Tính minimax và tính cofinite của môđun đối đồng điều địa phương

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

❚❘❺◆ ❚❍➚ ❚❍❯ ❍❖⑨■

❚➑◆❍ ▼■◆■▼❆❳ ❱⑨ ❚➑◆❍ ❈❖❋■◆■❚❊ ❈Õ❆ ▼➷✣❯◆
✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✶✽


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

❚❘❺◆ ❚❍➚ ❚❍❯ ❍❖⑨■

❚➑◆❍ ▼■◆■▼❆❳ ❱⑨ ❚➑◆❍ ❈❖❋■◆■❚❊ ❈Õ❆ ▼➷✣❯◆
✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆●
◆❣➔♥❤✿ ✣↕✐ sè ✈➔ ❧þ t❤✉②➳t sè
▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✹

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❈→♥ ❜ë ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
P●❙✳❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❍♦➔♥❣

❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✶✽
✐


▲❮■ ❈❆▼ ✣❖❆◆
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔
tr✉♥❣ t❤ü❝ ✈➔ ❦❤æ♥❣ trò♥❣ ❧➦♣ ✈î✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ♠å✐ sü
❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷ñ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤
❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳
❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✻ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✶✽
❚→❝ ❣✐↔
❚r➛♥ ❚❤à ❚❤✉ ❍♦➔✐

❳→❝ ♥❤➟♥
❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❝❤✉②➯♥ ♠æ♥

❳→❝ ♥❤➟♥
❝õ❛ ❝→♥ ❜ë ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝

✐✐


ớ ỡ

ữủ t t ữợ sỹ ữợ ừ
P ổ ữủ tọ ỏ trồ t ỡ
s s tợ t ỳ ồ qỵ tứ tr ỳ ồ tr
ở số t ú tổ tỹ t ỡ trữ t ỡ
ổ ỡ Pỏ ồ ữ P t

tổ t sợ õ ồ
ổ tọ ỏ t ỡ tợ tt t ổ ồ
t t ợ ỳ t t t t
ỡ t ổ ổ q t ú ù tổ tr sốt q tr
ồ t t tổ t ờ sr ợ ồ
ữỡ tr
ổ ỡ tt ở ú ù tổ
t t tr q tr ồ
ổ ữủ ỷ ỡ tợ tt t tr t
tổ ữủ ồ t ự t




▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥
▲í✐ ❝↔♠ ì♥
▼ð ✤➛✉
❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ▼æ✤✉♥ ◆♦❡t❤❡r ✈➔ ▼æ✤✉♥ ❆rt✐♥ ✳
✶✳✸ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹ ▼æ✤✉♥ Ext ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✺ ▼æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣

✳
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✐✐
✐✐✐
✶
✺

✳ ✺
✳ ✻
✳ ✽
✳ ✶✵
✳ ✶✷

❈❤÷ì♥❣ ✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✶ ✈➔ t➼♥❤ ♠✐♥✐♠❛① ❝õ❛ ♠æ✤✉♥
✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣
✶✺
✷✳✶ ▼æ✤✉♥ ♠✐♥✐♠❛① ✈➔ ♠æ✤✉♥ ❝♦❢✐♥✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ♠ët ✈➔ t➼♥❤ ❝❤➜t ♠✐♥✐♠❛① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

❈❤÷ì♥❣ ✸ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✷ ✈➔ t➼♥❤ ▲❛s❦❡r ②➳✉

✷✼

✸✳✶ ▼æ✤✉♥ ▲❛s❦❡r ②➳✉ ✈➔ ♠æ✤✉♥ ❝♦❢✐♥✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ❤❛✐ ✈➔ t➼♥❤ ❝❤➜t ▲❛s❦❡r ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✐✈

▼ð ✤➛✉
❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✭❝â ✤ì♥ ✈à✮✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔
M ❧➔ R ✲ ♠æ✤✉♥ ❦❤→❝ 0✳ ❱î✐ ♠é✐ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ i ❝❤♦ tr÷î❝✱ t❛ ❝â ♠æ✤✉♥
✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ t❤ù i ❝õ❛ M ✤è✐ ✈î✐ ❣✐→ ❧➔ ✐✤➯❛♥ I ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛
❜ð✐ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✭①❡♠ ❬✶✶❪ ❤♦➦❝ ❬✽❪✮ ♥❤÷ s❛✉✿
i
n
HIi (M ) = −
lim
→ ExtR (R/I , M ).
n≥1

