



A ề ó tử ❝ù❝ t✐Ó✉✳


➜Ĩ ➤Ị ❝❐♣ ➤Õ♥ ♠ét ✈➭✐ tÝ♥❤ ❝❤✃t ❝đ❛ ♠➠➤✉♥ ❆rt✐♥✱ s❛✉ ➤➞② t❛ sÏ ♥❤➽❝
❧➵✐ ❦❤➳✐ ♥✐Ö♠ ➤é ❞➭✐ ❝đ❛ ♠➠➤✉♥✳

✶✳✶✳✸ ➜Þ♥❤ ♥❣❤Ü❛✳

❈❤♦

R ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥ ❦❤➳❝ ❦❤➠♥❣ ✈➭ M ❧➭ ♠ét

R✲♠➠➤✉♥✳
✭✐✮ ▼ét ❞➲②

M0 ⊆ M1 ⊆ · · · ⊆ Mn = M ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ M

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét
✭✐✐✮ ❳Ý❝❤

✳

①Ý❝❤

0 = 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♥❣ ❤❛✐ ♠➠➤✉♥ ❝♦♥ ❧➭ 0 ✈➭ ❝❤Ý♥❤
♥ã✳
✭✐✐✐✮

➜é ❞➭✐

❝đ❛

M ✱ ❦Ý ❤✐Ư✉ ❧➭

❞➭✐ ❝ñ❛ ❝➳❝ ①Ý❝❤ ❝ã ❞➵♥❣

R (M )✱ ❧➭ ❝❐♥ tr➟♥ ➤ó♥❣ ❝đ❛ ❝➳❝ ➤é

0 = M0 ⊂ M1 ⊂ . . . ⊂ Mn = M, tr♦♥❣ ➤ã

Mi = Mi+1 ✈í✐ ♠ä✐ i = 0, 1, . . . , n − 1.
▼ét

R✲♠➠➤✉♥ M ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã

➤é ❞➭✐ ữ

ế

M ó ít t

ột ợ t r trờ ❤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♥❤ ❧➭ ➤é ❞➭✐

M ✳ tế ữ ỗ t

M ề ó ộ ợt q ộ

ủ ợ t
ị ❧ý✳
✭✐✮ ◆Õ✉
✭✐✐✮ ◆Õ✉

❚❛ ❝ã ❝➳❝ ♣❤➳t ❜✐Ĩ✉ s❛✉ ❧➭ ➤ó♥❣✳

R ❧➭ ✈➭♥❤ ❆rt✐♥ t❤× ♠ä✐ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝đ❛ R ề tố
R rt tì R ó ữ ❤➵♥ ✐➤➟❛♥ 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 ĩ ❝❐♥ tr➟♥

❝ñ❛ ➤é ❝❛♦ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè tr♦♥❣

R

dim(R) = Sup{ht p| p ∈ Spec(R)}.
❈❤✐Ị✉ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭

❝❤✐Ị✉ ❑r✉❧❧ ủ

R ế dim R ữ tì ó

ộ ❞➭✐ ❝ñ❛ ❞➲② ✐➤➟❛♥ ♥❣✉②➟♥ tè ❞➭✐ ♥❤✃t tr♦♥❣
✭✐✐✐✮ ❈❤♦

M = 0 ❧➭ ♠ét R✲♠➠➤✉♥✳ ❑❤✐ ➤ã

R✳

❝❤✐Ị✉ ❑r✉❧❧ ❝đ❛ ♠➠➤✉♥

M ✱ ❦ý ❤✐Ö✉ ❧➭ dim M ✱ ❧➭ dim R/ Ann M ✳ ◆Õ✉ M ❧➭ ♠➠➤✉♥ ❦❤➠♥❣ t❤×
t❛ q✉② ➢í❝

dim M = −1✳

✶✳✶✳✻ ▼Ư♥❤ ➤Ị✳
✈➭

R = 0 ❧➭ ✈➭♥❤ ❆rt✐♥ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ R ❧➭ ✈➭♥❤ ◆♦❡t❤❡r

dim R = 0.

✶✳✶✳✼ ❇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 = 0.

✺

❑❤✐ ➤ã


✶✳✷

▼➠➤✉♥ ❊①t ✈➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣

✶✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳

▼ét

❣✐➯✐ ①➵ ➯♥❤

❝đ❛

M ❧➭ ♠ét ❞➲② ❦❤í♣

. . . −→ P2 −→ P1 −→ P0 −→ M −→ 0
tr♦♥❣ ➤ã ỗ

Pi

ị ĩ



M, N R−♠➠➤✉♥ ✈➭ n ≥ 0 ❧➭ ♠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

2
1
ExtnR (M, N ). ❈ơ t❤Ĩ✱ ♥Õ✉ . . . −→ P2 −→
P1 −→
P0 −→ M −→ 0 ❧➭

♠ét ❣✐➯✐ ①➵ ➯♥❤ ❝ñ❛

M, t➳❝ ➤é♥❣ ❤➭♠ tư Hom(−, N ) ✈➭♦ ❞➲② ❦❤í♣ tr➟♥

t❛ ❝ã ♣❤ø❝
u∗

u∗

1
2
0 −→ Hom(P0 , N ) −→
Hom(P1 , N ) −→
Hom(P2 , N ) −→ . . .

❑❤✐ ➤ã

ExtnR (M, N ) = Ker u∗n+1 / Im u∗n ❧➭ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ t❤ø n

❝đ❛ ♣❤ø❝ tr➟♥ ✭♠➠➤✉♥ ♥➭② ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ✈✐Ư❝ ❝❤ä♥ ❣✐➯✐ ①➵ ➯♥❤
❝đ❛

M ✮✳

✶✳✷✳✸ ệ ề
s ớ ọ

ế

M, N

ữ s tì

ExtnR (M, N ) ❧➭ ❤÷✉ ❤➵♥

n.

❑Õt q✉➯ ❞➢í✐ ➤➞② ❝❤♦ t❛ tí t ữ

Ext tử ị


▼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✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ợ ớ tệ ở rt
ữ ♥❛② ➜è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ➤➲ 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 ❧➭ ✐➤➟❛♥ ❝đ❛ R. ớ 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 (−) ❧➭ ❤➭♠ tö ❦❤í♣ tr➳✐ tõ ♣❤➵♠ trï ❝➳❝ R−♠➠➤✉♥ ➤Õ♥
♣❤➵♠ trï ❝➳❝

R−♠➠➤✉♥✳ ❍➭♠ tư ΓI (−) ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ tư I−①♦➽♥✳
✼


▼ét

❣✐➯✐ ♥é✐ ①➵

❝đ❛

N ❧➭ ♠ét ❞➲② ❦❤í♣

0 −→ N −→ E0 −→ E1 −→ E2 −→ . . .
tr♦♥❣ ➤ã ỗ

Ei ộ ú ý ỗ ề ❝ã ❣✐➯✐ ♥é✐ ①➵✳

✶✳✷✳✼ ➜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

1
0
E1 −→
E2 . . .
0 −→ N −→ E0 −→

❧➭ ❣✐➯✐ ♥é✐ ①➵ ❝ñ❛

N, t➳❝ ➤é♥❣ ❤➭♠ tö ΓI (−) t❛ ❝ã ♣❤ø❝
u∗

u∗

0
1
0 −→ 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 ❝➡ ❜➯♥ ❝đ❛ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳
✶✳✷✳✽ ▼Ư♥❤ ➤Ị✳
✭✐✮

N

❧➭ ♠ét

R−♠➠➤✉♥✳

HI0 (N ) ∼
= ΓI (N ).

