ôỔn định của một số tập idean nguyên tố gắn kết liên quan đến dãy đối chính quy chiều lớn hơn k

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆

❱➹ ❚❍➚ ❚❍➑ ❙■◆❍

✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ❚❾P ■✣➊❆◆
◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ▲■➊◆ ◗❯❆◆ ✣➌◆
❉❶❨ ✣➮■ ❈❍➑◆❍ ◗❯❨ ❈❍■➋❯ ▲❰◆ ❍❒◆ ❦

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❇➻♥❤ ✣à♥❤ ✲ ♥➠♠ ✷✵✶✾


❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆

❱➹ ❚❍➚ ❚❍➑ ❙■◆❍

✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ❚❾P ■✣➊❆◆
◆●❯❨➊◆ ❚➮ ●➁◆ ❑➌❚ ▲■➊◆ ◗❯❆◆ ✣➌◆
❉❶❨ ✣➮■ ❈❍➑◆❍ ◗❯❨ ❈❍■➋❯ ▲❰◆
số ỵ tt số
số

ữớ ữợ P ❍Ú❯ ❑❍⑩◆❍


✐

▼ö❝ ❧ö❝

▼Ð ✣❺❯

✷

✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✹

✶✳✶
✶✳✷
✶✳✸
✶✳✹

❚➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❉➣② ❝❤➼♥❤ q✉② ✈➔ ❞➣② ❧å❝ ❝❤➼♥❤ q✉②
❉➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ ❦ ✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳

✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
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✳
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✳
✳

✳
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✳
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✳
✳
✳
✳

✳
✳
✳
✳

✳
✳
✳
✳

✹
✶✵

✶✹
✶✾

✷ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ❚❾P ■✣➊❆◆ ◆●❯❨➊◆ ❚➮
●➁◆ ❑➌❚ ▲■➊◆ ◗❯❆◆ ✣➌◆ ❉❶❨ ✣➮■ ❈❍➑◆❍ ◗❯❨
❈❍■➋❯ ▲❰◆ ❍❒◆ ❦
✷✼
✷✳✶ ❑➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t
t

AttR (0:A (x1 n1 , ..., xi ni ) R)
i=0

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✷✳✷ ❑➳t q✉↔ ê♥ ✤à♥❤ ❝❤♦ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t
t
n
AttR (TorR
i (R/I, (0:A J )))
i=0

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✹✶


✶

▲í✐ ❝↔♠ ì♥
✣➸ ❤♦➔♥ t❤➔♥❤ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ✧ ê♥ ✤à♥❤
❝õ❛ ♠ët sè t➟♣ ✐❞✤❡❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❧✐➯♥ q✉❛♥ ✤➳♥ ❞➣② ✤è✐ ❝❤➼♥❤ q✉②
❝❤✐➲✉ ❧ỵ♥ ❤ì♥ ❦ ✧✱❧í✐ ✤➛✉ t✐➯♥ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❣û✐ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✤➳♥
t❤➛② ❣✐→♦ ❚❙✳ P❤↕♠ ❍ú✉ ❑❤→♥❤✳ ❚❤➛② ✤➣ trü❝ t✐➳♣ ❝❤➾ ❜↔♦ ữợ
tổ tr sốt q tr ự tỉ✐ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳
◆❣♦➔✐ r❛ tỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ ❚❤➛②✱ ❈ỉ ❑❤♦❛ ❚♦→♥
❚r÷í♥❣ ✣↕✐ ❍å❝ ◗✉② ◆❤ì♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥
✤ó♥❣ t❤í✐ ❤↕♥ q✉② ✤à♥❤✳
❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝↔♠ ì♥ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❜➯♥ tæ✐✱
✤ë♥❣ ✈✐➯♥ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ✈➔ ❧✉➟♥ ✈➠♥ ♥➔②✳
◆❣➔② ✻ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✾

❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥

❱➹ ❚❍➚ ❚❍➑ ❙■◆❍


✷

▼Ð ✣❺❯
❚r♦♥❣ s✉èt ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❝❤♦ (R, m) tr
ữỡ ợ ỹ ✤↕✐ m✳ I ✱ J ❧➔ ❤❛✐ ✐✤➯❛♥ ❝õ❛ R, M Rổ ỳ
s A Rổ rt
ỵ t❤✉②➳t ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠ỉ✤✉♥ ❆rt✐♥ ✤÷đ❝ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞
✤÷❛ ✈➔♦ ♥➠♠ ✶✾✼✸ ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ✤è✐ ợ ỵ tt t
sỡ ừ ổ ỳ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r✳ ▼ët R−♠ỉ✤✉♥ N
✤÷đ❝ ❣å✐ ❧➔ t❤ù ❝➜♣ ♥➳✉ N = 0 ✈➔ ✈ỵ✐ ♠å✐ r ∈ R✱ ♣❤➨♣ ♥❤➙♥ ❜ð✐ r tr➯♥ N ❧➔
t♦➔♥ ❝➜✉ ❤♦➦❝ ❧ơ② ❧✐♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② AnnR(N ) = p ❧➔ ♠ët ✐✤➯❛♥
♥❣✉②➯♥ tè✳ ❑❤✐ ✤â✱ t❛ ♥â✐ N ❧➔ p−t❤ù ❝➜♣✳

▼ët ❜✐➵✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥ K ❧➔ ♠ët ♣❤➙♥ t➼❝❤ t❤➔♥❤ tê♥❣ ❤ú✉
❤↕♥ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❝♦♥ K = K1 + . . . + Kn, tr♦♥❣ ✤â Ki ❧➔ pi−t❤ù ❝➜♣✳ ▼å✐
❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ K ✤➲✉ ❝â t❤➸ ✤÷❛ ✤÷đ❝ ✈➲ ❞↕♥❣ tè✐ t❤✐➸✉ tù❝ ❧➔ ❝→❝
pi ✤ỉ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈ỵ✐ ♠å✐ i = 1, . . . , n ✈➔ ❦❤æ♥❣ ❝â Ki ♥➔♦ t❤ø❛✳ ❑❤✐ ✤â
t➟♣ {p1, . . . , pn} ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ừ ổ K ỵ
AttR(K) ồ ♠æ✤✉♥ ❆rt✐♥ ✤➲✉ ❝â ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣✳
◆➠♠ ✶✾✼✾✱ ▼✳ ❇r♦❞♠❛♥♥ ❬✺❪ ❝❤ù♥❣ ♠✐♥❤ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t
AssR (M/J n M ) ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ◆❤÷ ♠ët ❦➳t q✉↔ ✤è✐ ♥❣➝✉✱ ❘✳ ❨✳ ❙❤❛r♣
❬✶✺❪ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t AttR (0:AJ n) ê♥ ✤à♥❤
❦❤✐ n ✤õ ❧ỵ♥✳ ❚ø ✤➙②✱ ♠ët ọ ữủ t r ợ ộ (x1, ..., xr ) ❝→❝
♣❤➛♥ tû ❝õ❛ ✈➔♥❤ R✱ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t AttR(0 :A (xn1 , ..., xnr ))
❝â ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥ ❦❤ỉ♥❣❄ ❚✉② ♥❤✐➯♥✱ ❑❛t③♠❛♥ ❬✺❪ ①➙② ❞ü♥❣ ♠ët ✈➼ ❞ö
✈➲ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ (R, m) ❝â ❝❤✐➲✉ ❜➡♥❣ ✺ ✈➔ ❤❛✐ ♣❤➛♥ tû x, y ∈ m s❛♦
2
❝❤♦ AssR(H(x,y)R
(R)) ❧➔ t➟♣ ✈æ ❤↕♥✳ ❉♦ ✤â✱
AssR (R/(xn , y n )R) ❧➔ t➟♣ ✈æ
n∈N
❤↕♥✳ ❙✉② r❛ AttR(0 :A (xn, yn)R) ❧➔ t➟♣ ✈æ ❤↕♥✱ tr♦♥❣ ✤â A = E(R/m) ❧➔
n∈N
❜❛♦ ♥ë✐ ①↕ ❝õ❛ R/m ❧➔ ♠ët R−♠æ✤✉♥ ❆rt✐♥✳ ❉♦ ✤â✱ t➟♣ AttR(0 :A (xn, yn)R)
❦❤ỉ♥❣ ê♥ ✤à♥❤ ❦❤✐ n ✤õ ❧ỵ♥✳ ❱➻ ✈➟②✱ ❝❤ó♥❣ t❛ ❝➛♥ t➻♠ ✤✐➲✉ ❦✐➺♥ ❝õ❛ A ✈➔
1