❈→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❧î♣ ♠æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ ❝â t❤➸ ①❡♠ t❤➯♠
tr♦♥❣ ❝✉è♥ s→❝❤ ❬✽❪✳
▼ët ✤à♥❤ ❧þ q✉❛♥ trå♥❣ tr♦♥❣ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ ❧➔ ✧◆❣✉②➯♥ ❧þ
✤à❛ ♣❤÷ì♥❣ ✲ t♦➔♥ ❝ö❝ ❝❤♦ ❝❤✐➲✉ ❤ú✉ ❤↕♥ ❝õ❛ ❝→❝ ♠æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛
♣❤÷ì♥❣✧ ✭①❡♠ ❬✶✵✱ ✣à♥❤ ❧þ ✶❪ ✲ ❜➔✐ ❜→♦ ❝õ❛ ●✳ ❋❛❧t✐♥❣s✮ ♣❤→t ❜✐➸✉✿ ✧❱î✐ ♠ët
i
sè ♥❣✉②➯♥ ❞÷ì♥❣ r ✤➣ ❝❤♦✱ ❝→❝ Rp✲♠æ✤✉♥ HIR
(Mp ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈î✐ ♠å✐
i ≤ r ✈➔ ♠å✐ p ∈ Spec R ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ R✲♠æ✤✉♥ HIi (M ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤
✈î✐ ♠å✐ i ≤ r✧✳
❈â ♠ët ❞↕♥❣ tr➻♥❤ ❜➔② ❦❤→❝ ❝❤♦ ♣❤→t ❜✐➸✉ ❝õ❛ ♥❣✉②➯♥ ❧þ ✤à❛ ♣❤÷ì♥❣ ✲
t♦➔♥ ❝ö❝ ❝õ❛ ❋❛❧t✐♥❣s ♠➔ t❛ q✉❛♥ t➙♠ ð ✤➙②✱ ❧✐➯♥ q✉❛♥ ✤➳♥ sü ❦❤→✐ q✉→t ❤â❛
❝❤✐➲✉ ❤ú✉ ❤↕♥ fI (M ) ❝õ❛ M ✤è✐ ✈î✐ I ✱ tr♦♥❣ ✤â
p

fI (M ) := inf{i ∈ N | HIi (M )

❦❤æ♥❣ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤},

ð ✤➙② t❛ q✉② ÷î❝ r➡♥❣ inf(∅) = ∞✳ ❑❤✐ ✤â
0 :R HIi (M ) }

fI (M ) := inf{i ∈ N | I

= inf{i ∈ N | I n HIi (M ) = 0
✶

✈î✐ ♠å✐ n ∈ N};

✭†✮


ỗ tớ ú õ ỵ ữỡ t ử ừ ts ữủ ổ
tự s
fI (M ) = inf{fIRp (Mp ) | p Spec R}
= inf{fIRp (Mp ) | p Supp(M/IM )



dim R/ p 0},

ỵ r ố ỳ số t
ổ ố ỗ ữỡ ợ t ổ ỳ s
số õ ổ ố ỗ q ữỡ õ t
tố tr ỡ s
rr ợ t
ỳ n ừ M ố ợ I fIn(M ) ữủ

ổ tự
fIn (M ) = inf{fIRp (Mp ) | p Supp(M/IM )



dim(R/ p) n}.



ú ỵ r fIn(M ) số ữỡ t õ fI0(M ) = fI (M )
ứ õ ởt ọ tỹ ữủ t r t t t ừ ổ ố
ỗ ữỡ ợ ỳ ừ M ố ợ I
t s
fI1 (M ) = inf{i N | HIi (M )

ổ }

fI2 (M ) = inf{i N | HIi (M )

ổ sr }

õ ú ổ t q ừ rr tr
tr ớ ọ tr ử t t q tự t ừ ồ
ự ữủ r số i ọ t HIi (M ) ổ ổ
ợ số fI1(M ) ỵ t q tự ừ ồ



❧➔ ❝❤➾ r❛ r➡♥❣ sè ♥❣✉②➯♥ i ♥❤ä ♥❤➜t s❛♦ ❝❤♦ HIi (M ) ❦❤æ♥❣ ❧➔ ♠æ✤✉♥ ▲❛s❦❡r
②➳✉ ❜➡♥❣ ✈î✐ fI2(M ) ❦❤✐ R ❧➔ ✈➔♥❤ ♥û❛ ✤à❛ ♣❤÷ì♥❣ ✭①❡♠ ✣à♥❤ ❧þ ✸✳✷✳✸✮✳

❈æ♥❣ ❝ö ✤➸ ❤å ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ t❤ù ♥❤➜t ♥➯✉ tr➯♥ ❧➔ ✤à♥❤ ❧þ
s❛✉ ✤➙②✿

✣à♥❤ ❧þ ✶✳ ✭❬✹✱ ✣à♥❤ ❧þ ✶✳✶❪✮ ❈❤♦

❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ I ❧➔ ♠ët ✐✤➯❛♥
❝õ❛ R ✈➔ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â R✲♠æ✤✉♥ HIi (M ) ❧➔
♠✐♥✐♠❛① ✈➔ I ✲❝♦❢✐♥✐t❡ ✈î✐ ♠å✐ i < fI1(M ) ✈➔ HIf (M )(M ) ❦❤æ♥❣ ❧➔ ♠✐♥✐♠❛①✳
❍ì♥ ♥ú❛✱ ✈î✐ ♠é✐ ♠æ✤✉♥ ❝♦♥ ♠✐♥✐♠❛① N ❝õ❛ HIf (M )(M )✱ t❤➻ R✲♠æ✤✉♥
f (M )
HomR (R/I, HI
(M )/N ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳
R