✭✐✐✮ ❱í✐

N = N/ΓI (N ) t❛ ❝ã HIn (N )
= HIn (N ) ớ n 1.

ế
ỗ



0 N −→ N −→ N −→ 0

❧➭ ❞➲② ❦❤í♣ ♥❣➽♥ t❤× ✈í✐

n ❝ã ➤å♥❣ ❝✃✉ ♥è✐ HIn (N ) −→ HIn+1 (N ) s❛♦ ❝❤♦ t❛ ❝ã ❞➲② ❦❤í♣

❞➭✐

0 −→ 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 )✳
✽


ệ ề

sử

(R, m)

0=M

ị tr



R ữ ❤❛♥ s✐♥❤ ❝❤✐Ò✉ d✳ ❑❤✐ ➤ã t❐♣
Σ = {N : N

❧➭ ♠➠ ➤✉♥ ❝♦♥ ❝đ❛

M

✈➭

dim N < d}

❝ã ♣❤➬♥ tư ❧í♥ ♥❤✃t t❤❡♦ q✉❛♥ ❤Ư ❜❛♦ ❤➭♠✱ ❣✐➯ sư ❧➭

N ✳ ➜➷t G = M/N.

t❛ ❝ã
✭✐✮

dim G = d;

✭✐✐✮

G ❦❤➠♥❣ ❝ã ♠➠➤✉♥ ❝♦♥ ❝❤✐Ò✉ ♥❤á ❤➡♥ d❀

Ass G = {p ∈ Ass(M ) : dim R/p = d}❀
d
∼ d
✭✐✈✮ H (G) = H (M ).
✭✐✐✐✮

m

m

✶✳✷✳✶✵ ➜Þ♥❤ ❧ý

❧➭

R ✲♠➠➤✉♥✳

✭❚Ý♥❤ ➤é❝ ❧❐♣ ✈í✐ ✈➭♥❤ ❝➡ së✮✳

R✲➤➵✐ sè ✈➭ M
R✲♠➠➤✉♥H i (M ) ∼
=

❈❤♦

❑❤✐ ➤ã t❛ ❝ã ❝➳❝ ➤➻♥❣ ❝✃✉ ♥❤÷♥❣

R

❧➭

IR

HIi (M ) ✈í✐ ♠ä✐ i ≥ 0✳
❑❤✐

R ❧➭ R✲➤➵✐ sè ♣❤➻♥❣ t❛ ò ó ị ý s ị ý


ị ❧ý
♣❤➻♥❣ ✈➭

M

❧➭

✭➜Þ♥❤ ❧ý ❝❤✉②Ĩ♥ ❝➡ së ♣❤➻♥❣✮✳

R✲♠➠➤✉♥✳

❑❤✐ ➤ã t❛ ❝ã

R ✲➤➻♥❣

❈❤♦
❝✃✉

R

❧➭

R✲➤➵✐

sè

HIi (M ) ⊗R R ∼
=

i
HIR
(M ⊗R R ) ✈í✐ ♠ä✐ i ≥ 0✳

❈❤Ø ❝ã ❤÷✉ ❤➵♥ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ❦❤➳❝ ❦❤➠♥❣✳ ❍➡♥ ♥÷❛✱ ❝❤Ø

sè ❜Ð ♥❤✃t ✈➭ ❧í♥ ♥❤✃t ❦❤➳❝ ❦❤➠♥❣ ❝ß♥ ❝❤♦ t❛ ❝➳❝ ➤➷❝ tr➢♥❣ ✈Ị ➤é s➞✉
✈➭ ❝❤✐Ị✉ ❝đ❛ ♠➠➤✉♥✳ ❚r➢í❝ ❤Õt t❛ ♥❤➽❝ ❧➵✐ ❝➳❝ ❦Õt q✉➯ ♥ỉ✐ t✐Õ♥❣ s❛✉ ➤➞②
❝đ❛ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✈Ị tÝ♥❤ tr✐Ưt t✐➟✉ ✈➭ ❦❤➠♥❣ tr✐Ưt t✐➟✉ ❝đ❛ ♠➠➤✉♥ ➤è✐
➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳

✾


✶✳✷✳✶✷ ➜Þ♥❤ ❧ý✳

❬✶✱ ➜Þ♥❤ ❧ý ✻✳✶✳✷✱ ➜Þ♥❤ ❧ý ✻✳✶✳✹❪ ✭✐✮ ❈❤♦ M

◆♦❡t❤❡r✳ ❑❤✐ ➤ã✱

✭✐✐✮

●✐➯ sư

M

HIi (M ) = 0,
❧➭

✈í✐ ♠ä✐

❧➭

R✲♠➠➤✉♥

i > dim M.

R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✱ ❦❤➳❝ ❦❤➠♥❣ ✈➭ ❝❤✐Ị✉ ❑r✉❧❧

dim M = d. ❑❤✐ ➤ã Hmd (M ) = 0.
▼ét ♣❤➬♥ tư

0 = a ∈ R ➤➢ỵ❝ ❣ä✐ ❧➭

♣❤➬♥ tư

M −❝❤Ý♥❤

q✉②

♥Õ✉

am = 0 ❦Ð♦ t❤❡♦ m = 0 ✈í✐ ♠ä✐ m ∈ M. ▼ét ❞➲② ❝➳❝ ♣❤➬♥ tö
a1 , . . . , an ❝đ❛ R ➤➢ỵ❝ ❣ä✐ ❧➭ M −❞➲② ❝❤Ý♥❤ q✉② ♥❣❤❒♦ ♥Õ✉ ai ❧➭ ♣❤➬♥ tö
❝❤Ý♥❤ q✉② ❝đ❛

M/(a1 , . . . , ai−1 )M ✈í✐ ♠ä✐ i = 1, . . . , n. ▼ét ❞➲② ❝➳❝

♣❤➬♥ tö a1 , . . . , an
♠ét

∈ R ➤➢ỵ❝ ❣ä✐ ❧➭ M −❞➲② ❝❤Ý♥❤ q✉② ♥Õ✉ a1 , . . . , an ❧➭

M −❞➲② ❝❤Ý♥❤ q✉② ♥❣❤❒♦ ✈➭ M/(a1 , . . . , an )M = 0 I ột

ủ

R ó ỗ ❞➲② ❝❤Ý♥❤ q✉② ❝đ❛ M tr♦♥❣ I ❝ã t❤Ĩ ♠ë ré♥❣

t❤➭♥❤ ♠ét ❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐✱ ✈➭ ❝➳❝ ❞➲② ❝❤Ý♥❤ q✉② tè✐ ➤➵✐ ❝ñ❛

M

tr♦♥❣

I ❝ã ❝❤✉♥❣ ➤é ❞➭✐✳ ➜é ❞➭✐ ❝❤✉♥❣ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭

M

tr♦♥❣

I ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❧➭ depth(I, M ). ➜é s➞✉ ❝ñ❛ M tr♦♥❣ ✐➤➟❛♥ ❝ù❝