r


✤➸
AttR (0 :A (xn1 , ..., xnr )R) ❧➔ t➟♣ ❤ú✉ ❤↕♥✳
n ,...,n ∈N

◆❤➔♥ ✈➔ ❍♦➔♥❣ ❬✶✸❪ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ A−✤è✐ ❞➣② ❝❤✐➲✉ > k ♥❤÷ s❛✉✿ ▼ët
❞➣② (x1, . . . , xr ) ❝→❝ ♣❤➛♥ tû ❝õ❛ m ✤÷đ❝ ❣å✐ ❧➔ A−✤è✐ ❞➣② ❝❤✐➲✉ > k ♥➳✉ xi ∈
p ✈ỵ✐ ♠å✐ p ∈ (AttR (0 :A (x1 , . . . , xi−1 )R))>k ✈ỵ✐ ♠å✐ i = 1, . . . , r. ❚✐➳♣ t❤❡♦✱
◆❤➔♥ ✈➔ ❍♦➔♥❣ ❬✶✸❪ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ (
AttR (0 :A (xn1 , ..., xnr )R))≥k
n ,...,n ∈N
❧➔ t➟♣ ❤ú✉ ❤↕♥✱ tr♦♥❣ ✤â (x1, ..., xr ) ❧➔ ♠ët A−✤è✐ ❞➣② ❝❤✐➲✉ > k✳
❉✉♥❣ ✈❛ ◆❤➔♥ ❬✶✷❪ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ dimR(0 :A I) > k t❤➻ ♠å✐ A−✤è✐
❞➣② ❝❤✐➲✉ > k tr♦♥❣ I ❝â t❤➸ ❦➨♦ ❞➔✐ ✤➳♥ ❝ü❝ ✤↕✐ ✈➔ ♠å✐ A−✤è✐ ❞➣② ❝❤✐➲✉ > k
❝ü❝ ✤↕✐ tr♦♥❣ I ✤➲✉ ❝â ❝ị♥❣ ✤ë ❞➔✐✳ ✣ë ❞➔✐ ❝❤✉♥❣ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ở rở
> k ừ A tr I ỵ ❤✐➺✉ ❧➔ Width>k (I, A)✳ ◆➳✉ dimR(0 :A I) ≤ k t
ợ ồ số ữỡ r õ t t➻♠ ✤÷đ❝ ♠ët A−✤è✐ ❞➣② ❝❤✐➲✉ > k
tr♦♥❣ I ❝â ở r r trữớ ủ t q ữợ Width>k (I, A) = +∞✳
❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ AttR(0 :A J n) t❛ ❝â t❤➸ ❝❤➾ r❛ r➡♥❣
Width>k (I, (0 :A J n )) ê♥ ✤à♥❤ n ừ ợ
ỗ õ ữỡ ❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥
t❤ù❝ ❝ì ❜↔♥ ✈➲ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✱ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t✱
❞➣② ❝❤➼♥❤ q✉② ✈➔ ❞➣② ❧å❝ ❝❤➼♥❤ q✉②✱ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ✈➔ ✤ë
rë♥❣ ❝❤✐➲✉ > k✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝õ❛ t➟♣ ♥❣✉②➯♥ tè
t
❣➢♥ ❦➳t AttR(0 :A (xn1 , ..., xni )R) ✈➔ ❦➳t q✉↔ ê♥ ✤à♥❤ ❝❤♦ t➟♣ ♥❣✉②➯♥ tè
(x1 , ..., xr )

1

1

r

r

1

1

1

❣➢♥ ❦➳t

i=0
t

i=0

i

✳

n
AttR (TorR
i (R/I, (0 :A J )))

r

r


✹

❈❤÷ì♥❣ ✶

❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤➸
❧➔♠ ❝ì sð ❝❤♦ ❈❤÷ì♥❣ ✷ ♥❤÷ t➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✱ t➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t✱
❞➣② ❝❤➼♥❤ q✉②✱ ❞➣② ❧å❝ ❝❤➼♥❤ q✉② ✈➔ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ > k✳ ❚r♦♥❣
t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ ❧✉ỉ♥ ❣✐↔ t❤✐➳t r➡♥❣ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥
✈➔ ❝â ✤ì♥ ✈à 1 = 0✳

✶✳✶ ❚➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t
❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✈➲ t➟♣ ♥❣✉②➯♥ tè ❧✐➯♥
❦➳t ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥✳ ▼ët ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛

✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè t ừ M tỗ t ởt tỷ
x ∈ M ✱ x = 0 s❛♦ ❝❤♦ AnnR (x) = p✳
❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ỵ AssR(M ) Ass(M )
ữ
R

AssR (M ) = {p ∈ SpecR | ∃x ∈ M, x = 0, p = AnnR (x)}.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ P❤➛♥ tû a ừ R ữủ ồ ởt ữợ ừ ổ
ừ Rổ M tỗ t x M,

x=0

s ax = 0.

ữợ ừ ổ ừ M ữủ ỵ ❧➔ ZDR(M ). ◆❤÷ ✈➟②✱
ZDR (M ) = {a ∈ R | ∃x, 0 = x ∈ M, ax = 0} .

ú ỵ ổ õ AssR(R/p) = {p} , ✈ỵ✐ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè
❜➜t ❦➻ ❝õ❛ R

ỵ ỵ R ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ R−♠æ✤✉♥

❦❤→❝ ❦❤æ♥❣✳ ❑❤✐ ✤â

✭✐✮ P❤➛♥ tû ❝ü❝ ✤↕✐ tr♦♥❣ ❤å ❝→❝ ✐✤➯❛♥ F = {AnnR(x) | 0 = x ∈ M } ❧➔ ♠ët
✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳ ✣➦❝ ❜✐➺t✱ Ass (M ) = ∅ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
M = 0✳
✭✐✐✮ ❚➟♣ ữợ ừ ừ M ủ ừ tt ❝↔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥
❦➳t ❝õ❛ M ✱ tù❝ ❧➔
p.

ZDR (M ) =
p∈AssR (M )

❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ p = Ann(x) ❧➔ ♠ët ♣❤➛♥ tû
❝ü❝ ✤↕✐ ❝õ❛ F t❤➻ p ❧➔ ♠æt ✐✤➯❛♥ ♥❣✉②➯♥ tè✳
❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ a, b ∈ R ♠➔ ab ∈ p, b ∈/ p t❛ ❝â abx = 0, bx = 0✳ ❉♦
0 = bx ∈ M ♥➯♥ Ann(bx) ∈ F ✳ ❱➻ Ann(x) ❧➔ ♣❤➛♥ tû ❝ü❝ ✤↕✐ ❝õ❛ F ✈➔
Ann(x) ⊂ Ann(bx) ❞♦ ✤â Ann(bx) = Ann(x)✳ ❍ì♥ ♥ú❛✱ t❛ ❝â abx = 0 ♥➯♥
a ∈ Ann(bx)✳ ❉♦ ✤â a ∈ Ann(x)✳ ❙✉② r❛ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❱➟② p ❧➔ ✐✤➯❛♥
♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳
❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ AssR (M ) = ∅ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M = 0✳
sỷ M = 0 õ tỗ t 0 = x ∈ M ✈➔ ❞♦ ✤â F = ∅✳ ❱➻ R tr

tỗ t ởt tỷ ỹ ✤↕✐ p ∈ F ✳ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ p ∈ AssR(M )
♥➯♥ AssR(M ) = ∅✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ AssR(M ) = t tỗ t 0 = x ∈ M s❛♦
❝❤♦ AssR(x) = p✱ ✈ỵ✐ p ♥❣✉②➯♥ tè✳ ❉♦ ✤â M = 0✳
✭✐✐✮ ●✐↔ sû α ∈ ZDR(M )✱ t❛ ❝❤ù♥❣ ♠✐♥❤ α ∈
p✳ ❚❤➟t ✈➟②✱ ✈➻
p∈Ass (M )
ZDR (M ) tỗ t x = 0 s❛♦ ❝❤♦ αx = 0✳ ❙✉② r❛ α ∈ Ann(x)✳ ❱➻ t❤➳
R




tỗ t p Ass(M ) s Ann(x) p✳ ❙✉② r❛ α ∈
ZDR (M ) ⊆

p.
p∈AssR (M )

❉♦ ✤â✱

p.
p∈AssR (M )

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû α ∈
p t❛ s➩ ự ởt ữợ ừ ừ
pAss (M )
M t
p t tỗ t↕✐ p ∈ Ass(M ) s❛♦ ❝❤♦ α ∈ p✳
p∈Ass (M )
õ tỗ t x = 0, x M ✤➸ αx = 0✳ ❙✉② r❛ α ∈ ZDR(M ). ❉♦ ✤â✱

R

R

p ⊆ ZDR (M ).
p∈AssR (M )

❱➟② ZDR(M ) =

p.
p∈AssR (M )

❇ê ✤➲ ✶✳✶✳✺✳ ❈❤♦ R ❧➔ ✈➔♥❤ ✈➔ M, N ❧➔ ❝→❝ R−♠æ✤✉♥✳ ❑❤✐ ✤â
✭✐✮ ◆➳✉ N ⊂ M t❤➻ AssR(N ) ⊂ AssR(M )✳
✭✐✐✮ ❈❤♦ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè tr➯♥ ✈➔♥❤ R✳ ❑❤✐ ✤â p AssR(M )
tỗ t ởt Rổ ❝♦♥ N ❝õ❛ M s❛♦ ❝❤♦ N ∼= R/p.
❈❤ù♥❣ ♠✐♥❤✳ sỷ p AssR(N ) õ tỗ t↕✐ 0 = x ∈ N ✤➸ p =
AnnR (x)✳ ❱➻ x ∈ N ⊂ M ♥➯♥ x ∈ M ✳ ❉♦ ✤â p ∈ AssR (M )✳ ❱➟② AssR (N ) ⊂
AssR (M )✳
✭✐✐✮ ●✐↔ sû p ∈ AssR(M ) õ tỗ t 0 = x M s❛♦ ❝❤♦ p = AnnR(x)✳
✣➦t N = Rx ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M ✳
❚❛ ①➨t →♥❤ ①↕ f ♥❤÷ s❛✉
f : R −→ Rx
a −→ ax.

❘ã r➔♥❣ f ❧➔ Rt ử ỵ ổ t ❝â N ∼=
R/Kerf = R/Ann(x) = R/p✳
◆❣÷đ❝ ❧↕✐✱ ❞♦ AssR(R/p) = {p} = AssR(N ) tỗ t tỷ x = 0✱
x ∈ N ⊆ M, ✤➸ p = AnnR (x)✳ ❱➟② p ∈ AssR (M )✳
❇ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

ỵ r M ởt ♠æ✤✉♥ tr➯♥ ✈➔♥❤ ❣✐❛♦

❤♦→♥ ◆♦❡t❤❡r R✱ S ❧➔ t➟♣ ❝♦♥ ♥❤➙♥ ✤â♥❣ ❝õ❛ R✳ ❑❤✐ ✤â

AssS −1 R S −1 M = pS −1 R | p ∈ AssR (M ) , p ∩ S = ∅ .