1
I

1
I

1
I

❑❤→✐ ♥✐➺♠ ♠æ✤✉♥ I ✲❝♦❢✐♥✐t❡ tr♦♥❣ ✤à♥❤ ❧þ tr➯♥ ✤÷ñ❝ ❣✐î✐ t❤✐➺✉ ❜ð✐ ❘✳
❍❛rts❤♦r♥❡ ♥➠♠ ✶✾✼✵ ✭①❡♠ ❬✶✷❪✮ ✈➔ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ R✲♠æ✤✉♥ M
✤÷ñ❝ ❣å✐ ❧➔ I ✲❝♦❢✐♥✐t❡ ♥➳✉ Supp(M ) ⊆ V (I) ✈➔ ExtiR(R/I, M ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤
✈î✐ ♠å✐ i ≥ 0✳
▼ët tr♦♥❣ ❝→❝ ❝æ♥❣ ❝ö ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝❤➼♥❤ t❤ù ❤❛✐ ❝õ❛
❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲❙❡❞❣❤✐ ❬✹❪ ❧➔ ✤à♥❤ ❧þ ❞÷î✐ ✤➙②✿

✣à♥❤ ❧þ ✷✳ ✭❬✹✱ ✣à♥❤ ❧þ ✶✳✷❪✮ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ R✱ M ❧➔

♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ t ≥ 1 ❧➔ ♠ët sè ♥❣✉②➯♥ s❛♦ ❝❤♦ ❝→❝ R✲♠æ✤✉♥
HI0 (M ), . . . , HIt−1 (M ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✤à❛ ♣❤÷ì♥❣ ✈î✐ ♠å✐ p ∈ Supp(M/IM )
♠➔ dim(R/p) > 1✳ ❑❤✐ ✤â✱ ❝→❝ R✲♠æ✤✉♥ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈î✐ ♠å✐ i ≤ t
✈➔ R✲♠æ✤✉♥ HomR(R/I, HIt (M )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳

❚ø ♥❤ú♥❣ ❦➳t q✉↔ tr➯♥ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲❙❡❞❣❤✐ ❬✹❪ ✤➣ ✤÷❛ r❛ ❝→❝
❤➺ q✉↔ ❝õ❛ ✤à♥❤ ❧þ ✷✱ ✤â ❧➔ ♠ët sè ♠ð rë♥❣ ❝❤♦ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲
◆❛❣❤✐♣♦✉r tr♦♥❣ ❬✼❪✱ ❉❡❧❢✐♥♦✲▼❛r❧❡② tr♦♥❣ ❬✾❪ ✈➔ ❑✳ ■✳ ❨♦s❤✐❞❛ tr♦♥❣ ❬✶✾❪ ✤è✐
✈î✐ ♠ët ✈➔♥❤ ◆♦❡t❤❡r tò② þ✳
✸


✣à♥❤ ❧þ ✸✳ ❬✹✱ ✣à♥❤ ❧þ ✶✳✸❪ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r✱ ■ ❧➔ ✐✤➯❛♥ ❝õ❛ R✱ M

❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ s❛♦ ❝❤♦ dim(M/IM ) ≤ 1✳ ❑❤✐ ✤â R✲♠æ✤✉♥ HIt (M )
❧➔ I ✲❝♦❢✐♥✐t❡ ✈î✐ ♠å✐ sè ♥❣✉②➯♥✳

▼ët ❦➳t q✉↔ ❝❤➼♥❤ ❦❤→❝ ♥ú❛ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪ ✤â ❧➔✿ ◆➳✉ (R, m) ❧➔
✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r ✤➛② ✤õ✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥
❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ R✲♠æ✤✉♥ ExtjR(R/I, HIi (M )) ❧➔ ▲❛s❦❡r ②➳✉ ✈î✐ ♠å✐
i < fI3 (M ) ✈➔ ✈î✐ ♠å✐ j ≥ 0✳ ❍ì♥ ♥ú❛✱ ✈î✐ ♠é✐ ♠æ✤✉♥ ❝♦♥ ▲❛s❦❡r ②➳✉ N ❝õ❛
f (M )
f (M )
HI
(M )✱ t❤➻ t❛ ❝â R✲♠æ✤✉♥ HomR (R/I, HI
(M )/N ) ❝ô♥❣ ❧➔ ▲❛s❦❡r ②➳✉
✭①❡♠ ✣à♥❤ ❧þ ✸✳✷✳✻✮✳
❚ø ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✤➣ t❤✉ ✤÷ñ❝ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r✲