➤➵✐ ❞✉② ♥❤✃t
r➺♥❣

➤é s➞✉

❝đ❛

m ➤➢ỵ❝ ❣ä✐ ❧➭ ➤é s➞✉ ❝đ❛ M ✈➭ ❦Ý ❤✐Ư✉ ❧➭ depth M. ❈❤ó ý

depth M

dim M. ❚õ ➤ã t❛ ❝ã dim M = Sup{i : Hmi (M ) = 0},

✈➭ ❜➺♥❣ q✉② ♥➵♣ t❤❡♦ ➤é ❞➭✐ ❞➲② ❝❤Ý♥❤ q✉② t❛ ❝ã

depth M = inf{i :

Hmi (M ) = 0} ✭①❡♠ ❬✶✱ ❍Ö q✉➯ ✻✳✷✳✽❪✮✳
▼➷❝ ❞ï

M ữ s ì ố ồ ề

ị HIi (M ) ữ s✐♥❤ ❝ị♥❣ ❦❤➠♥❣ ❧➭ ♠➠➤✉♥
❆rt✐♥✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt✱ ố ồ ề ị ớ
ự tì ❝➳❝ ♠➠➤✉♥ ➤ã ❧➭ ❆rt✐♥✳
✶✳✷✳✶✸ ➜Þ♥❤ ❧ý✳
❑❤✐ ➤ã

❬✶✱ ➜Þ♥❤ ❧ý ✼✳✶✳✸❪

●✐➯ sư

M

❧➭

R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤✳

R✲♠➠➤✉♥ Hmi (M ) ❧➭ ❆rt✐♥ ✈í✐ ♠ä✐ sè tù ♥❤✐➟♥ i✳
✶✵


P❤➬♥ ❝ß♥ ❧➵✐ ❝đ❛ ♠ơ❝ ♥➭② ❞➭♥❤ ➤Ĩ ♥❤➽❝ ❧➵✐ ♠ét sè ❦✐Õ♥ t❤ø❝ ✈Ò ➤è✐

♥❣➱✉ ▼❛t❧✐s ✈➭ ➤è✐ ♥❣➱✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❑ý ❤✐Ư✉

E(k) ❧➭ ❜❛♦ ♥é✐ ①➵ ❝đ❛

R✲♠➠➤✉♥ k ✈í✐ k = R/m✳ ❚❛ ❦Ý ❤✐Ư✉ DR (−) t❤❛② Hom(, E(k))
ớ ỗ

R M t ọ DR (M ) ❧➭

❬✶✱ ❈❤ó ý ✶✵✳✷✳✷❪

➤è✐ ♥❣➱✉ ▼❛t❧✐s

❝đ❛

M ✳ ❚❤❡♦

AnnR (M ) = AnnR (DR (M )). ❑Õt q✉➯ s❛✉ ➤➞② ❝ã t❤Ĩ

①❡♠ tr♦♥❣ ❬✶✱ ➜Þ♥❤ ❧ý ✶✵✳✷✳✶✷❪✳
✶✳✷✳✶✹ ➜Þ♥❤ ❧ý

✭➜Þ♥❤ ❧ý ➤è✐ ♥❣➱✉ ▼❛t❧✐s✮✳

➤Þ❛ ♣❤➢➡♥❣ ◆♦❡t❤❡r ➤➬② ➤đ ✈➭

❈❤♦

(R, m) ❧➭ ✈➭♥❤ ❣✐❛♦ ❤♦➳♥

M ✱ A ❧➭ ❝➳❝ R✲♠➠➤✉♥✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤

➤Ị s❛✉ ❧➭ ➤ó♥❣✳

✭✐✮

◆Õ✉

M

❧➭

♠➠➤✉♥

◆♦❡t❤❡r

t❤×

DR (M )

❧➭

♠➠➤✉♥

❆rt✐♥

✈➭

M∼
= DR (DR (M ))✳

✭✐✐✮

◆Õ✉

A

❧➭ ♠➠➤✉♥ ❆rt✐♥ t❤×

DR (A)

❧➭ ♠➠➤✉♥ ◆♦❡t❤❡r ✈➭

A ∼
=

DR (DR (A))✳
❑❤✐ R ❧➭ ✈➭♥❤ ➤➬② ➤đ✱ ➜Þ♥❤ ❧ý ➤è✐ ♥❣➱✉ ▼❛t❧✐s ❝❤♦ t❛ t➢➡♥❣ ø♥❣ ❣✐÷❛
♣❤➵♠ trï ❝➳❝

R✲♠➠➤✉♥ ❆rt✐♥ ✈➭ ♣❤➵♠ trï ❝➳❝ R✲♠➠➤✉♥ ◆♦❡t❤❡r✳ ➜Þ♥❤

❧ý ➜è✐ ♥❣➱✉ ➤Þ❛ ♣❤➢➡♥❣ ❬✶✱ ị ý t ố ệ ữ ố
ồ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✈➭ ❤➭♠ tư

Ext✳ ❚r➢í❝ ❤Õt t❛ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠

✈➭♥❤ ●♦r❡♥st❡✐♥ t❤❡♦ ❬✷❪✳
✶✳✷✳✶✺ ➜Þ♥❤ ♥❣❤Ü❛✳

❤✐Ư✉

✭✐✮ ●✐➯ sư

M ❧➭ R✲♠➠➤✉♥✳ ❈❤✐Ị✉ ♥é✐ ①➵ ❝đ❛ M ✱ ❦ý

inj dim M ❤♦➷❝ inj dimR M ❧➭ sè ♥❣✉②➟♥ ♥❤á ♥❤✃t n s❛♦ ❝❤♦ tå♥

t➵✐ ❣✐➯✐ ♥é✐ ①➵
t➵✐ sè

E• ❝đ❛ M ♠➭ E m = 0 ✈í✐ ♠ä✐ m > n✳ ◆Õ✉ ❦❤➠♥❣ tå♥

n ♥❤➢ ✈❐② t❛ ➤Þ♥❤ ♥❣❤Ü❛ ❝❤✐Ị✉ ♥é✐ ①➵ ❝đ❛ M ❧➭ ✈➠ ❝ï♥❣✳

✭✐✐✮ ▼ét ✈➭♥❤ ◆♦❡t❤❡r✱ ➤Þ❛ ♣❤➢➡♥❣

R ➤➢ỵ❝ ❣ä✐ ❧➭

inj dim R < ∞. ▼ét ✈➭♥❤ tr ợ ọ


rst

rst

ế

ế ị

♣❤➢➡♥❣ ❤ã❛ ❝ñ❛ ♥ã t➵✐ ♠ä✐ ✐➤➟❛♥ tè✐ ➤➵✐ ❧➭ ✈➭♥❤ ●♦r❡♥st❡✐♥✳
✶✳✷✳✶✻ ➜Þ♥❤ ❧ý

✭➜Þ♥❤ ❧ý ➤è✐ ♥❣➱✉ ➤Þ❛ ♣❤➢➡♥❣✮✳

●✐➯ sư

➤å♥❣ ❝✃✉ ❝đ❛ ♠ét ✈➭♥❤ ➤Þ❛ ♣❤➢➡♥❣ ●♦r❡♥st❡✐♥

f : R −→ R
s✐♥❤✳ ❑❤✐ ➤ã

❧➭ t♦➭♥ ❝✃✉ ✈➭♥❤✳

●✐➯ sö

M

(R, m)

(R , m )

❧➭ ♠ét

❧➭ ➯♥❤

❝❤✐Ị✉

R✲♠➠➤✉♥

n

✈➭

❤÷✉ ❤➵♥

ExtjR (M, R ) ❧➭ R✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ✈➭ t❛ ❝ã ➤➻♥❣ ❝✃✉✿
Hmi (M ) ∼
= DR (ExtnR −i (M, R )).