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ♥➯♥ S −1R ❝ô♥❣ ❧➔ ♠ët ✈➔♥❤
❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✳ ❉♦ ✤â✱ t➟♣ AssS R S −1M ❧➔ ①→❝ ✤à♥❤✳
▲➜② p ∈ AssR(M ) s❛♦ p S = õ tỗ t m = 0, m ∈ M
s❛♦ ❝❤♦ p = (0 :R m)✳ ❚❛ ❝â pS −1R = (0 :S R m/1) ∈ Spec(S −1R)✳ ❉♦
✤â✱ pS −1R ∈ AssS R S −1M ✳ ❙✉② r❛ pS −1R | p ∈ AssR (M ) , p ∩ S = ∅ ⊆
−1

−1

−1

AssS −1 R S −1 M .

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû q ∈ AssS R S −1M ✳ ❑❤✐ ✤â✱ q ∈ Spec(S −1)R ✈➔ ❝â
❞✉② ♥❤➜t p ∈ Spec(R) s❛♦ ❝❤♦ p ∩ S = ∅ ✈ỵ✐ q = pS −1R. t tỗ t
m M, s S s ❝❤♦ q = (0 :S R m/s). ❱➻ s/1 ❧➔ ♣❤➛♥ tû ❦❤↔ ♥❣❤à❝❤ ❝õ❛
S −1 R ♥➯♥ q = (0 :S R m/1). ❱➻ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ♥➯♥ p ❧➔ ♠ët ✐✤➯❛♥
❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ✤÷đ❝ s✐♥❤ ❜ð✐ ❝→❝ ♣❤➛♥ tû p1, p2, . . . , pn.
❑❤✐ ✤â✱ pim/1 = 0S M , ✈ỵ✐ ♠å✐ i = 1, 2, . . . n. ▼➦t ❦❤→❝✱ ✈ỵ✐ ộ i = 1, 2, . . . n
t tỗ t↕✐ ♠ët si ∈ S s❛♦ ❝❤♦ sipim = 0.

✣➦t t := s1 . . . sn ∈ S ✳ ❑❤✐ ✤â✱ tpim = 0, ✈ỵ✐ ♠å✐ i = 1, 2, . . . n. ❙✉② r❛
ptm = 0 ❞♦ ✤â p ⊆ (0 :R tm). ◆❣÷đ❝ ❧↕✐✱ ❧➜② r ∈ (0 :R tm). ❑❤✐ ✤â rtm = 0
♥➯♥ (rt/1)(m/1) = 0S M . ❙✉② r❛ rt/1 ∈ (0 :S R m/1) = pS −1R.
❱➻ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ♥➯♥ rt ∈ p. ❍ì♥ ♥ú❛✱ t ∈ S ⊆ R/p ♥➯♥ r ∈ p
s✉② r❛ p ⊇ (0 :R tm). ❉♦ ✤â p = (0 :R tm), s✉② r❛ p ∈ AssR(M ). ❱➟②
−1

−1

−1

−1

−1

−1

pS −1 R | p ∈ AssR (M ) , p ∩ S = ∅ ⊇ AssS −1 R S −1 M .

❍➺ q✉↔ ✶✳✶✳✼✳ ❈❤♦ M ❧➔ ♠ët R ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥
tè ❝õ❛ R✳ ❑❤✐ ✤â✱

AssRp (Mp ) = {qRp | q ∈ AssR (M ) , q ⊆ p} .

ỵ ỵ


✽

❈❤♦ ✈➔♥❤ R ✈➔

f

✭✶✳✶✮

g

0 −→ M −→ M −→ M −→ 0

❧➔ ❞➣② ❦❤ỵ♣ ❝→❝ R−♠ỉ✤✉♥✳ ❑❤✐ ✤â AssR(M ) ⊂ AssR(M ) ∪ AssR(M

).

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ p ∈ AssR(M )✱ t❤❡♦ ❇ê ✤➲ ✶✳✶✳✺ t❤➻ M ❝❤ù❛ ♠ët
♠æ✤✉♥ ❝♦♥ N ∼= R/p✱ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ①❡♠ N = R/p✳
❱ỵ✐ ♠å✐ x ∈ N ✈➔ x = 0 t❛ ❝â x = a + p, a ∈/ p✳ ❉♦ ✤â AnnR(x) = p.
✰✮ ❚r÷í♥❣ ủ N M = {0} õ tỗ t 0 = x ∈ N ∩ M ✳
❱➻ 0 = x ∈ N ♥➯♥ AnnR(x) = p✳ ❍ì♥ ♥ú❛✱ ✈➻ x ∈ M ♥➯♥ t❛ ✤÷đ❝
p ∈ AssR (M ).

✰✮ ❚r÷í♥❣ ❤đ♣ N ∩ M

= {0}

✭✶✳✷✮

✳ ❚❛ ✤✐ ①➨t →♥❤ ①↕
g|N : N −→ M .

❱➻ ✭✶✳✶✮ ❧➔ ❞➣② ❦❤ỵ♣ ✈➔ Kerg|N = Kerg∩N ♥➯♥ Kerg|N = M ∩N = {0} . ❙✉② r❛
g|N ✤ì♥ ❝➜✉ ❞♦ ✤â N ⊂ M ✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✶✳✺ t❤➻ AssR (N ) ⊂ AssR (M )✳

❍ì♥ ♥ú❛✱ ✈➻ AssR(N ) = p ♥➯♥
p ∈ AssR (M ).

✭✶✳✸✮

❑➳t ❤ñ♣ ✭✶✳✷✮ ✈➔ ✭✶✳✸✮ t❛ ✤÷đ❝ p ∈ Ass(M ) ∪ AssR(M )✳ ❱➟② AssR(M )
AssR (M ) AssM (M ).

ỵ ữủ ự

ỵ ỵ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M = 0 ❧➔

R−♠æ✤✉♥

❤ú✉ s õ tỗ t ộ
0 = M0 ⊂ M1 ⊂ . . . ⊂ Mn = M

tr♦♥❣ ✤â ❝→❝ Mi ✈ỵ✐✱ i = 1, n, ❧➔ R−♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ i t❛
❝â Mi/Mi−1 ∼= R/pi✱ ✈ỵ✐ pi ∈ SpecR✳


✾

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ M = 0 ♥➯♥ AssR(M ) = ∅✳ ❈❤å♥ p1 ∈ AssR(M ) ❜➜t ❦ý✳
❚❤❡♦ ❇ê ✤➲ tỗ t M1 ổ ừ M M1 ∼= R/p1 ❤❛②
M1 /M0 ∼
= R/p1 .

◆➳✉ M1 = M t❤➻ AssR(M/M1) = ∅✱ t❛ ❧↕✐ ❝❤å♥ p2 ∈ Ass(M/M1),
tợ tỗ t M2/M1 ổ ừ M/M1 s❛♦ ❝❤♦ M2/M1 ∼= R/p2✳

▲➔♠ t✐➳♣ tư❝ q✉→ tr➻♥❤ ♥❤÷ tr➯♥ tr➯♥ t❛ s➩ ✤÷đ❝ ♠ët ❞➣② ❝→❝ R− ♠ỉ✤✉♥
❝♦♥ ❝õ❛ M ♥❤÷ s❛✉
0 = M0 ⊂ M1 ⊂ . . . ⊂ Mn ⊂ . . . ⊂ M

tr♦♥❣ ✤â Mi/Mi−1 ∼= R/pi✳
❉➣② tr➯♥ ❧➔ ♠ët ❞➣② ❞ø♥❣ ✈➻ M ổ tr tr R t s tỗ
t ♠ët sè n ∈ N ✤➸ M = Mn✳
❱➟② Mn = M õ ự

ỵ ỵ R ởt ◆♦❡t❤❡r ✈➔ M ❧➔

♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â AssR(M ) ❧➔ t➟♣ ❤ú✉ ❤↕♥✳

❈❤ù♥❣ ♠✐♥❤✳ ❚r÷í♥❣ ❤đ♣ M = 0 t❤➻ AssR(M ) = ∅ ❞♦ ✤â AssR(M ) ❧➔ t➟♣
❤ú✉ ❤↕♥ ✳
❚r÷í♥❣ ❤đ♣ M = 0✱ t❤❡♦ ỵ t s tỗ t ởt ộ ❝❤✉②➲♥
❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ M ♥❤÷ s❛✉
0 = M0 ⊂ M1 ⊂ . . . ⊂ Mn = M

✈ỵ✐ Mi/Mi−1 ∼= R/pi✱ pi ∈ SpecR✳
❳➨t ❞➣②
✱

0 −→ Mn−1 −→ M −→ M/Mn−1 −→ 0

❱➻ ❞➣② tr➯♥ ❧➔ ❞➣② ❦❤ỵ♣ ♥➯♥ ử ỵ t ữủ
AssR (M ) AssR (Mn−1 ) ∪ AssR (M/Mn−1 )

tr♦♥❣ ✤â AssR(M/Mn−1) = Ass(R/pn) = {pn} .
❳➨t ❞➣② t❤ù ❤❛✐ ♥❤÷ s❛✉

✶✵

0 −→ Mn−2 −→ Mn−1 −→ Mn−1 /Mn−2 −→ 0.

❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ✤÷đ❝ ❜❛♦ ❤➔♠ t❤ù❝ s❛✉
AssR (Mn−1 ) ⊂ Ass(Mn−2 ) ∪ Ass(Mn−1 /Mn−2 )

✈➔ AssR(Mn−1/Mn−2) = AssR(R/pn−1) = {pn−1} . ❚❤ü❝ ❤✐➺♥ t✐➳♣ tư❝ ♥❤÷
✈➟② t❛ ữủ ợ
0 M1 M2 M2 /M1 −→ 0

✈➔ t❛ ❝â AssR(M2) ⊂ AssR(M1) ∪ AssR(M2/M1)✳
❉♦ AssR(M2/M1) = AssR(R/p2) = {p2} ✈➔ AssR(M1) = AssR(R/p1) = {p1}
♥➯♥ AssR(M ) ⊂ {p1} ∪ {p2} ∪ . . . ∪ {pn} = {p1, p2, . . . , pn} . AssR(M )
t ỳ
ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳

✶✳✷ ❚➟♣ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t
❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ t❛ s➩ ♥❤➢❝ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♠æ✤✉♥ t❤ù
❝➜♣✱ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ♠æ✤✉♥✱ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ♠æ✤✉♥✱
♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ✈➔ ✤à♥❤ ♥❣❤➽❛ ✈➲
✤è✐ ♥❣➝✉ ▼❛t❧✐s✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ▼ët R−♠æ✤✉♥ M ✤÷đ❝ ❣å✐ ❧➔ t❤ù ❝➜♣ ♥➳✉ M = 0 ✈➔
♥➳✉ ợ ồ x R tỹ ỗ x,M : M −→ M ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♣❤➨♣
♥❤➙♥ ❝õ❛ x tr M t ụ

ú ỵ ◆➳✉ M ❧➔ ♠ët R−♠æ✤✉♥ t❤ù ❝➜♣ t❤➻

AnnR (M )

✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛ M

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ▼ët R−♠æ✤✉♥ M ❧➔ t❤ù ❝➜♣ ✈➔ p =
M

✤÷đ❝ ❣å✐ ❧➔ p−t❤ù ❝➜♣✳

❧➔ ♠ët

AnnR (M )

t❤➻


✶✶

▼➺♥❤ ✤➲ ✶✳✷✳✹✳ ✭❬✽❪✱ ❚r❛♥❣ ✷✻✮ ▼ët ♠ỉ✤✉♥ t❤÷ì♥❣ ❦❤→❝ ❦❤æ♥❣ ❝õ❛ ♠æ✤✉♥

p−t❤ù

❝➜♣ ❧➔ ♠æ✤✉♥ p−t❤ù ❝➜♣✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M ❧➔ R−♠æ✤✉♥ p−t❤ù ❝➜♣✱ N ❧➔ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü
❝õ❛ M ❦❤✐ ✤â M/N ❧➔ ♠ỉ✤✉♥ t❤÷ì♥❣ ❦❤→❝ ❦❤ỉ♥❣ ❝õ❛ M ✳ ❱ỵ✐ ♠å✐ x ∈ R
①↔② r❛ ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉
✰✮ ❚r÷í♥❣ ❤đ♣ x ∈ p t❤➻ ϕx,M : M M ụ õ tỗ t↕✐ n ∈ N
s❛♦ ❝❤♦ xnM = 0✳ ❙✉② r❛ xn(M/N ) = (xnM + N )/N = 0 ❞♦ ✤â

ϕx,M/N : M/N −→ M/N

❧➔ ❧ơ② ❧✐♥❤✳
✰✮ ❚r÷í♥❣ ❤đ♣ x ∈/ p t❤➻ ϕx,M : M −→ M ❧➔ t♦➔♥ ❝➜✉✱ ❞♦ ✤â xM = M ✳ ❙✉② r❛
x(M/N ) = (xM +N )/N = (M +N )/N = M/N ✱ ❞♦ ✤â ϕx,M/N : M/N −→ M/N
❧➔ t♦➔♥ ❝➜✉✳
❍ì♥ ♥ú❛✱ t❛ ❝â AnnR(M/N ) = p✳ ❚❤➟t ✈➟②✱ ❧➜② ♠ët ❣✐→ trà x ∈
AnnR (M/N ) ❜➜t ❦ý✱ ❦❤✐ õ tỗ t n N s xn (M/N ) = 0 s✉② r❛
(xn M +N )/N = 0 ❞♦ ✤â xn M +N = N ✳ ❉♦ xn M = 0 ♥➯♥ x ∈ AnnR (M ) = p✳
❙✉② r❛ AnnR(M/N ) ⊆ p✳ ◆❣÷đ❝ ❧↕✐✱ ❧➜② ♠ët ❣✐→ trà x ❜➜t ❦ý s❛♦ ❝❤♦ x ∈
AnnR (M ) = p õ tỗ t k > 0 ✤➸ xk M = 0 s✉② r❛ xk (M/N ) = 0✳ ❉♦
✤â xk ∈ AnnR (M/N ) s✉② r❛ x ∈ AnnR (M/N )✳ ❉♦ ✤â AnnR (M/N ) = p✳
❱➟② M/N ❧➔ ♠ët p−t❤ù ❝➜♣✳

❇ê ✤➲ ✶✳✷✳✺✳ ✭❬✽❪✱ ❚r❛♥❣ ✷✼✮ ▲✐♥❤ ❤â❛ tû ❝õ❛ ♠ët ♠æ✤✉♥ p−t❤ù ❝➜♣ ❧➔
♠ët ✐✤➯❛♥ p−♥❣✉②➯♥ sì✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M ❧➔ R−♠ỉ✤✉♥ p−t❤ù ❝➜♣✳ ●✐↔ sû ab ∈ AnnR(M ) ✈➔
bn ∈
/ AnnR (M )✱ ✈ỵ✐ ♠å✐ n✳ ❱➻ M ❧➔ R−♠ỉ✤✉♥ ptự bM = M
tỗ t ởt số tü ♥❤✐➯♥ n s❛♦ ❝❤♦ bn ∈ AnnR (M )✳ ▼➦t ❦❤→❝✱ t❤❡♦ ♥❤÷ t❛
❣✐↔ t❤✐➳t t❤➻ bn ∈/ AnnR(M ) ♥➯♥ bM = M ✳ ❱➻ ab ∈ AnnR(M ) ♥➯♥ abM = 0
s✉② r❛ aM = 0 ✈➔ ❞♦ ✤â a ∈ AnnR (M )✳ ◆❤÷ ✈➟②✱ AnnR(M ) ❧➔ ♥❣✉②➯♥ sì✳


✶✷

❍ì♥ ♥ú❛✱ p = AnnR (M ) ♥➯♥ AnnR(M ) ❧➔ p−t❤ù ❝➜♣✳
❇ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

❱➼ ❞ư ✶✳✷✳✻✳ R ữỡ ợ p ♥❣✉②➯♥ tè ❝ü❝ ✤↕✐

✈➔ ♠å✐ ♣❤➛♥ tû tr♦♥❣ p ✤➲✉ ❧➔ ❧ơ② ❧✐♥❤ t❤➻ R ❝❤➼♥❤ ❧➔ R−♠ỉ✤✉♥ p−t❤ù ❝➜♣✳

❇ê ✤➲ ✶✳✷✳✼✳ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥ ✈➔ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛

R✱ M1 , M2 , . . . , Mr

❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ p−t❤ù ❝➜♣ ❝õ❛ M ✳
❑❤✐ ✤â✱ P = M1 + M2 + · · · + Mr ❝ô♥❣ ❧➔ p−t❤ù ❝➜♣ ❝õ❛ M ✳

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ x ∈ R ①↔② r❛ ❤❛✐ trữớ ủ s
rữớ ủ x p t ợ ♠å✐ i t❛ ❝â ϕx,M : Mi → Mi ❧➔ ụ õ
tỗ t ni s xn Mi = 0✳
❱ỵ✐ n = Max {n1, n2, . . . , nr }✱ t❤➻ xnMi = 0 ✈ỵ✐ ♠å✐ i✱ ❞♦ ✤â xnP = 0✳ ❙✉②
r❛ ϕx,P : P −→ P ❧➔ ❧ơ② ❧✐♥❤✳
✰✮ ❚r÷í♥❣ ❤đ♣ x ∈/ p t❤➻ ✈ỵ✐ ♠å✐ i t❛ ❝â✱ ϕx,M : Mi → Mi ❧➔ t♦➔♥ ❝➜✉✱ ❦❤✐
✤â xMi = Mi ✈ỵ✐ ♠å✐ i✳ ❉♦ ✤â xP = P s✉② r❛ ϕx,P : P −→ P ❧➔ t♦➔♥ ❝➜✉✳
❱➟② P = M1 + M2 + · · · + Mr ❧➔ R−♠æ✤✉♥ p−t❤ù ❝➜♣✳
i

i

i

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✽✳ ✭✐✮ ▼ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M ❧➔ ♠ët ♣❤➙♥ t➼❝❤
t❤➔♥❤ tê♥❣ ❤ú✉ ❤↕♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ pi t❤ù
❝➜♣ Mi✳ ◆➳✉ M = 0 ❤♦➦❝ M ❝â ♠ët ❜✐➵✉ ❞✐➵♥ t❤ù ❝➜♣ t❤➻ t❛ ♥â✐ M
❜✐➵✉ ❞✐➵♥ ✤÷đ❝✳
M = M1 + M2 + · · · + Mr

✭✐✐✮ ▼ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M ✤÷đ❝ ❣å✐ ❧➔ tè✐ t❤✐➸✉ ♥➳✉ ❝→❝ ♠æ✤✉♥
❝♦♥ t❤ù ❝➜♣ M1, M2, . . . , Mr t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥
✭✶✮ ❈→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè AnnR(Mi) ♣❤➙♥ ❜✐➺t✳
✭✷✮ ❑❤æ♥❣ ❝â Mi ♥➔♦ ♥➡♠ tr♦♥❣ tê♥❣ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝á♥ ❧↕✐✳
▼å✐ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M ✤➲✉ ✤÷❛ ✤÷đ❝ ✈➲ ❞↕♥❣ tè✐ t❤✐➸✉✳ ❚➟♣
❤đ♣ {p1, . . . , pn} ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ M ỵ
AttRM.