❙❡❞❣❤✐ ♥❤÷ tr➯♥ ✤➙②✱ ✤➲✉ ✤÷❛ ✤➳♥ ❜➔✐ t♦→♥ ①❡♠ ①➨t ✈î✐ ✤✐➲✉ ❦✐➺♥ ♥➔♦ ✤➸ ❝❤♦
t➟♣ ❤ñ♣ AssR(HIi (M )) ❧➔ ❤ú✉ ❤↕♥ ❦❤✐ i = fIj (M ) ✭❝❤➥♥❣ ❤↕♥ ✈î✐ j = 1, 2, 3✮✳
▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔
♥❤÷ ✤➣ ♥➯✉ tr➯♥✱ ❝→❝ ❦✐➳♥ t❤ù❝ ♥➔② ❞ü❛ tr➯♥ ❜➔✐ ❜→♦ ❝❤➼♥❤ ❧➔ ❜➔✐ ❜→♦ ❬✹❪✿
❑✳ ❇❛❤♠❛♥♣♦✉r✱ ❘✳ ◆❛❣❤✐♣♦✉r ❛♥❞ ▼✳ ❙❡❞❣❤✐✱ ▼✐♥✐♠❛①♥❡ss ❛♥❞ ❈♦❢✐♥✐t❡
♣r♦♣❡rt✐❡s ♦❢ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣② ♠♦❞✉❧❡s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ❱♦❧✳ ✹✶
✭✷✵✶✸✮✱ P♣✳ ✷✼✾✾✲✷✽✶✹✳ ✭❉❖■✿ ✶✵✳ ✶✵✽✵✴✵✵✾✷✼✽✼✷✳✷✵✶✷✳✻✻✷✼✵✾✮✳ ❇➯♥ ❝↕♥❤ ✤â
✤➸ ✈✐➺❝ tr➻♥❤ ❜➔② ✤÷ñ❝ ✤➛② ✤õ ✈➔ rã þ ❤ì♥✱ ❧✉➟♥ ✈➠♥ t❤❛♠ ❦❤↔♦ t❤➯♠ ♥❤✐➲✉
❦✐➳♥ t❤ù❝ ð ❜➔✐ ❜→♦ ❬✺❪✱ ❬✻❪✱ ❬✼❪✱ ❬✶✼❪✱✳ ✳ ✳ ❀ ✈➔ ❝→❝ ❝✉è♥ s→❝❤ ❬✽❪ ✈➔ ❬✶✺❪✳
▲✉➟♥ ✈➠♥ ✤÷ñ❝ ❜è ❝ö❝ ❧➔♠ ❜❛ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥
t❤ù❝ ❝ì sð ❝➛♥ t❤✐➳t ✤➸ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❝❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✶ ❝õ❛ ♠æ✤✉♥ M ✤è✐ ✈î✐ ✐✤➯❛♥ I
tr♦♥❣ ♠è✐ ❧✐➯♥ ❤➺ ✈î✐ t➼♥❤ ❝❤➜t ♠✐♥✐♠❛① ❝õ❛ ♠æ✤✉♥✳ ❈❤÷ì♥❣ ✸ ❝õ❛ ❧✉➟♥ ✈➠♥
t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ✈➲ ❝❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ✷ ❝õ❛ M ✤è✐ ✈î✐ ✐✤➯❛♥ I ✈➔ t➼♥❤ ❝❤➜t
▲❛s❦❡r ②➳✉ ❝õ❛ ♠æ✤✉♥✳
3
I

3
I

✹


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
Ð ❝❤÷ì♥❣ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❈→❝ ❦✐➳♥
t❤ù❝ ð ❝❤÷ì♥❣ ♥➔② ✤÷ñ❝ tr➻♥❤ ❜➔② ❞ü❛ ✈➔♦ ❝→❝ ❝✉è♥ s→❝❤ ❬✽❪ ✈➔ ❬✶✺❪✳

✶✳✶ ■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ✭■✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✮✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥✱ p ❧➔ ✐✤➯❛♥

♥❣✉②➯♥ tè ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â p ✤÷ñ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M
♥➳✉ tç♥ t↕✐ ♠ët ♣❤➛♥ tû 0 = x ∈ M s❛♦ ❝❤♦ AnnR(x) = p✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝
✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ AssR(M ) ❤♦➦❝ Ass(M )✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭✣❛ t↕♣ ❝õ❛ ✐✤➯❛♥✮✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✱ ❦❤✐ ✤â ✤❛

t↕♣ ❝õ❛ I ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ V (I) ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

V (I) = {p ∈ Spec(R) | I ⊆ p} .

▼➺♥❤ ✤➲ ✶✳✶✳✸✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥ ✈➔ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â t❛ ❝â
✐✮ AssR(0 :M I) = AssR(M ) ∩ V (I)✳
✐✐✮ AssR(M/(0 :M I)) ⊆ AssR(M )✳
✺


✐✐✐✮ ❈❤♦ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ R✲♠æ✤✉♥ M ✳ ❑❤✐ ✤â
Ass(M ) ⊆ Ass(N ) ∪ Ass(M/N ).

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹ ✭❚➟♣ ❣✐→ ❝õ❛ ♠æ✤✉♥✮✳ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥✳ ❚❛ ✤➦t
SuppR (M ) = {p ∈ Spec(M ) | Mp = 0} .

❑❤✐ ✤â SuppR(M ) ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❣✐→ ❝õ❛ R✲♠æ✤✉♥ M ✳

▼➺♥❤ ✤➲ ✶✳✶✳✺✳ ✐✮ ❈❤♦ p ∈ Spec(R)✳ ❑❤✐ ✤â p ∈ AssR(M ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉
M

❝â ♠ët ♠æ✤✉♥ ❝♦♥ ✤➥♥❣ ❝➜✉ ✈î✐ R/p✳

✐✐✮ ❈❤♦ 0 → M

→ M → M” → 0

❧➔ ❞➣② ❦❤î♣ ❝→❝ R✲♠æ✤✉♥✳ ❑❤✐ ✤â

SuppR (M ) ⊆ SuppR (M ) = SuppR (M ) ∪ SuppR (M ”).