✶✷


❈❤➢➡♥❣ ✷

❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ✈➭ t❐♣ ✐➤➟❛♥
♥❣✉②➟♥ tè ❣➽♥ ❦Õt
✷✳✶

▼➠➤✉♥ t❤ø ❝✃♣ ✈➭ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣

✷✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳

♣❤Ð♣ ♥❤➞♥ ❜ë✐

▼ét

R✲♠➠➤✉♥ A ➤➢ỵ❝ ❣ä✐ ❧➭

♥Õ✉

A = 0 ✈➭

x tr➟♥ A ❧➭ t♦➭♥ ❝✃✉ ❤♦➷❝ ❧ị② ❧✐♥❤ ✈í✐ ♠ä✐ x ∈ R. ❚r♦♥❣

tr➢ê♥❣ ❤ỵ♣ ♥➭② t❐♣ ❝➳❝ ♣❤➬♥ tư

x ∈ R s❛♦ ❝❤♦ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ x tr➟♥ A

❧➭ ❧ò② ❧✐♥❤ ❧➭♠ t❤➭♥❤ ♠ét ✐➤➟❛♥ ♥❣✉②➟♥ tè
✷✳✶✳✷ ❇ỉ ➤Ị✳

t❤ø ❝✃♣

p ✈➭ t❛ ❣ä✐ A ❧➭ p− t❤ø ❝✃♣✳

✭✐✮ ❚æ♥❣ trù❝ t✐Õ♣ ❝đ❛ ❤÷✉ ❤➵♥ ❝➳❝ ♠➠➤✉♥

p✲t❤ø

❝✃♣ ❧➭

p✲t❤ø ❝✃♣✳
✭✐✐✮ ❚❤➢➡♥❣ ❦❤➳❝
✭✐✐✐✮ ◆Õ✉ ❆ ❧➭

❈❤ø♥❣ ♠✐♥❤✳

♥❤➞♥ ❜ë✐

0 ❝ñ❛ ♠ét ♠➠➤✉♥ p t❤ø ❝✃♣ ❧➭ p✲t❤ø ❝✃♣✳

p✲t❤ø ❝✃♣ t❤× Ann A ❧➭ p✲ ♥❣✉②➟♥ s➡✳

✭✐✮ ❈❤♦

A1 , . . . , An ❧➭ p✲t❤ø ❝✃♣✳ ❈❤♦ x ∈ p. ❑❤✐ ➤ã ♣❤Ð♣

x tr➟♥ A1 , . . . , An ❧➭ ❧ò② ❧✐♥❤✳ ❉♦ ➤ã ∃t ∈ N s❛♦ ❝❤♦ xt Ai =

n
i=1 Ai ❧➭ ❧ò②
❧✐♥❤✳ ❈❤♦ x ∈
/ p. ❑❤✐ ➤ã xAi = Ai , ∀i. ❙✉② r❛ x( ni=1 Ai ) = ni=1 Ai ,
n
n
tø❝ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ x tr➟♥
i=1 Ai ❧➭ t♦➭♥ ❝✃✉✳ ❱❐②
i=1 Ai ❧➭ p ✲t❤ø

0, ∀i. ❱× t❤Õ xt (

n
i=1 Ai )

= 0, tø❝ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ x tr➟♥

✶✸

❝✃♣✳
✭✐✐✮ ❈❤♦

A ❧➭ p✲t❤ø ❝✃♣✱ B ❧➭ ♠➠➤✉♥ ❝♦♥ ❝ñ❛ A s❛♦ ❝❤♦ A/B = 0. ❈❤♦

x ∈ p. ❑❤✐ ➤ã ∃t ∈ N s❛♦ ❝❤♦ xt A = 0, s✉② r❛
xt (A/B) = (xt A + B)/B = 0,
tø❝ ♣❤Ð♣ ♥❤➞♥ ❜ë✐
s✉② r❛
❜ë✐

x tr➟♥ A/B ❧➭ ❧ò② ❧✐♥❤✳ ❈❤♦ x ∈
/ p. ❑❤✐ ➤ã xA = A,

x(A/B) = (xA + B)/B = (A + B)/B = A/B, tø❝ ♣❤Ð♣ ♥❤➞♥

x tr➟♥ A/B ❧➭ t♦➭♥ ❝✃✉✳ ❱❐② A/B ❧➭ p✲t❤ø ❝✃♣✳

✭✐✐✐✮ ❚❛ ❝ã

1.A = A, s✉② r❛ 1 ∈
/ Ann A, s✉② r❛ Ann A = R. ❈❤♦

xy ∈ Ann A ♠➭ x ∈
/ Ann A✳ ●✐➯ sö y n A = 0, ∀n✳ ❱× A ❧➭ t❤ø ❝✃♣ ♥➟♥
♣❤Ð♣ ♥❤➞♥ ❜ë✐ y tr➟♥ A ❧➭ t♦➭♥ ❝✃✉✱ tø❝ yA
❱× t❤Õ

= A✳ ❙✉② r❛ 0 = xyA = xA✳

x ∈ Ann A✱ ✈➠ ❧ý✳ ❱❐② y n ∈ Ann A ✈í✐ n ♥➭♦ ➤ã✳ ❱❐② Ann A

❧➭ p✲♥❣✉②➟♥ s➡✳
✷✳✶✳✸ ❇ỉ ➤Ò✳

❈❤♦ ❆ ❧➭

p✲t❤ø

❝✃♣ ✈➭

S

❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ tr♦♥❣

R✳

❑❤✐

➤ã

ϕ

✭✐✮ ➜å♥❣ ❝✃✉ ❝❤Ý♥❤ t➽❝

A −→ S −1 A

❝❤♦ ❜ë✐

m −→ m/1

❧➭ t♦➭♥

❝✃✉✳
✭✐✐✮ ◆Õ✉
✭✐✐✐✮ ◆Õ✉

S ∩ p = ∅ t❤× S −1 A = 0.
S ∩ p = ∅ t❤× S −1 A ❧➭ S −1 p✲t❤ø ❝✃♣✳

❈❤ø♥❣ ♠✐♥❤✳

✭✐✮ ❈❤♦

m/s ∈ S −1 A, m ∈ A, s ∈ S. ◆Õ✉ s ∈ p t❤×

∃n ∈ N s❛♦ ❝❤♦ sn A = 0. ❉♦ ➤ã
m/s = sn m/sn s = 0/sn s = 0/1 = ϕ(0).
◆Õ✉

s∈
/ p t❤× sA = A✳ ❉♦ m ∈ A ♥➟♥ m ∈ sA. ❱× t❤Õ ∃y ∈ A s❛♦ ❝❤♦

m = sy, s✉② r❛ m/s = sy/s = y/1 = ϕ(y). ❱❐② ϕ ❧➭ t♦➭♥ ❝✃✉✳
✶✹


✭✐✐✮ ◆Õ✉

S ∩ p = ∅ t❤× ∃s ∈ S ∩ p. ❱× s ∈ S ♥➟♥ ∃n ∈ N s❛♦ ❝❤♦

sn A = 0. ❉♦ ➤ã ✈í✐ ∀m/t ∈ S −1 A, m ∈ A, t ∈ S t❤×
m/t = sn m/sn t = 0/sn t = 0/1.
❱× t❤Õ