✶✸

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✾✳ ✭❬✽❪✱ ❚r❛♥❣ ✸✺✮ ▼ët R−♠ỉ✤✉♥ M ✤÷đ❝ ❣å✐ ❧➔ ❜➜t ❦❤↔

tê♥❣ ♥➳✉ M ❦❤→❝ ❦❤æ♥❣ ✈➔ tê♥❣ ❝õ❛ ❤❛✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ M ❧✉æ♥ ❧➔
♠ët ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ M ✳

❇ê ✤➲ ✶✳✷✳✶✵✳ ✭❬✽❪✱ ❚r❛♥❣ ✸✺✮ ◆➳✉ M ❧➔ R−♠æ✤✉♥ ❆rt✐♥ ❦❤→❝ ❦❤æ♥❣ ✈➔
❜➜t ❦❤↔ tê♥❣ t❤➻ M ❧➔ ♠æ✤✉♥ t❤ù ❝➜♣✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sỷ M ổ ổ tự õ tỗ t↕✐ ♣❤➛♥ tû
x ∈ R s❛♦ ❝❤♦ M = xM ✈➔ xn M = 0✱ ✈ỵ✐ ♠å✐ n > 0. ❱➻ M ❧➔ R−♠æ✤✉♥
❆rt✐♥ ♥➯♥ ❞➣② ❝→❝ ♠æ✤✉♥ ❝♦♥ {xnM }n0 ừ M ứ õ tỗ t
số tü ♥❤✐➯♥ k s❛♦ ❝❤♦ xk M = xk+1M = · · · = x2k M = . . . ✳
✣➦t M1 = Ker(ϕx ,M ) ✈➔ M2 = xk M. ❑❤✐ ✤â✱ M1 ✈➔ M2 ❧➔ ❤❛✐ ♠æ✤✉♥
❝♦♥ ❝õ❛ M ✳ ❱➻ xk M1 = 0 ✈➔ xk M = 0 ♥➯♥ M1 = M ✱ ✈➔ ✈➻ M = xM ♥➯♥
M2 = M. ❉♦ ✤â M1 , M2 ❧➔ ❤❛✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ M ✳
●✐↔ sû u ∈ M ❜➜t ❦ý✱ ✈➻ xk u ∈ xk M = x2k M tỗ t v M s❛♦ ❝❤♦
xk u = x2k v ✱ s✉② r❛ xk (u − xk v) = 0✳ ❉♦ ✤â u − xk v ∈ M1 .
❚❛ ❝â u = xk v + (u − xk v) ∈ M2 + M1. ❙✉② r❛✱ M = M1 + M2✳ ✣✐➲✉ ♥➔② ❧➔

♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t M ❧➔ ❜➜t ❦❤↔ tê♥❣✳ ❱➟② M ổ tự
k

ỵ r ồ R−♠æ✤✉♥ ❆rt✐♥ ❦❤→❝ ❦❤æ♥❣ ✤➲✉ ❝â

♠ët ❜✐➵✉ ❞✐➵♥ t❤ù ❝➜♣✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M ❧➔ R−♠æ✤✉♥ ❆rt✐♥ ❦❤æ♥❣ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝✳ ❳➨t t➟♣
❝→❝ ♠ỉ✤✉♥ ❝♦♥ ❦❤→❝ ❦❤ỉ♥❣✱ ❦❤ỉ♥❣ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ❝õ❛ M ✳ ❚➟♣ ♥➔② ❦❤→❝
ré♥❣ ✈➻ ❝â ❝❤ù❛ M.
❱➻ M ❧➔ R−♠æ✤✉♥ ❆rt✐♥ ♥➯♥ t➟♣ ♥➔② ❝â ♣❤➛♥ tû ❝ü❝ t✐➸✉ ❧➔ N ✳ ❉♦ N
❧➔ ♠æ✤✉♥ ❦❤æ♥❣ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ♥➯♥ N ❦❤ỉ♥❣ ❧➔ ♠ỉ✤✉♥ t❤ù ❝➜♣✳ ❱➻ N ❧➔
♠æ✤✉♥ ❆rt✐♥ ❦❤→❝ ❦❤æ♥❣ ✈➔ ❦❤æ♥❣ ❧➔ ♠æ✤✉♥ t❤ù ❝➜♣ ♥➯♥ N ❧➔ tê♥❣ ❝õ❛
❤❛✐ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü N1 ✈➔ N2✳ ❍ì♥ ♥ú❛✱ ❞♦ t➼♥❤ ❝❤➜t ❝ü❝ t✐➸✉ ❝õ❛ N
tr♦♥❣ t➟♣ t❛ ✤❛♥❣ ①➨t ♥➯♥ N1✱ N2 ❧➔ ❝→❝ ♠ỉ✤✉♥ ❜✐➵✉ ❞✐➵♥ ✤÷đ❝✳ ❱➻ N ❧➔
tê♥❣ ❝õ❛ ❤❛✐ ♠ỉ✤✉♥ ❜✐➵✉ ❞✐➵♥ ✤÷đ❝ N1✱ N2 ♥➯♥ N ❝ơ♥❣ ❧➔ ♠ët ♠æ✤✉♥ ❜✐➵✉




ữủ t ợ ồ N ð tr➯♥✳
❱➟② M ❧➔ ♠ỉ✤✉♥ ❜✐➵✉ ❞✐➵♥ ✤÷đ❝✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✷✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M ❧➔ ♠ët
R−

♠æ✤✉♥ ✳ ✣è✐ ♥❣➝✉ ▼❛t❧✐s ❝õ❛ M ❧➔ ♠æ✤✉♥

D (M ) = HomR (M, E (R/m))

tr♦♥❣ ✤â E (R/m) ở ừ R/m

ú ỵ r ✸✸✮ ❈❤♦ M, N ❧➔ ❝→❝ R−♠æ✤✉♥ ✳ ●✐↔ sû r➡♥❣
M

❧➔ ♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â✱ t❛ ❝â ✤➥♥❣ ❝➜✉
M ⊗R D (N ) ∼
= D (HomR (M, N )) .

❇ê ✤➲ ✶✳✷✳✶✹✳ ❈❤♦ M ❧➔ R−♠æ✤✉♥✳ ❑❤✐ ✤â✱
Ann(D(M )) = Ann(M ).

▼➺♥❤ ✤➲ ✶✳✷✳✶✺✳ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✤à❛ ♣❤÷ì♥❣✱ ◆♦❡t❤❡r✱
✤➛② ✤õ✳ ❑❤✐ ✤â✱

✭✐✮ ◆➳✉ N ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ D(N ) ❧➔ R−♠æ✤✉♥ ❆rt✐♥✳
✭✐✐✮ ◆➳✉ N ❧➔ R−♠æ✤✉♥ ❆rt✐♥ t❤➻ D(N ) ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳

▼➺♥❤ ✤➲ ✶✳✷✳✶✻✳ ✭❬✸❪ ❚r❛♥❣ ✷✵✻✮ ❈❤♦ R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✱ ✤➛② ✤õ✱ M
❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ A ❧➔ R−♠æ✤✉♥ ❆rt✐♥✳ ❑❤✐ ✤â✱
✭✐✮
✭✐✐✮

AttR (D(M )) = AssR (M ).
AttR (A) = AssR (D(A)).

✶✳✸ ❉➣② ❝❤➼♥❤ q✉② ✈➔ ❞➣② ❧å❝ ❝❤➼♥❤ q✉②
❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❞➣② ❝❤➼♥❤ q✉②✱ ❞➣② ❧å❝ ❝❤➼♥❤
q✉② ✈➔ ♠ët sè t➼♥❤ ❝❤➜t t❤❡♦ ❝✉è♥ s→❝❤ ❬✶✶❪✳

✶✺

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ❈❤♦ M ❧➔ R−♠æ✤✉♥✳ ▼ët ♣❤➛♥ tû x = 0 ❝õ❛ R ✤÷đ❝

❣å✐ ❧➔ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ❝õ❛ M ❤❛② M −❝❤➼♥❤ q✉② ♥➳✉ xm = 0 ✈ỵ✐ ♠å✐
m ∈ M, m = 0✳
▼ët ❞➣② ❝→❝ ♣❤➛♥ tû x1, x2, .., xr ❝õ❛ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞➣② ❝❤➼♥❤ q✉②
❝õ❛ M ❤❛② M −❞➣② ❝❤➼♥❤ q✉② ♥➳✉ t❤ä❛ ♠➣♥ ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✭✐✮ M/(x1, . . . , xr )M = 0✳
✭✐✐✮ xi ❧➔ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ❝õ❛ M/(x1, . . . , xi−1)M ✱ ✈ỵ✐ ♠å✐ i = 1, r✳
◆➳✉ ❞➣② x1, , .., xr ❝❤➾ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✐✐✮ t❤➻ t❛ ♥â✐ x1, , .., xr ❧➔ ♠ët
M −❞➣② ❝❤➼♥❤ q✉② ②➳✉✳
❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ x ∈ R ❧➔ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ❝õ❛
M ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x ∈
/ p ✈ỵ✐ ♠å✐ p ∈ AssR (M ). ◆❣♦➔✐ r❛✱ ♥➳✉ (R, m) ❧➔ ✈➔♥❤
✤à❛ ♣❤÷ì♥❣✱ x1, x2, .., xr ∈ m ✈➔ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤æ♥❣
t❤➻ ✤✐➲✉ ❦✐➺♥ ✭✐✮ ❧✉æ♥ t❤ä❛ ♠➣♥ ❜ð✐ ❇ê ✤➲ ◆❛❦❛②❛♠❛✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ M ❧➔ R−♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝❤ó♥❣ t❛ ❝â t❤➸ t➼♥❤
✤ë ❞➔✐ ❝õ❛ ♠ët ❞➣② ❝❤➼♥❤ q✉② ❝ü❝ ✤↕✐ ❝õ❛ M tr♦♥❣ I ♥❤÷ s

ỵ ỵ M ♠ët R−♠æ✤✉♥ ❤ú✉ ❤↕♥

s✐♥❤ ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛ R s❛♦ ❝❤♦ IM = M ✳ ❑❤✐ ✤â✱ ♠å✐ ❞➣② ❝❤➼♥❤ q✉② ❝ü❝
✤↕✐ ❝õ❛ M tr♦♥❣ I ✤➲✉ ❝â ❝ò♥❣ ✤ë ❞➔✐ n ✤÷đ❝ ❝❤♦ ❜ð✐
n = inf i | Exti R (R/I, M ) = 0 .