✐✐✐✮ AssR(M ) ⊆ SuppR(M ) ✈➔ ♥➳✉ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r t❤➻ ♠é✐ ♣❤➛♥ tû ❝ü❝ t✐➸✉
❝õ❛ t➟♣ SuppR(M ) ✤➲✉ t❤✉ë❝ ✈➔♦ t➟♣ AssR(M )✳
✐✈✮ ◆➳✉ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r t❤➻ AssR(M ) ❧➔ t➟♣
❤ú✉ ❤↕♥✳ ❍ì♥ ♥ú❛ AssR(M ) ⊆ V (AnnR(M )) ✈➔ ♠é✐ ♣❤➛♥ tû tè✐ t✐➸✉ ❝õ❛
V (AnnR (M )) ✤➲✉ t❤✉ë❝ AssR (M )✳ ❱➻ t❤➳ AnnR (M ) ❧➔ ❣✐❛♦ ❝õ❛ ❝→❝ ✐✤➯❛♥
♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✳
✈✮ ◆➳✉ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻
V (AnnR (M )) = SuppR (M ).

✈✐✮ ◆➳✉ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ♠ët ✈➔♥❤ R t❤➻ SuppR(R/I) = V (I)✳

✶✳✷ ▼æ✤✉♥ ◆♦❡t❤❡r ✈➔ ▼æ✤✉♥ ❆rt✐♥
▼æ✤✉♥ ◆♦❡t❤❡r ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❧î♣ ♠æ✤✉♥ ❝ì ❜↔♥ ♥❤➜t ❝õ❛ ✣↕✐ sè
❣✐❛♦ ❤♦→♥✳ ❙❛✉ ✤➙② t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â✳
✻


R ởt M ởt Rổ õ
s tữỡ ữỡ

ỳ s ồ ổ ừ M ỳ s
t N1, N2, . . . ổ ừ M
N1 N2 . . . t tỗ t m 1 s Nn = Nm ợ ồ n m
tố ồ t rộ ổ ừ M õ
tỷ tố
Rổ M tọ ởt tr tữỡ ữỡ tr ồ
ổ tr


R õ ỡ ợ Rổ
0 M M M 0

õ M tr M M tr
ộ Rổ ỳ s tr tr R ởt Rổ tr
ởt Rổ tr S ởt t õ ừ R t
S 1 M ởt S 1 Rổ tr

ố ừ ổ tr ổ rt
M ởt Rổ õ s tữỡ
ữỡ
N1, N2, . . . ổ ừ M
N1 N2 . . . t tỗ t m 1 s Nn = Nm ợ ồ n m
ỹ t ồ t rộ ổ ừ M ổ õ
tỷ ỹ t



R✲♠æ✤✉♥ M

t❤ä❛ ♠➣♥ ♠ët tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ ❣å✐ ❧➔

♠æ✤✉♥ ❆rt✐♥✳ ❚❛ ♥â✐ R ❧➔ ✈➔♥❤ ❆rt✐♥ ♥➳✉ ♥â ❧➔ ♠ët R✲♠æ✤✉♥ ❆rt✐♥✳ ❚ù❝ ❧➔✱
R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❞✳❝✳❝ tr➯♥ t➟♣ ❝→❝ ✐✤➯❛♥ ❤♦➦❝ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♠å✐
t➟♣ ❦❤→❝ ré♥❣ ❝→❝ ✐✤➯❛♥ ❝õ❛ R ✤➲✉ ❝â ♣❤➛♥ tû ❝ü❝ t✐➸✉✳
▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ❆rt✐♥✳

▼➺♥❤ ✤➲ ✶✳✷✳✹✳
✐✮ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ ❞➣② ❦❤î♣ ♥❣➢♥ ❝→❝ R✲♠æ✤✉♥
0 → M → M → M” → 0

❑❤✐ ✤â M ❧➔ ❆rt✐♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ M ✈➔ M ” ❧➔ ❆rt✐♥✳
✐✐✮ ▼é✐ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ❆rt✐♥ R ❧➔ ♠ët R✲♠æ✤✉♥ ❆rt✐♥✳
✐✐✐✮ ▼é✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè tr♦♥❣ ♠ët ✈➔♥❤ ❆rt✐♥ R ❧➔ ♠ët ✐✤➯❛♥ ❝ü❝ ✤↕✐✳