S −1 A = 0✳
S ∩ p = ∅ ♥➟♥ S −1 p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ S −1 R. ❈❤♦ a/s ∈

✭✐✐✐✮ ❱×

S −1 p ✈í✐ a ∈ p, s ∈ S ✳ ❑❤✐ ➤ã ∃n ∈ N s❛♦ ❝❤♦ an A = 0. ❙✉② r❛
(a/s)n (b/t) = an b/sn t = 0/sn t = 0/1
✈í✐ ♠ä✐

b/t ∈ S −1 A. ❱× t❤Õ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ a/s tr➟♥ S −1 A ❧➭ ❧ị② ❧✐♥❤✳

❈❤♦

a/s ∈
/ S −1 p ✈í✐ a ∈ R ✈➭ s ∈ S. ❑❤✐ ➤ã a ∈
/ p. ❉♦ A ❧➭ p✲t❤ø ❝✃♣

♥➟♥

aA = A. ❱× tế ớ ỗ x/t S 1 A, t S, x ∈ A✱ ❧✉➠♥ tå♥ t➵✐

y ∈ A s❛♦ ❝❤♦ x = ay. ❙✉② r❛
x/t = ay/t = (a/s)(sy/t) ∈ a/sS −1 A.
❉♦ ➤ã

a/sS −1 A = S −1 A, tø❝ ❧➭ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ a/s tr➟♥ S −1 A ❧➭ t♦➭♥

❝✃✉✳ ❱❐②

S −1 A ❧➭ S −1 p✲t❤ø ❝✃♣✳

✷✳✶✳✹ ❇ỉ ➤Ị✳
❧➭

❈❤♦

A ❧➭ R✲♠➠➤✉♥✱ p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝ñ❛ R ✈➭ A1 , . . . , Ar

p✲♠➠➤✉♥ ❝♦♥ t❤ø ❝✃♣ ❝ñ❛ A. ❑❤✐ ➤ã B = A1 + . . . + Ar

❝♦♥ t❤ø ❝✃♣ ❝đ❛

❈❤ø♥❣ ♠✐♥❤✳

❧➭

p✲♠➠➤✉♥

A.

❍✐Ĩ♥ ♥❤✐➟♥

B = 0✳ ❳Ðt ➳♥❤ ①➵
r

Ai −→ A1 + . . . + Ar

ϕ:
i=1

❝❤♦ ❜ë✐

ϕ(x1 , . . . , xr ) −→ x1 + . . . + xr . ❉Ô t❤✃② ϕ ❧➭ t♦➭♥ ❝✃✉✳ ❙✉② r❛

B ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❝đ❛

r
i=1 Ai . ❚❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✷ ✭✐✮ ✈➭ t❤❡♦ ❣✐➯ t❤✐Õt✱

✶✺


t❛ ❝ã

r
i=1 Ai ❧➭

p✲t❤ø ❝✃♣✳ ❉♦ ➤ã t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✷ ✭✐✐✮ t❛ s✉② r❛ B ❧➭

p✲t❤ø ❝✃♣✳
▼➠➤✉♥

✷✳✶✳✺ ➜Þ♥❤ ♥❣❤Ü❛✳

A ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ ➤➢ỵ❝ ♥Õ✉ A ❝ã ❜✐Ĩ✉

❞✐Ơ♥ t❤➭♥❤ tæ♥❣ ❝➳❝ ♠➠➤✉♥ ❝♦♥

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 = A1 + . . . + An ❧➭

♠ét ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ ❝đ❛
❧➭

A. ◆Õ✉ tå♥ t➵✐ i = j s❛♦ ❝❤♦ Ai ✈➭ Aj ➤Ị✉

p−t❤ø ❝✃♣ t❤× Ai + Aj ❝ị♥❣ ❧➭ p−t❤ø ❝✃♣✳ ❱× t❤Õ✱ ❜➺♥❣ ❝➳❝❤ ❧♦➵✐ ➤✐

❝➳❝ t❤➭♥❤ ♣❤➬♥ t❤ø ❝✃♣ t❤õ❛ ✈➭ ❣❤Ð♣ ❧➵✐ ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ t❤ø ❝✃♣ ø♥❣
✈í✐ ❝ï♥❣ ♠ét ✐➤➟❛♥ tố t ó tể rút ọ ỗ ể ễ t❤ø ❝✃♣
t❤➭♥❤ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ tè✐ t❤✐Ĩ✉✳

✷✳✷

❙ù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣

❚r➢í❝ ❤Õt ❣✐➯ t❤✐Õt
t❤✐Ĩ✉

A ❧➭ ♠➠➤✉♥ ❜✐Ĩ✉ ❞✐Ơ♥ ➤➢ỵ❝ ✈í✐ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐

A = A1 + . . . + An ✱ tr♦♥❣ ➤ã Ai ❧➭ pi ✲t❤ø ❝✃♣✱ ∀i = 1, n✳
Ann A ❧➭ ❣✐❛♦ ❝đ❛ ♥❤÷♥❣ ✐➤➟❛♥ pi −♥❣✉②➟♥ s➡ ✈➭

✷✳✷✳✶ ❇ỉ ➤Ị✳

Ass(R/ Ann A) ⊆ {p1 , . . . , pn }
❈❤ø♥❣ ♠✐♥❤✳

➜➷t

Ann Ai = qi ✈í✐ i = 1, . . . , n. ❚❤❡♦ ❇æ ➤Ò ✷✳✶✳✷✭✐✐✐✮✱

qi ❧➭ ✐➤➟❛♥ pi −♥❣✉②➟♥ s➡✳ ❚❛ ❝ã
n

Ann A = Ann(

n

Ai ) =

i=1

Ann Ai =
i=1

✶✻

n

qi .
i=1


❱×

qi ❧➭ pi ✲♥❣✉②➟♥ s➡✱ ♥➟♥ ❜➺♥❣ ❝➳❝❤ ❜á ➤✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ t❤õ❛ tr♦♥❣

♣❤➞♥ tÝ❝❤

Ann A =

n
i=1 qi t❛ ➤➢ỵ❝ ♣❤➞♥ tÝ❝❤ ♥❣✉②➟♥ s➡ t❤✉ ❣ä♥ ❝đ❛

Ann A✳ ➜➳♥❤ sè ❧➵✐ ❝➳❝ ❝❤Ø sè✱ t❛ ❝ã t❤Ó ❣✐➯ t❤✐Õt
Ass(R/ Ann A) = {p1 , . . . , pr } ⊆ {p1 , . . . , pn }
✭ë ➤➞②

n ≥ r).