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✸✳ ❈❤♦ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛

s❛♦ ❝❤♦ IM = M ✳ ❑❤✐ ✤â ✤ë ❞➔✐ ❝❤✉♥❣ ❝õ❛ ♠å✐ ❞➣② ❝❤➼♥❤ q✉② ❝ü❝ ✤↕✐

❝õ❛ M tr♦♥❣ I ✤÷đ❝ ❣å✐ ❧➔ ✤ë s➙✉ ❝õ❛ M tr♦♥❣ I ✈➔ ❦➼ ❤✐➺✉ ❜ð✐ depth(I, M )✳
◆➳✉ IM = M t t q ữợ depth(I, M ) = +
ữ ✈➟②✱ ✤ë s➙✉ ❝õ❛ M tr♦♥❣ I ✤÷đ❝ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ s❛✉
R

depth(I, M) = inf i | Exti R (R/I, M ) = 0 .


✶✻

❚r♦♥❣ tr÷í♥❣ ❤đ♣ (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ t❤➻
depth(m, M ) ✤÷đ❝ ❣å✐ ❧➔ ✤ë s➙✉ ❝õ❛ M ✈➔ ❦➼ ❤✐➺✉ ❜ð✐ depth(M ).
▼➺♥❤ ✤➲ s❛✉ ✤➙② ❣✐ó♣ ❝❤ó♥❣ t❛ ❝â ♠ët ❝ỉ♥❣ t❤ù❝ q✉② ♥↕♣ ✤➸ t➼♥❤ ✤ë s➙✉✳

▼➺♥❤ ✤➲ ✶✳✸✳✹✳ ✭❬✶✶❪✱ ▼➺♥❤ ✤➲ ✶✳✷✳✶✵✮ ❈❤♦ M ❧➔ R−♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳

❑❤✐ ✤â✱ ♥➳✉ x1, x2, . . . , xr ❧➔ ♠ët ❞➣② ❝❤➼♥❤ q✉② ❝õ❛ M tr♦♥❣ I t❤➻
depth(I, M/(x1 , x2 , . . . , xr )M ) = depth(I, M ) − r.

❇ê ✤➲ ✶✳✸✳✺✳ ❈❤♦ M ❧➔ ♠ët R−♠æ✤✉♥ ✳ ❑❤✐ ✤â✱ ♥➳✉ a1, . . . , an ❧➔ ♠ët
M −❞➣② ❝❤➼♥❤ q✉② ✈➔ a1 α1 +· · ·+an αn = 0 ✈ỵ✐ αi ∈ M

✈ỵ✐ ♠å✐ i = 1, n.

t❤➻ αi ∈ a1M +· · ·+anM,

❈❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ n✳
✰✮ ❱ỵ✐ n = 1✱ ❣✐↔ sû a1 ❧➔ ♣❤➛♥ tû M −❝❤➼♥❤ q✉② ✈➔ a1α1 = 0, α ∈ M ✳ ❙✉②
r❛ α1 = 0 ❞♦ ✤â α1 ∈ a1M ✳
✰✮ ●✐↔ sû ❜ê ✤➲ tr➯♥ ✤ó♥❣ ✈ỵ✐ n − 1✱ tù❝ ❧➔ ♥➳✉ a1, . . . , an−1 ❧➔ ♠ët M −❞➣②

❝❤➼♥❤ q✉② ✈➔ a1α1 +· · ·+an−1αn−1 = 0 ✈ỵ✐ αi ∈ M t❤➻ αi ∈ a1M +· · ·+an−1M,
✈ỵ✐ ♠å✐ i = 1, n − 1✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♥â ❝ơ♥❣ ✤ó♥❣ ✈ỵ✐ n✱ ❣✐↔ sû a1, . . . , an
❧➔ ♠ët M −❞➣② ❝❤➼♥❤ q✉② ✈➔ a1α1 + · · · + anαn = 0 ✈ỵ✐ αi ∈ M ✱ t❛ ✤✐
❝❤ù♥❣ ♠✐♥❤ αi ∈ M t❤➻ αi ∈ a1M + · · · + anM, ✈ỵ✐ ♠å✐ i = 1, n✳ ❚❤➟t ✈➟②✱ ✈➻
a1 α1 +· · ·+an αn = 0 ♥➯♥ an αn = −(a1 α1 +· · ·+an−1 αn−1 ) ∈ a1 M +· · ·+an−1 M
❞♦ ✤â an(αn + (a1M + · · · + an−1M )) = 0✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❣✐↔ t❤✐➳t a1, . . . , an
❧➔ ♠ët M −❞➣② ❝❤➼♥❤ q✉② ♥➯♥ an ❧➔ ♣❤➛♥ tû M/ (a1, ..., an−1) M −❝❤➼♥❤ q✉②✱
❞♦ ✤â αn + (a1, ..., an−1) M = 0 s✉② r❛ αn ∈ (a1, ..., an−1) M ⊂ (a1, ..., an) M ✱
s✉② r❛ αn = a1β1 + a2β2 + ... + an−1βn−1, βi ∈ M, ✈ỵ✐ ♠å✐ i = 1, n − 1✳ ❑❤✐ ✤â✱
a1 α1 + · · · + an αn = a1 α1 + · · · + an−1 αn−1 + an (a1 β1 + a2 β2 + ... + an−1 βn−1 ) = 0

❤❛② a1 (α1 + anβ1) + a2 (α2 + anβ2) + ... + an−1 (αn−1 + anβn−1) = 0✳ ▼➦t ❦❤→❝✱
❞♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ ♥➯♥ t❛ ❝â ai (αi + anβi) ∈ (a1M + · · · + an−1M )✱ ✈ỵ✐ ♠å✐
i = 1, n − 1✳ ❱➟② αi ∈ a1 M + · · · + an M, ✈ỵ✐ ♠å✐ i = 1, n✳
❱➟② ❜ê ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳




ỵ a1, . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉② t❤➻ aδ1 , . . . , aδn ❧➔ M −❞➣②
1

n

❝❤➼♥❤ q✉②✳

❈❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ỵ tr t ự r ♥➳✉
a1 , . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉② t❤➻ aδ1 , . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉②✳ ❚❤➟t ✈➟②✱
❣✐↔ sû aδ1 , . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉②✳ ✣➦t M1 = M/a1δ M ❦❤✐ ✤â a2, . . . , an
❧➔ M1−❞➣② ❝❤➼♥❤ q✉②✱ s✉② r❛ aδ2 , . . . , an ❧➔ M1−❞➣② ❝❤➼♥❤ q✉②✳ ❈ù t✐➳♣ tư❝

q✉→ tr➻♥❤ ♥❤÷ ✈➟②✱ t❛ ✤÷đ❝ aδn ❧➔ M/ a1δ , a2δ , . . . , an−1δ M −❞➣② ❝❤➼♥❤
q✉② ✈ỵ✐ M = a1δ + a2δ + · · · + anδ M ✳
❇➙② ❣✐í t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤✱ ♥➳✉ a1, . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉② t❤➻
aδ1 , . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉② ❜➡♥❣ q✉② ♥↕♣✳ ❱ỵ✐ δ = 1 ❞➣② tr➯♥ ✤ó♥❣✳
❱ỵ✐ δ > 1✱ ❣✐↔ sû ❞➣② tr➯♥ ✤ó♥❣ ✈ỵ✐ δ − 1✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❞➣② tr➯♥ ✤ó♥❣ ✈ỵ✐
δ✳
❉♦ a1 ❧➔ ♠ët ♣❤➛♥ tû ❝❤➼♥❤ q✉② ❝õ❛ M ♥➯♥ t❛ ❝â aδ1 ❧➔ ♠ët ♣❤➛♥ tû ❝❤➼♥❤ q✉②
❝õ❛ M ✳ ❱ỵ✐ i > 1✱ t❛ ❝❤ù♥❣ ♠✐♥❤ aδ1 , . . . , ai ❧➔ ♠ët M −❞➣② ❝❤➼♥❤ q✉②✳ ❱ỵ✐ ω ∈
M ✱ ❣✐↔ sû ai (ω+(aδ1 , . . . , ai−1 )M ) = 0✱ s✉② r❛ ai ω = aδ1 ξ1 +· · ·+ai−1 ξi−1 , ξi ∈ M ✳
δ−1
❱➻ (aδ1, . . . , ai−1)M ⊂ (aδ−1
1 , . . . , ai−1 )M ♥➯♥ ai ω ∈ (a1 , . . . , ai−1 )M. ❍ì♥
♥ú❛✱ ❞♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ (a1δ−1, . . . , ai−1, ai) ❧➔ M −❞➣② ❝❤➼♥❤ q✉② ♥➯♥
δ−1
ω ∈ (aδ−1
1 , . . . , ai−1 )M ✳ ❉♦ ✤â ω = a1 η1 + · · · + ai−1 ηi−1 ηi ∈ M ✳ ❙✉② r❛ ai ω =
δ−1
a1 aδ−1
1 η1 + · · · + ai ai−1 ηi−1 ✳ ❱➟② 0 = a1 (a1 ξ1 − a1 η1 ) + · · · + ai−1 (ξi−1 − ai ηi−1 ).
❉♦ aδ−1
1 , . . . , ai−1 ❧➔ M −❞➣② ❝❤➼♥❤ q✉② ♥➯♥ →♣ ❞ư♥❣ ❇ê ✤➲ ✶✳✸✳✺✱ t❛ ✤÷đ❝
a1 ξ1 − ai η1 ∈ aδ−1
1 M + · · · + ai−1 M. ❙✉② r❛ ai η1 ∈ a1 M + · · · + ai−1 M.
❱➻ a1, . . . , ai−1, ai ❧➔ M −❞➣② ❝❤➼♥❤ q✉② ♥➯♥ η1 ∈ a1M + · · · + ai−1M ❤❛②
η1 = a1 η1 + · · · + ai−1 ηi−1 ✳ ❉♦ ✤â✱
1

1

2

n

1

1

2

2

n−1

n

1

δ−1
ω = aδ1 η1 + aδ−1
1 a2 η2 + · · · + ai−1 ai−1 ηi−1 + a2 η2 + · · · + ai−1 ηi−1 .
δ−1
= aδ1 η1 + a2 (aδ−1
1 η2 + η2 ) + · · · + ai−1 (ai−1 ηi−1 + ηi−1 ).