✶✳✸ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣
▲þ t❤✉②➳t ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ✤÷ñ❝ ✤÷❛ r❛ ❜ð✐ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ①❡♠ ♥❤÷
❧➔ ✤è✐ ♥❣➝✉ ✈î✐ ❧þ t❤✉②➳t ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ❝❤♦ ❝→❝ ♠æ✤✉♥ ◆♦❡t❤❡r✳ ❙❛✉ ✤➙②
t❛ ♥❤➢❝ ❧↕✐ ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳
✐✮ ▼ët R✲♠æ✤✉♥ M ✤÷ñ❝ ❣å✐ ❧➔ ♠æ✤✉♥ t❤ù ❝➜♣ ♥➳✉ t❤ä❛ ♠➣♥ M = 0 ✈➔ ✈î✐
♠å✐ x ∈ R ♣❤➨♣ ♥❤➙♥ ❜ð✐ x tr➯♥ M ❧➔ t♦➔♥ ❝➜✉ ❤♦➦❝ ❧ô② ❧✐♥❤✳ ❚r♦♥❣ tr÷í♥❣
❤ñ♣ ♥➔② t➟♣ p = {x ∈ R | xnM = 0, ✈î✐ ∈ N} ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ✈➔ t❛ ❣å✐ M
❧➔ p✲t❤ù ❝➜♣✳
✐✐✮ ▼ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ M = N1 + N2 + . . . + Nn
t❤➔♥❤ tê♥❣ ❤ú✉ ❤↕♥ ❝→❝ ♠æ✤✉♥ ❝♦♥ pi✲t❤ù ❝➜♣ Ni✳ ◆➳✉ M = 0 ❤♦➦❝ M ❝â ♠ët
✽


❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ t❤➻ t❛ ♥â✐ M ❧➔ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝✳ ◆➳✉ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè pi
✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈➔ ❦❤æ♥❣ ❝â ❤↕♥❣ tû Ni ♥➔♦ t❤ø❛ ✈î✐ ♠å✐ i = 1, 2, . . . , n

t❤➻ ❜✐➸✉ ❞✐➵♥ ♥➔② ✤÷ñ❝ ❣å✐ ❧➔ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ✭❤❛② t❤✉ ❣å♥✮✳
✐✐✐✮ ▼å✐ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ M ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ❞↕♥❣ tè✐ t✐➸✉✳ ❑❤✐ ✤â t➟♣
❤ñ♣ {p1, . . . , p2} ❧➔ ✤ë❝ ❧➟♣ ✈î✐ ✈✐➺❝ ❝❤å♥ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛ M ✈➔
♥â ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ M ✱ ❦➼ ❤✐➺✉ ❧➔ AttR(M )✳
❈→❝ ❤↕♥❣ tû Ni ✤÷ñ❝ ❣å✐ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ù ❝➜♣ ❝õ❛ M ✈î✐ n = 1, . . . , n✳

▼➺♥❤ ✤➲ ✶✳✸✳✷✳
✐✮ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✱ M ❧➔ ♠ët R✲♠æ✤✉♥ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝✳ ❑❤✐
✤â M = 0 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AttR(M ) = ∅✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② t➟♣ ❝→❝ ✐✤➯❛♥
♥❣✉②➯♥ tè tè✐ t✐➸✉ ❝õ❛ R ❝❤ù❛ AnnR(M ) ❝❤➼♥❤ ❧➔ t➟♣ ❝→❝ ♣❤➛♥ tû tè✐ t✐➸✉ ❝õ❛
AttR (M )✳
✐✐✮ ❈❤♦ ❞➣② ❦❤î♣ s❛✉ ❝→❝ R✲♠æ✤✉♥ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝
0 → M → M → M” → 0

❑❤✐ ✤â t❛ ❝â
AttR (M ”) ⊆ AttR (M ) ⊆ AttR (M ) ∪ AttR (M ”).

▼➺♥❤ ✤➲ ✶✳✸✳✸✳ ◆➳✉ R✲♠æ✤✉♥ M ❧➔ ❜✐➸✉ ❞✐➵♥ ✤÷ñ❝ t❤➻ t➟♣ AttR(M ) ❝❤➾ ♣❤ö

t❤✉ë❝ ✈➔♦ M ♠➔ ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛
M ✳ ❈❤♦ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✱ ❦❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ t÷ì♥❣ ✤÷ì♥❣
✐✮ p ∈ Att(M ).
✐✐✮ M ❝â ♠æ✤✉♥ t❤÷ì♥❣ ❧➔ p✲t❤ù ❝➜♣✳
✐✐✐✮ M ❝â ♠æ✤✉♥ t❤÷ì♥❣ Q s❛♦ ❝❤♦

√

Q = p.

✐✈✮ M ❝â ♠æ✤✉♥ t❤÷ì♥❣ Q s❛♦ ❝❤♦ AnnR(Q) = p.

✾


✶✳✹ ▼æ✤✉♥ Ext
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ✐✮ ✭▼æ✤✉♥ ①↕ ↔♥❤✮ ▼ët R✲♠æ✤✉♥ P ✤÷ñ❝ ❣å✐ ❧➔ ①↕ ↔♥❤
♥➳✉ ✈î✐ ♠é✐ t♦➔♥ ❝➜✉ f ✿ M → N ✈➔ ♠é✐ ✤ç♥❣ ❝➜✉ g✿ P
✤ç♥❣ ❝➜✉ h : P → M s❛♦ ❝❤♦ g = f h✳