✷✳✷✳✷ ❇ỉ ➤Ị✳

❈❤♦

Q

❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❦❤➳❝ ✵ ❝đ❛

♠ét ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉

Q = Q1 + . . . Qr

A✳

✈í✐

❑❤✐ ➤ã tå♥ t➵✐

Qi

❧➭

qi ✲t❤ø

❝✃♣✱

i = 1, . . . , n ✈➭
{q1 , . . . , qr } ⊆ {p1 , . . . , pn }.
❈❤ø♥❣ ♠✐♥❤✳

❱✐Õt

Q = A/P ✈í✐ P ❧➭ ♠➠➤✉♥ ❝♦♥ ❝đ❛ A✳ ❚❛ ❝ã
n

Q=(

n

Ai )/P =
i=1

❚❤❡♦ ➜Þ♥❤ ❧ý ➤➻♥❣ ❝✃✉ ♠➠➤✉♥

(Ai + P )/P.
i=1

(Ai + P )/P ∼
= Ai /(Ai ∩ P ). ❱× t❤Õ✱ ♥Õ✉

(Ai + P )/P = 0 t❤× t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✷✭✐✐✮✱ (Ai + P )/P ❧➭ pi −t❤ø ❝✃♣✳
❉♦ ➤ã ❜➺♥❣ ❝➳❝❤ ❜á ➤✐ ♥❤÷♥❣ t❤➭♥❤ ♣❤➬♥ t❤ø ❝✃♣ t❤õ❛ tr♦♥❣ tỉ♥❣ tr➟♥
t❛ ➤➢ỵ❝ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉ ❝đ❛

Q. ➜➳♥❤ sè ❧➵✐ t❤ø tù ❝➳❝ ❝❤Ø sè✱

t❛ ❝ã t❤Ó ❣✐➯ t❤✐Õt

Q = A/P = (A1 + P )/P + . . . + (Ar + P )/P
❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉ ❝đ❛

❱í✐ ỗ

Q ớ (Ai + P )/P pi tứ ❝✃♣✳

Q t❛ ❦Ý ❤✐Ö✉

ℵ(A) = {x ∈ R : ∃n ∈ N ➤Ó xn A = 0}
✶✼


ớ ỗ
ủ

I ủ R t í ệ V (I) ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè

R ❝❤ø❛ I.

✷✳✷✳✸ ➜Þ♥❤ ❧ý✳

✭➜Þ♥❤ ❧ý ❞✉② ♥❤✃t t❤ø ♥❤✃t✮✳ ❈❤♦

p

❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè✳

❑❤✐ ➤ã ❝➳❝ ♣❤➳t ❜✐Ó✉ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿
✭✐✮

p ∈ {p1 , . . . , pn }.

✭✐✐✮
✭✐✐✐✮
✭✐✈✮

A ❝ã ♠➠➤✉♥ t❤➢➡♥❣ ❧➭ p✲t❤ø ❝✃♣✳
A ❝ã ♠➠➤✉♥ t❤➢➡♥❣ Q s❛♦ ❝❤♦ ℵ(Q) = p.
A ❝ã ♠➠➤✉♥ t❤➢➡♥❣ Q s❛♦ ❝❤♦ p ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ✭t❤❡♦ q✉❛♥

❤Ư ❜❛♦ ❤➭♠✮ tr♦♥❣ t❐♣

❈❤ø♥❣



V (Ann Q).

ớ ỗ i = 1, . . . , n, t❛ ➤➷t Pi =

Aj ✳ ❱×
j=i

n

Ai ❦❤➠♥❣ t❤õ❛ tr♦♥❣ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ A =

Ai , ♥➟♥ A/Pi = 0. ❍➡♥
i=1

♥÷❛✱

A/Pi = (Ai + Pi )/Pi ∼
= Ai /(Ai ∩ Pi ).
❱× t❤Õ

A/Pi ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❦❤➳❝ ✵ ❝đ❛ Ai . ❚❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✷ ✭✐✐✮✱ t❛

s✉② r❛

A/Pi ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ pi ✲t❤ø ❝✃♣ ❝ñ❛ A✳

✭✐✐✮

⇒ ✭✐✐✐✮✳ ●✐➯ sư Q ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ p✲t❤ø ❝✃♣ ❝đ❛ A✳ ❑❤✐ ➤ã tõ ➤Þ♥❤

♥❣❤Ü❛ ♠➠➤✉♥ t❤ø ❝✃♣ t❛ ❝ã ♥❣❛② tÝ♥❤ ❝❤✃t
✭✐✐✐✮

ℵ(Q) = p.

⇒ ✭✐✈✮✳ ●✐➯ sö Q ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❝ñ❛ A s❛♦ ❝❤♦ ℵ(Q) = p. ❚❛ ❝ã
Rad(Ann Q) = {x ∈ R : ∃n ∈ N ➤Ó xn ∈ Ann Q}
= {x ∈ R : ∃n ∈ N ➤Ó xn Q = 0}
= ℵ(Q) = p

✶✽


▼➷t ❦❤➳❝✱

Rad(Ann Q) ❧➭ ❣✐❛♦ ❝ñ❛ t✃t ❝➯ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝❤ø❛

Ann Q✱ ♥➟♥ p ❧➭ ✐➤➟❛♥ tè✐ t❤✐Ó✉ tr♦♥❣ t❐♣ V (Ann Q).
✭✐✈✮

⇒ ✭✐✮✳ ❈❤♦ Q ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❝đ❛ A s❛♦ ❝❤♦ p ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉

tr♦♥❣ t❐♣

V (Ann Q). ❚❤❡♦ ❇ỉ ➤Ị ✷✳✷✳✷ ✈➭ ❜➺♥❣ ✈✐Ư❝ ➤➳♥❤ ❧➵✐ t❤ø tù ❝➳❝

❝❤Ø sè t❛ ❝ã t❤Ĩ ❣✐➯ t❤✐Õt

Q ❝ã ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉

m

Qi , Qi ❧➭ pi − t❤ø ❝✃♣, i = 1, . . . , m

Q=
i=1

✈í✐ ♠ét sè tù ♥❤✐➟♥

m

n ♥➭♦ ➤ã✳ ➜➷t Ann(Qi ) = qi , ❦❤✐ ➤ã t❤❡♦

❇æ ➤Ị ✷✳✶✳✷ ✭✐✐✐✮✱ qi ❧➭ ✐➤➟❛♥ pi ✲♥❣✉②➟♥ s➡✱
t❤✐Ĩ✉ tr♦♥❣ t❐♣

∀i = 1, m. ❱× p ❧➭ ✐➤➟❛♥ tè✐

V (Ann Q) ♥➟♥ p ∈ Ass(R/ Ann Q). ❱× t❤Õ✱ t❤❡♦ ❇ỉ ➤Ị

✷✳✷✳✶ t❛ ❝ã

p ∈ Ass(R/ Ann Q) ⊆ {p1 , . . . , pm } ⊆ {p1 , . . . , pn }.

✷✳✷✳✹ ➜Þ♥❤ ♥❣❤Ü❛✳

❚❤❡♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ t❐♣

{p1 , . . . , pn } ❝❤Ø ♣❤ơ t❤✉é❝

A ♠➭ ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈➭♦ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ĩ✉ ❝đ❛ A✳ ❱× t❤Õ t❛
❣ä✐ ♥ã ❧➭

t❐♣ ❝➳❝ ✐➤➟❛♥ ❣➽♥ ❦Õt

❝đ❛

A, ✈➭ ❦Ý ❤✐Ư✉ ❧➭ Att A.