∈ aδ1 M + a2 M + · · · + ai−1 M.

❙✉② r❛ ai ❧➔ ♣❤➛♥ tû ❝❤➼♥❤ q✉② ❝õ❛ M/(aδ1, a2, · · · , ai−1)M ✳ ◆❤÷ ✈➟②✱ ♥➳✉


✶✽

❧➔ M −❞➣② ❝❤➼♥❤ q✉② t❤➻ aδ1 , a2, . . . , an ❧➔ M −❞➣② ❝❤➼♥❤ q✉② t❤➻
aδ1 , a2 , . . . , an ❧➔ M q ỵ ữủ ự
a1 , · · · , an

1

1

❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ❞➣② ❧å❝ ❝❤➼♥❤ q✉② ❧➔ ♠ët tr♦♥❣
♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ♠ð rë♥❣ ❝õ❛ ❞➣② ❝❤➼♥❤ q✉② ✤÷đ❝ ◆✳ ❚✳ ữớ
r P ợ t ✶✾✼✽✱ ❬✹❪✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✼✳ ▼ët ❞➣② ❝→❝ ♣❤➛♥ tû x1, x2, . . . , xn ∈ m ✤÷đ❝ ❣å✐ ❧➔
❞➣② ❧å❝ ❝❤➼♥❤ q✉② ❝õ❛ M ❤❛② M ✲❞➣② ❧å❝ ❝❤➼♥❤ q✉② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐
p ∈ Ass (M/ (x1 , x2 , . . . , xi−1 ) M ) \ {m} t❤➻ xi ∈
/ p, ✈ỵ✐ ♠å✐ i = 1, 2, . . . , n.
▼ët ♣❤➛♥ tû x ∈ m ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ❧å❝ ❝❤➼♥❤ q✉② ✤è✐ ✈ỵ✐ M ❤❛② ♣❤➛♥
tû M −❧å❝ ❝❤➼♥❤ q✉② ♥➳✉ x ∈/ p ✈ỵ✐ ♠å✐ p ∈ AssR(M)\{m}.
❚ø ✤à♥❤ ♥❣❤➽❛ tr➯♥ t❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉✿
✭✐✮

x∈m

❧➔ M −♣❤➛♥ tû ❧å❝ ❝❤➼♥❤ q✉② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x ∈/

p
p∈AssR (M )\{m}

✭✐✐✮ ◆➳✉ x1, x2, . . . , xn ∈ m ❧➔ M ✲❞➣② ❝❤➼♥❤ q✉② t❤➻ ♥â ❝ô♥❣ ❧➔ M −❞➣② ❧å❝

❝❤➼♥❤ q✉②✳
✭✐✐✐✮ ❉➣② x1, x2, . . . , xn ∈ m ❧➔ M ✲❞➣② ❧å❝ ❝❤➼♥❤ q✉② ♥➳✉ xi ❧➔ ♣❤➛♥ tû ❧å❝
❝❤➼♥❤ q✉② ❝õ❛ M/ (x1, x2, . . . , xi−1) M ✱ ✈ỵ✐ ♠å✐ i = 1, 2, . . . , n.
✭✐✈✮ ▼ët ❞➣② ❝→❝ ♣❤➛♥ tû x1, x2, . . . , xn ∈ m ❧➔ M ✲❞➣② ❧å❝ ❝❤➼♥❤ q✉② ♥➳✉ ✈➔
❝❤➾ ♥➳✉ x1 ❧➔ M −♣❤➛♥ tû ❧å❝ ❝❤➼♥❤ q✉② ✈➔ x2, . . . , xn ❧➔ M/x1M −❞➣②
❧å❝ ❝❤➼♥❤ q✉②✳

▼➺♥❤ ✤➲ ✶✳✸✳✽✳ ❈❤♦ ❞➣② ❝→❝ ♣❤➛♥ tû x1, x2, . . . , xn ∈ m✳ ❑❤✐ ✤â
✭✐✮

❧➔ M −❞➣② ❧å❝ ❝❤➼♥❤ q✉② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x1/1, x2/1, . . . , xn/1
❧➔ Mp−❞➣② ❝❤➼♥❤ q✉② ✈ỵ✐ ♠å✐ p ∈ Supp(M ) \ {m} ❝❤ù❛ x1, x2, . . . , xn✳

x1 , x2 , . . . , xn

✭✐✐✮ ◆➳✉ x1, x2, . . . , xn ❧➔ M ✲❞➣② ❧å❝ ❝❤➼♥❤ q✉② t❤➻ xα1 , xα2 , . . . , xαn ❝ơ♥❣ ❧➔
M ✲❞➣② ❧å❝ ❝❤➼♥❤ q✉② ✈ỵ✐ ♠å✐ α1 , α2 , . . . , αn ∈ N✳
1

2

n


✶✾

❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❍✐➸♥ ♥❤✐➯♥✳
✭✐✐✮ ❚❤❡♦ ✭✐✮ t❛ ❝â x1/1, x2/1, . . . , xn/1 ❧➔ Mp✲❞➣② ❝❤➼♥❤ q✉② ✈ỵ✐ ♠å✐ p ∈
{p | p ∈ Supp (M ) \ {m} , x1 , x2 , ..., xn ∈ p}✳ ❙✉② r❛ xα1 /1, xα2 /1, . . . , xαn /1 ❧➔
Mp ✲❞➣② ❝❤➼♥❤ q✉② ✈ỵ✐ ♠å✐ p ∈ {p | p ∈ Supp (M ) \ {m} , x1 , x2 , ..., xn ∈ p}✳

❉♦ ✤â✱ t❤❡♦ ✭✐✮ t❛ ❧↕✐ ❝â xα1 , xα2 , . . . , xαn ❧➔ M ✲❞➣② ❧å❝ ❝❤➼♥❤ q✉②✳
❱➟② t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
1

1

2

2

n

n

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✾✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ t❤ü❝ sü ❝õ❛ R✳ ❑❤✐ ✤â
✭✐✮ ▼ët

❞➣② ❧å❝ ❝❤➼♥❤ q✉② x1, x2, . . . , xn tr♦♥❣ I ❣å✐ ❧➔ tè✐ ✤↕✐ ♥➳✉
x1 , x2 , . . . , xn , xn+1 ❦❤æ♥❣ ❧➔ M −❞➣② ❧å❝ ❝❤➼♥❤ q✉② ✈ỵ✐ ❜➜t ❦➻ xn+1 ∈ I.
M−

✭✐✐✮ ✣ë s➙✉ ❧å❝ ❝õ❛ M tr♦♥❣ I ❧➔ ✤ë ❞➔✐ ❝õ❛ ♠ët M −❞➣② ❧å❝ ❝❤➼♥❤ q✉② tè✐
✤↕✐ ❜➜t ❦ý tr♦♥❣ I ✱ ❦➼ ❤✐➺✉ ❧➔ f depth(I, M ).
◗✉② ữợ tr I ổ tỗ t M ồ ❝❤➼♥❤ q✉② tè✐ ✤↕✐ ♥➔♦ ❝↔
t❤➻ f depth(I, M ) = ∞.

✶✳✹ ❉➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ ❦
❑❤→✐ ♥✐➺♠ ♣❤➛♥ tû ✤è✐ ❝❤➼♥❤ q✉② ✈➔ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ✤➣ ✤÷đ❝ ❆✳❖♦✐s❤✐
❬✶✽❪ ✤÷❛ r❛ ✈➔♦ ♥➠♠ ✶✾✼✻✳ ◆❤➔♥ ✈➔ ❍♦➔♥❣ ❬✶✸❪ ✤➣ ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ ♥➔②
❧➯♥ t❤➔♥❤ ❞➣② ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k✱ ✈ỵ✐ k ≥ −1 ❧➔ ♠ët sè ♥❣✉②➯♥✳

❙❛✉ ✤â✱ tr♦♥❣ ❬✶✷❪ ◆❤➔♥ ✈➔ ❉✉♥❣ ✤➣ tr➻♥❤ ❜➔② t❤➯♠ ❝→❝ ♠➺♥❤ ✤➲ ✈➲ ❞➣②
✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k✳ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ ❧✉ỉ♥ ①➨t (R, m) ❧➔ ✈➔♥❤
◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ A ❧➔ R−♠æ✤✉♥ ❆rt✐♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M ❧➔ ♠ët

♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✱ A ❧➔ ♠ët R−♠æ✤✉♥ ❆rt✐♥ ✈➔ k ≥ −1 ❧➔ ♠ët sè
♥❣✉②➯♥✳ ❑❤✐ ✤â✱

R−

✭✐✮ ▼ët ♣❤➛♥ tû x ∈ m ữủ ồ tỷ Aố q ợ
ỡ k ♥➳✉ x ∈/ p✱ ✈ỵ✐ ♠å✐ p ∈ AttR A t❤ä❛ dim (R/p) > k.


✷✵

✭✐✐✮ ▼ët ❞➣② (x1, ..., xn) ❝→❝ ♣❤➛♥ tû tr♦♥❣ m ✤÷đ❝ ❣å✐ ❧➔ ♠ët A−❞➣② ✤è✐
❝❤➼♥❤ q✉② ❝❤✐➲✉ > k ♥➳✉ xi ❧➔ ♣❤➛♥ tû (0 :A (x1, . . . , xi−1)R)−✤è✐ ❝❤➼♥❤
q✉② ❝❤✐➲✉ > k tù❝ ❧➔ xi ∈/ p ✈ỵ✐ ♠å✐ p ∈ AttR(0 :A (x1, . . . , xi−1)R) t❤ä❛
♠➣♥ dim(R/p) > k✱ ✈ỵ✐ ộ i = 1, n

ú ỵ r trữớ ủ k = −1✱ t❤➻ ♣❤➛♥ tû A−✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉

❧ỵ♥ ❤ì♥ k ❝❤➼♥❤ ❧➔ ♣❤➛♥ tû A−✤è✐ ❝❤➼♥❤ q✉② ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❆✳❖♦✐s❤✐ ❬✶✽❪✳
❉♦ ✤â✱ ♠ët ♣❤➛♥ tû x ∈ R ❧➔ ♣❤➛♥ tû A−✤è✐ ❝❤➼♥❤ q✉② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
(0 :A x) = 0 ✈➔ xA = A.