→ N✱

❧✉æ♥ tç♥ t↕✐

✐✐✮ ✭●✐↔✐ ①↕ ↔♥❤✮ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥✳ ▼ët ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ R✲♠æ✤✉♥ M
❧➔ ♠ët ❞➣② ❦❤î♣
f2

f1

f0

ϕ

... −
→ P2 −
→ P1 −
→ P0 →
− M →0

tr♦♥❣ ✤â Pi ❧➔ ❝→❝ R✲♠æ✤✉♥ ①↕ ↔♥❤ ✈î✐ ♠å✐ i ≥ 0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✷✳ ✐✮ ✭▼æ✤✉♥ ♥ë✐ ①↕✮ ▼ët R✲♠æ✤✉♥ E ✤÷ñ❝ ❣å✐ ❧➔ ♥ë✐ ①↕ ♥➳✉

✈î✐ ♠å✐ ✤ì♥ ❝➜✉ f : N → M ✈➔ ✤ç♥❣ ❝➜✉ g : N → E ✱ ❧✉æ♥ tç♥ t↕✐ ✤ç♥❣ ❝➜✉
h : M → E s❛♦ ❝❤♦ g = hf ✳
✐✐✮ ✭●✐↔✐ ♥ë✐ ①↕✮ ▼ët ❣✐↔✐ ♥ë✐ ①↕ ❝õ❛ R✲♠æ✤✉♥ M ❧➔ ♠ët ❞➣② ❦❤î♣
ϕ

f0

f1

f2

0→M →
− E0 −
→ E1 −
→ E2 −
→ ...

tr♦♥❣ ✤â Ei ❧➔ ❝→❝ R✲♠æ✤✉♥ ♥ë✐ ①↕ ✈î✐ ♠å✐ i ≥ 0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✸ ✭▼æ✤✉♥ Ext✮✳ ❈❤♦ N ❧➔ R✲♠æ✤✉♥✳ ❳➨t ❤➔♠ tû ♣❤↔♥ ❜✐➳♥✱

❦❤î♣ tr→✐ Hom(−, N )✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥✱ ❧➜② ♠ët ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ M
f2

f1

f0

ϕ

... −
→ P2 −
→ P1 −
→ P0 →
− M → 0.

❚→❝ ✤ë♥❣ ❤➔♠ tû Hom(−, N ) ✈➔♦ ❞➣② ❦❤î♣ tr➯♥ t❛ ❝â ✤è✐ ♣❤ù❝
f∗

f∗

f∗

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

∗
❑❤✐ ✤â ExtiR(M, N ) = Ker fi∗/ Im fi−1
✤÷ñ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ♠ð rë♥❣ t❤ù i ❝õ❛
M ✈➔ N ✳ ▼æ✤✉♥ ♥➔② ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❧ü❛ ❝❤å♥ ❣✐↔✐ ①↕ ↔♥❤ ❝õ❛ M ✳

❚❛ ①➨t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ Ext✳
✶✵


▼➺♥❤ ✤➲ ✶✳✹✳✹✳ ❈❤♦ M ✱ N ❧➔ ❝→❝ R✲♠æ✤✉♥✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✐✮ M ❧➔ ♠æ✤✉♥ ①↕ ↔♥❤✳
✐✐✮ ExtiR(M, N ) = 0 ✈î✐ ♠å✐ R✲♠æ✤✉♥ N ✈➔ ✈î✐ ♠å✐ i > 0✳
✐✐✐✮ Ext1R(M, N ) = 0 ✈î✐ ♠å✐ R✲♠æ✤✉♥ N ✳

▼➺♥❤ ✤➲ ✶✳✹✳✺✳
✐✮ Ext0R(M, N ) ∼
= Hom(M, N ) ✈î✐ M ✱ N ❧➔ ❝→❝ R✲♠æ✤✉♥✳
✐✐✮ ❈❤♦ M ❧➔ ♠æ✤✉♥ ①↕ ↔♥❤✱ N ❧➔ ♠æ✤✉♥ ❜➜t ❦➻ tr➯♥ R ❦❤✐ ✤â ExtnR(M, N ) = 0
✈î✐ ♠å✐ n ♥❣✉②➯♥ ❞÷ì♥❣✳
✐✐✐✮ ◆➳✉ M ✱ N ❧➔ R ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ ExtiR(M, N ) ❝ô♥❣ ❧➔ ❤ú✉ ❤↕♥ s✐♥❤
✈î✐ ♠å✐ i ≥ 0✳
✐✈✮ ❈❤♦ ❞➣② ❦❤î♣ ♥❣➢♥ 0 → N

→ N → N” → 0

❦❤✐ ✤â tç♥ t↕✐ ❞➣② ❦❤î♣ ❞➔✐

0 → Hom(N , M ) → Hom(N, M ) → Hom(N , M ) → Ext1R (N , M ) →
→ Ext1R (N, M ) → Ext1R (N , M ) → Ext2R (N , M ) → . . .

tr♦♥❣ ✤â ExtnR(N , M ) → Extn+1
R (N ”, M ) ❧➔ ✤ç♥❣ ❝➜✉ ♥è✐ ✈î✐ ♠å✐ n ≥ 0✳
✈✮ ❈❤♦ ❞➣② ❦❤î♣ ♥❣➢♥ 0 → N