❚õ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ t❛ ❝ã ♥❣❛② ❦Õt q✉➯ s❛✉ ➤➞②✳
✷✳✷✳✺ ❍Ư q✉➯✳

●✐➯ sư

A ❝ã ❤❛✐ ❜✐Ĩ✉ ❞✐Ơ♥ t❤ø ❝✃♣ tè✐ t❤✐Ó✉
m

A=

n

Ai =
i=1

❑❤✐ ➤ã
♠ä✐

Aj .
j=1

m = n ✈➭ s❛✉ ❦❤✐ ➤➳♥❤ sè ❧➵✐ ❝➳❝ ❝❤Ø sè t❤× ℵ(Ai ) = ℵ(Ai ) ✈í✐

i = 1, . . . , n.
✶✾


◆❣♦➭✐ r❛ ❝ị♥❣ tõ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ t❛ ❝ã ❤Ư q✉➯ ❤❛② ❞ï♥❣ s❛✉✳
✷✳✷✳✻ ❍Ö q✉➯✳
❝❤♦

p = Ann(Q).

❈❤ø♥❣ ♠✐♥❤✳

❝ã

p ∈ {p1 , ..., pn } ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ A ❝ã ♠➠➤✉♥ t❤➢➡♥❣ Q s❛♦

●✐➯ sö

A ❝ã ♠➠➤✉♥ t❤➢➡♥❣ Q s❛♦ ❝❤♦ p = Ann(Q). ❚❛

p ∈ {p1 , ..., pn } t ị ý ợ sử p = pi ✳ ➜➷t

P = A/Pi ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✷✳✸✳ ❚❛ ❝ã P ❧➭ pi ✲t❤ø
❝✃♣ ì

R tr pi ữ s ❉♦ ➤ã tå♥

t➵✐ sè ♥❣✉②➟♥ ❞➢➡♥❣

n s❛♦ ❝❤♦ pni P =✳ ❉♦ ✈❐② pi P = P. ❑Ð♦ t❤❡♦

Q := P/pi P ❧➭ pi t❤ø ❝✃♣ ❦❤➳❝ ❦❤➠♥❣✱ ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❝ñ❛ A✳ ❚❛ ❝ã
pi ⊆ Ann(Q) ⊆ Rad(Ann(Q)) = pi .
❉♦ ✈❐②

Ann(Q) = pi = p.

✷✳✷✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳

❣ä✐ ❧➭
❣ä✐

◆Õ✉

p ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ tr♦♥❣ t❐♣ Att A t❤× p ➤➢ỵ❝

✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ❝➠ ❧❐♣

p ❧➭

❝đ❛

✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt ♥❤ó♥❣

A. ❚r➢ê♥❣ ❤ỵ♣ ♥❣➢ỵ❝ ❧➵✐✱ t❛

❝đ❛

A. ◆Õ✉ pi ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥

tè ❣➽♥ ❦Õt ❝➠ ❧❐♣ ❝đ❛

A t❤× t❤➭♥❤ ♣❤➬♥ t❤ø ❝✃♣ Ai t➢➡♥❣ ø♥❣ ➤➢ỵ❝ ❣ä✐

❧➭

❝đ❛

t❤➭♥❤ ♣❤➬♥ ❝➠ ❧❐♣

✷✳✷✳✽ ▼Ư♥❤ ➤Ị✳

A.

R/ Ann A

✈➭

A

❝ã ❝ï♥❣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❝➠ ❧❐♣✱

tø❝ ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥ ♥❣✉②➟♥ tè ❣➽♥ ❦Õt tè✐ t❤✐Ĩ✉ ❝đ❛
♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ❝❤ø❛

❈❤ø♥❣ ♠✐♥❤✳

❈❤♦

A ❧➭ t❐♣ ❝➳❝ ✐➤➟❛♥

Ann A.

p ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐ t❤✐Ó✉ ❝❤ø❛ Ann A. ❱× A

❝ị♥❣ ❧➭ ♠➠➤✉♥ t❤➢➡♥❣ ❝đ❛

A ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✸ ✭✐✈✮ ⇒ ✭✐✮ t❛ ❝ã

p ∈ {p1 , . . . , pn } = Att A. ◆Õ✉ p ❦❤➠♥❣ ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ Att A
✷✵


t❤× tå♥ t➵✐

q ⊂ p s❛♦ ❝❤♦ q = p ✈➭ q ∈ Att A. ❚❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✸ ✭✐✮

⇒ ✭✐✈✮✱ tå♥ t➵✐ ♠➠➤✉♥ t❤➢➡♥❣ Q ❝ñ❛ A s❛♦ ❝❤♦ q ❧➭ ✐➤➟❛♥ ♥❣✉②➟♥ tè tè✐
t❤✐Ĩ✉ ❝❤ø❛

Ann Q. ❈❤ó ý r➺♥❣ Ann A ⊆ Ann Q. ❉♦ ➤ã q ∈ V (Ann A).

➜✐Ị✉ ♥➭② ❧➭ ♠➞✉ t❤✉➱♥ ✈í✐ tÝ♥❤ tè✐ t❤✐Ĩ✉ ❝đ❛
t❤✐Ĩ✉ ❝đ❛

Att A.

◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư
✷✳✷✳✸ ✭✐✮

p. ❱❐② p ❧➭ ♣❤➬♥ tư tè✐

p ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ Att A. ❚❤❡♦ ➜Þ♥❤ ❧Ý

⇒ ✭✐✈✮✱ tå♥ t➵✐ ♠➠➤✉♥ t❤➢➡♥❣ Q ❝ñ❛ A s❛♦ ❝❤♦ p ❧➭ ✐➤➟❛♥

♥❣✉②➟♥ tè tè✐ t❤✐Ĩ✉ ❝❤ø❛

Ann Q. ❈❤ó ý r➺♥❣ Ann A ⊆ Ann Q. ❉♦ ➤ã

p ∈ V (Ann A). ◆Õ✉ p ❦❤➠♥❣ ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ V (Ann A) t❤× tå♥
t➵✐

q ∈ V (Ann A) s❛♦ ❝❤♦ q ⊂ p ✈➭ q = p. ▲❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ tr➟♥

t❛ s✉② r❛
t❤Õ✱

q ∈ Att A. ➜✐Ò✉ ♥➭② ❧➭ ♠➞✉ t❤✉➱♥ ✈í✐ tÝ♥❤ tè✐ t❤✐Ĩ✉ ❝đ❛ p. ❱×

p ❧➭ ♣❤➬♥ tư tè✐ t❤✐Ĩ✉ ❝đ❛ V (Ann A).