❇ê ✤➲ ✶✳✹✳✸✳ ✭❬✽❪✮ ❚➟♣ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❝ü❝ t✐➸✉ ❝õ❛ AttR(M ) ❧➔ t➟♣ t➜t

❝↔ ♣❤➛♥ tû ❝ü❝ t✐➸✉ ❝õ❛ Var(AnnR(M ))✳ ❍ì♥ ♥ú❛✱

dimR (M ) = max {dim(R/p) | p ∈ AttR (M )} .

▼➺♥❤ ✤➲ ✶✳✹✳✹✳ ❬✶✷❪ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M ❧➔ ♠ët
R−♠æ✤✉♥

❑❤✐ ✤â✱

❤ú✉ ❤↕♥ s✐♥❤✱ A ❧➔ R−♠æ✤✉♥ ❆rt✐♥ ✈➔ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳

✭✐✮ ◆➳✉ dim(A/IA) ≤ k t tỗ t ởt tỷ Aố q
> k tr♦♥❣ I ✳
✭✐✐✮ ◆➳✉ x ∈ m ❧➔ ♠ët ♣❤➛♥ tû A−✤è✐ ❝❤➼♥❤ q✉② ✈ỵ✐ ❝❤✐➲✉ > k tr♦♥❣ I t
dim(A/xA) k
ự sỷ tỗ t ♠ët ♣❤➛♥ tû p ∈ AttR (A) s❛♦ ❝❤♦ I ⊆ p
✈➔ dim (R/p) > k✳ ❱➻ p ∈ AttR (A) tỗ t ởt ổ tữỡ
A/B = 0 ừ A s❛♦ ❝❤♦ p = AnnR (A/B) s✉② r❛ ✈ỵ✐ ♠å✐ x ∈ p t❤➻
x (A/B) = 0 ❤❛② xA ⊂ B ✱ ❞♦ ✤â pA ⊆ B ✳ ❱➔ ✈➻ I ⊆ p ♥➯♥ IA ⊆ pA ⊆ B
s✉② r❛ IA ⊆ B ✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✹✳✸ t❛ ✤÷đ❝ dim (A/IA) ≥ dim (A/B) =
dim (R/ AnnR (A/B)) = dim (R/p) > k ✱ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈ỵ✐
❣✐↔ t❤✐➳t dim (A/IA) ≤ k✳ ❉♦ ✤â I p✱ ✈ỵ✐ ♠å✐ p ∈ AttR (A) t❤ä❛
dim (R/p) > k, tỗ t ởt tỷ x ∈ m s❛♦ ❝❤♦


✷✶

✱ ✈ỵ✐ ♠å✐ p ∈ AttR (A) t❤ä❛ dim (R/p) > k. ❱➟② t❤❡♦ ✤à♥❤ ♥❣❤➽❛ x
❧➔ ♠ët ♣❤➛♥ tû A−✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k tr♦♥❣ I ✳

x∈
/p

✭✐✐✮ ●✐↔ sû r➡♥❣ ❆ ❝â ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t❤✐➸✉ ♥❤÷ s❛✉ A = A1 +A2 +
· · ·+An ✱ tr♦♥❣ ✤â Ai ❧➔ ❝→❝ pi −t❤ù ❝➜♣✳ ❱➻ ❣✐↔ t❤✐➳t x ❧➔ ♣❤➛♥ tû A−✤è✐
❝❤➼♥❤ q✉② ✈ỵ✐ ❝❤✐➲✉ > k ♥➯♥ x ∈/ pi ✈ỵ✐ ♠é✐ i t❤ä❛ ♠➣♥ dim(R/pi) >
k ✈➔ xAi = Ai ✳ ❉♦ ❞â A/xA ❧➔ ♠ët t❤÷ì♥❣ ❝õ❛ A/ x∈/ p Ai ữ ỵ
r A/ x/p Ai ởt tữỡ ừ x∈p Ai. ▼➔ dim( x∈p Ai) =
max dim (R/pi ) ≤ k ✳ ❱➟② dim(A/xA) ≤ k ✳
x∈p
▼➺♥❤ ✤➲ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤
i

i

i

i

i

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✺✳ ❈❤♦ M ❧➔ ♠ët R−♠ỉ✤✉♥ ✳ ❑❤✐ ✤â ❤➔♠ tû ❞➝♥ ①✉➜t

♣❤↔✐ t❤ù n ❝õ❛ Hom(−, M ) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ tû ♠ð rë♥❣ tự n ỵ
Extn R (, M )
t ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ N
∂n

∂1

∂0

X : . . . −−−→ Xn −−−→ Xn−1 −−→ . . . −−−→ X1 −−−→ X0 −−−→ N −−−→ 0.

❑❤✐ ✤â ♣❤ù❝ t❤✉ ❣å♥ ❝õ❛ N ❧➔
∂n

∂1

X : . . . −−−→ Xn −−−→ Xn−1 −−→ . . . −−−→ X1 −−−→ X0 −−−→ 0.

❚→❝ ✤ë♥❣ ❤➔♠ tû Hom (−, M ) ✈➔♦ ❞➣② ♣❤ù❝ tr t ữủ ỷ ợ s
0

1

Hom(X, M ) : 0 −−−→ Hom(X0 , M ) −−−−→ Hom(X1 , M ) −−−−→ . . .
δ n−1

−−−→ Hom(Xn−1 , M ) −−−−−→ Hom(Xn , M ) −−−→ . . . .

tr♦♥❣ ✤â δn = Hom(∂n, 1).
❑❤✐ ✤â ✈ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ n✱ t❛ ❝â
Extn (N, M ) = H n Hom X, M

= kerδ n /Imδ n−1 .

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✻✳ ❈❤♦ N ❧➔ ♠ët R−♠æ✤✉♥✳ ❑❤✐ ✤â ❤➔♠ tû ❞➝♥ ①✉➜t

tr→✐ t❤ù n ❝õ❛ ❤➔♠ tû − ⊗R N ✤÷đ❝ ồ tỷ tự n ỵ

TorR
n (, N )✳


✷✷

❱ỵ✐ ♠å✐ R−♠ỉ✤✉♥ M ✱ t❛ ❝â ♠ỉ✤✉♥ TorRn(M, N ) ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ①♦➢♥
t❤ù n ❝õ❛ M ✈➔ N ✳
❳➨t ❣✐↔✐ t❤ù❝ ①↕ ↔♥❤ ❝õ❛ M
∂n+1

∂n

∂1

ε

· · · −−→ Xn+1 −−−→ Xn −−−→ · · · −−→ X1 −−→ X0 −−−→ M −−→ 0.

❚❛ ✤÷đ❝ ❣✐↔✐ t❤ù❝ ①↕ ↔♥❤ t❤✉ ❣å♥
∂n

∂1

∂0

X : . . . −−−→ Xn −−−→ Xn−1 −−→ . . . −−−→ X1 −−−→ X0 −−−→ 0.

❚→❝ ✤ë♥❣ ❤➔♠ tû − ⊗ N ✈➔♦ ❞➣② ❦❤ỵ♣ tr➯♥ t❛ ❝â ♣❤ù❝
∂n ⊗1

∂1 ⊗1

∂0 ⊗1

· · · Xn ⊗ N −−−→ · · · −−→ X1 ⊗ N −−−→ X0 ⊗ N −−−→ 0.

❑❤✐ ✤â✱ TorRn(M, N ) = Ker(∂n ⊗ 1)/Im(∂n+1 ⊗ 1) ✳

❇ê ✤➲ ✶✳✹✳✼✳ ✭❬✶✷❪✮ ❈❤♦ n ≥ 1 ❧➔ ♠ët sè ♥❣✉②➯♥✳ ❑❤✐ ✤â ❤❛✐ ✤✐➲✉ ❦✐➺♥ s❛✉

❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮

dim Tori R (R/I, A) k,

ợ ồ i < n.

ỗ t ởt A ✤è✐ ❝❤➼♥❤ q✉② ❝❤✐➲✉ ❧ỵ♥ ❤ì♥ k tr♦♥❣ I ❝â ✤ë ❞➔✐
n✳

▼➺♥❤ ✤➲ ✶✳✹✳✽✳ ❬✶✷❪ ❈❤♦ A ❧➔ ♠ët R−♠æ✤✉♥ ❆rt✐♥✳ ❑❤✐ ✤â ♥➳✉ p ∈
AttR (A)

t❤➻ AnnR (0 :A p) = p.

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ AttRA = p ∩ R : p ∈ AttRA ♥➯♥ ✈ỵ✐ p ∈ AttR (A) t s
tỗ t p AttR A s p ∩ R = p. ❱➔ ✈➻ p ∈ AttR A ♥➯♥ p ∈ Var(AnnR A)
s✉② r❛ p ⊇ AnnR A. ▼➔ AnnR A = AnnR D(A) ♥➯♥ p ⊇ AnnR D(A) ❞♦ ✤â
p ∈ Var(AnnR D(A))✳ ❙✉② r❛ AnnR D(A)/pD(A) = p✳

❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ✤➛② ✤õ t❤➻ AnnR 0 :A p = p. ❉♦ ✤â
p ⊆ AnnR (0 :A p) ⊆ R ∩ AnnR 0 :A p = R ∩ p = p.

❱➟② AnnR (0 :A p) = p.