✷✳✷✳✾ ▼Ư♥❤ ➤Ị✳

❈❤♦

✭✐✮ P❤Ð♣ ♥❤➞♥ ❜ë✐

x ∈ R✳ ❑❤✐ ➤ã

n

x tr➟♥ A ❧➭ t♦➭♥ ❝✃✉ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x ∈
/

pi .
i=1
n

✭✐✐✮ P❤Ð♣ ♥❤➞♥ ❜ë✐

x tr➟♥ A ❧➭ ❧ò② ❧✐♥❤ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x ∈

pi , tø❝

i=1

❧➭

n

ℵ(A) =

pi .
i=1

n
❈❤ø♥❣ ♠✐♥❤✳

✭✐✐✮ ◆Õ✉ x

pi t❤× x ∈
/ pi , ∀i = 1, n✳ ❑❤✐ ➤ã ♣❤Ð♣ ♥❤➞♥

∈
/
i=1

❜ë✐

x tr➟♥ Ai ❧➭ t♦➭♥ ❝✃✉✱ ♥❣❤Ü❛ ❧➭ xAi = Ai , ∀i = 1, n. ❚❛ ❝ã
n

xA = x

n

Ai =
i=1

n

xAi =
i=1

✷✶

Ai = A.
i=1


n

❱× t❤Õ ♣❤Ð♣ ♥❤➞♥ ❜ë✐

x tr➟♥ A ❧➭ t♦➭♥ ❝✃✉✳ ◆Õ✉ x ∈

pi t❤× x ∈ pi ✈í✐
i=1

t

i ♥➭♦ ➤ã✳ ❑❤✐ ➤ã ∃ t ∈ N s❛♦ ❝❤♦ x Ai = 0. ❈ã
n
t

xt Aj ⊆

xt Ai =

xA=
i=1

i=j

Aj = A.
i=j

❱× t❤Õ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ xt tr➟♥ ❆ ❦❤➠♥❣ ❧➭ t♦➭♥ ❝✃✉✳ ❙✉② r❛ ♣❤Ð♣ ♥❤➞♥ ❜ë✐

x tr➟♥ ❆ ❦❤➠♥❣ ❧➭ t♦➭♥ ❝✃✉✳
n

✭✐✐✮ ◆Õ✉

pi t❤× x ∈ pi , ∀i = 1, n. ❑❤✐ ➤ã ♣❤Ð♣ ♥❤➞♥ ❜ë✐ x tr➟♥

x∈
i=1

Ai ❧➭ ❧ò② ❧✐♥❤ ✈í✐ ♠ä✐ i✱ ♥❣❤Ü❛ ❧➭ ∃t ∈ N s❛♦ ❝❤♦ xt Ai = 0, ∀i = 1, n.
n

❙✉② r❛

t

i=1

n

◆Õ✉

xt Ai = 0. ❱× t❤Õ ♣❤Ð♣ ♥❤➞♥ ❜ë✐ x tr➟♥ A ❧➭ ❧ò② ❧✐♥❤✳

xA=

pi , ♥❣❤Ü❛ ❧➭ x ∈
/ pi ✈í✐ i ♥➭♦ ➤ã✳ ❑❤✐ ➤ã xAi = Ai . ❙✉② r❛

x∈
/
i=1

xt Ai = Ai ✈í✐ ♠ä✐ t ∈ N. ❱× t❤Õ✱ ∀t ∈ N t❛ ❝ã
n
t

xt Ai ⊇ Ai = 0.

xA=
i=1

❱× ✈❐② ♣❤Ð♣ ♥❤➞♥ ❜ë✐

x tr➟♥ A ❦❤➠♥❣ ❧ị② ❧✐♥❤✳

✷✳✷✳✶✵ ▼Ư♥❤ ➤Ị✳

I

❈❤♦

❧➭ ✐➤➟❛♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝đ❛

R✳

❑❤✐ ➤ã ❝➳❝ ♣❤➳t

❜✐Ĩ✉ s❛✉ t➢➡♥❣ ➤➢➡♥❣✿
✭✐✮

A = IA.

✭✐✐✮ ❚å♥ t➵✐

x∈I

❈❤ø♥❣ ♠✐♥❤✳

sư ♥❣➢ỵ❝ ❧➵✐✱
✈✐Õt

✭✐✮

s❛♦ ❝❤♦

A = xA.

⇒ ✭✐✐✮ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ I ⊆ pi , ∀i = 1, n✳ ●✐➯

I ⊆ pi , ớ i ó ì I ữ s✐♥❤ ♥➟♥ t❛ ❝ã t❤Ó

I = (x1 , . . . , xk ). ớ ỗ k, ì xk I ⊆ pi ♥➟♥ ∃rk ∈ N s❛♦
✷✷


❝❤♦

xrk Ni = 0. ❈❤ä♥ r ❧➭ sè ❧í♥ ♥❤✃t tr♦♥❣ ❝➳❝ rk ✈➭ ➤➷t t = rk. ❑❤✐

I t Ni = 0. ❱× A = IA ♥➟♥ ❜➺♥❣ q✉② ♥➵♣ t❛ ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝

➤ã

A = I t A. ❉♦ ➤ã
n
t

A=I A=I

t

Ai = I t

Aj ⊆

i=1

➜✐Ò✉ ♥➭② ❧➭ ♠➞✉ t❤✉➱♥✳ ❱❐②
♥❣✉②➟♥ tè✱

Aj = A.
j=i

j=i

I ⊆ pi , ∀i = 1, n. ❚❤❡♦ ➜Þ♥❤ ❧Ý tr➳♥❤

I ∈
/ ∪ni=1 pi . ❙✉② r❛ tå♥ t➵✐ x ∈ I s❛♦ ❝❤♦ x ∈
/ pi ✈í✐ ♠ä✐ i.

❱× t❤Õ t❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✷✳✾✱ ♣❤Ð♣ ♥❤➞♥ ❜ë✐

x tr➟♥ A ❧➭ t♦➭♥ ❝✃✉✱ tø❝ ❧➭

xA = A.
✭✐✐✮

⇒ ✭✐✮ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳

✷✳✷✳✶✶ ❇ỉ ➤Ị✳
❝❤♦

❈❤♦

S

❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ tr♦♥❣

R✳ ➜➳♥❤ sè ❧➵✐ ❝➳❝ pi

s❛♦

S ∩ pi = ∅ ✈í✐ i = 1, . . . , r ✈➭ S ∩ pj = ∅ ✈í✐ i = r + 1, . . . , n. ➜➷t
r

L1 =

sA, L2 =
i=1

s∈S
✈➭

Ai

L3 ❧➭ tæ♥❣ t✃t ❝➯ ❝➳❝ ♠➠➤✉♥ ❝♦♥ p− t❤ø ❝✃♣ ❝ñ❛ A s❛♦ ❝❤♦ p∩S = ∅✳

❑❤✐ ➤ã

L1 = L2 = L3 .

ứ

ợ

ớ ỗ j

= r +1, . . . , n, ✈× S ∩pj = ∅ ♥➟♥ t❛ ❝ã t❤Ó ❝❤ä♥
k

xj ∈ S ∩ pj . ❱× xj ∈ pj ♥➟♥ tå♥ t➵✐ kj ∈ N s❛♦ ❝❤♦ xj j Aj = 0
n

✈í✐ ♠ä✐

k

j = r + 1, . . . , n. ➜➷t x =

xj j . ❱× S ❧➭ t❐♣ ➤ã♥❣ ♥❤➞♥ ♥➟♥
j=r+1

x ∈ S ✳ ❱× t❤Õ t❛ ❝ã
r

sA ⊆ xA = x

L1 =
s∈S

r

Ai ⊆
i=1

✷✸

Ai = L2 .
i=1