→ N → N” → 0

❦❤✐ ✤â tç♥ t↕✐ ❞➣② ❦❤î♣ ❞➔✐

0 → Hom(M, N ) → Hom(M, N ) → Hom(M, N ”) → Ext1R (M, N ”) →
→ Ext1R (M, N ) → Ext1R (M, N ”) → Ext2R (M, N ) → . . .

tr♦♥❣ ✤â ExtnR(M, N ”) → Extn+1
R (M, N ) ❧➔ ✤ç♥❣ ❝➜✉ ♥è✐ ✈î✐ ♠å✐ n ≥ 0✳

✶✶


✶✳✺ ▼æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣
▼æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❆✳ ●r♦t❤❡♥❞✐❝❦
✈➔♦ ❦❤♦↔♥❣ ♥➠♠ ✶✾✻✵✳ ❚r÷î❝ ❦❤✐ ✤➳♥ ✈î✐ ♠æ✤✉♥ ♥➔② t❛ ❣✐î✐ t❤✐➺✉ ✈➲ ❤➔♠ tû
a✲①♦➢♥✳
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶ ✭❍➔♠ tû a✲①♦➢♥✮✳ ❈❤♦ a ❧➔ ✐✤➯❛♥ ❝õ❛ R✱ ♠æ✤✉♥ ❝♦♥ a✲
①♦➢♥ ❝õ❛ R✲♠æ✤✉♥ M ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉ Γa(M ) = n≥1(0 :M an)✳
◆➳✉ h : M → N ❧➔ ✤ç♥❣ ❝➜✉ ❝→❝ R✲♠æ✤✉♥✱ ❦❤✐ ✤â t→❝ ✤ë♥❣ ❤➔♠ tû Γa(h)
✈➔♦ ✤ç♥❣ ❝➜✉ tr➯♥ t❛ ✤÷ñ❝ ✤ç♥❣ ❝➜✉ ❝↔♠ s✐♥❤ h∗ : Γa(M ) → Γa(N ) ❝❤♦ ❜ð✐
h∗ (m) = h(m)✳ ❑❤✐ ✤â Γa (−) ❧➔ ❤➔♠ tû ❤✐➺♣ ❜✐➳♥✱ t✉②➳♥ t➼♥❤✱ ❦❤î♣ tr→✐ tø
♣❤↕♠ trò ❝→❝ R✲♠æ✤✉♥ ✤➳♥ ♣❤↕♠ trò ❝→❝ R✲♠æ✤✉♥✳ ❍➔♠ tû Γa(−) ❣å✐ ❧➔ ❤➔♠
tû a✲①♦➢♥✳
❙❛✉ ✤➙② ❧➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ Γa(M )✳

▼➺♥❤ ✤➲ ✶✳✺✳✷✳

✐✮ Γ0(M ) = M ✈➔ ΓR(M ) = 0✳
✐✐✮ ◆➳✉ a ⊆ b t❤➻ Γb(M ) ⊆ Γa(M )✳

✐✐✐✮ Γa+b(M ) = Γa(M ) ∩ Γb(M )✳
✐✈✮ AssR(Γa(M )) = AssR(M ) ∩ V (a) ✈î✐ M ❧➔ R✲♠æ✤✉♥ ◆♦❡t❤❡r✳
✈✮ ◆➳✉ R ❧➔ ◆♦❡t❤❡r t❤➻ AssR(M/Γa(M )) = AssR(M ) \ V (a)✳
✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✸ ✭▼æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣✮✳ ❈❤♦ M ❧➔ R✲♠æ✤✉♥
❜➜t ❦➻✱ ❦❤✐ ✤â ❧✉æ♥ tç♥ t↕✐ ❣✐↔✐ ♥ë✐ ①↕ ❝õ❛ M ❝â ❞↕♥❣
d0

φ

d1

d2

di−1

di

0→M →
− E0 −
→ E1 −
→ E2 −
→ . . . −−→ E i −
→ ....

❚→❝ ✤ë♥❣ ❤➔♠ tû Γa(−) ✈➔♦ ❞➣② ❦❤î♣ tr➯♥ t❛ ✤÷ñ❝ ♣❤ù❝ s❛✉
d0

d1

d2

di−1

di

di+1

∗
∗
∗
∗
∗
∗
0 → Γa (E 0 ) −
→
Γa (E 1 ) −
→
Γa (E 1 ) −
→
. . . −−
→ Γa (E i ) −
→
Γa (E i+1 ) −−
→ ....

✶✷


❑❤✐ ♥â✐ Hai (M ) = Ker di∗/ Im di−1
✤÷ñ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣

∗
t❤ù i ❝õ❛ M ✤è✐ ✈î✐ ✐✤➯❛♥ a✳

▼➺♥❤ ✤➲ ✶✳✺✳✹✳ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ R✱ M ❧➔
R✲♠æ✤✉♥✳

❑❤✐ ✤â

✐✮ HI0(M ) ∼
= ΓI (M )✳
✐✐✮ ◆➳✉ M ❧➔ ♥ë✐ ①↕ t❤➻ HIi (M ) = 0 ✈î✐ ♠å✐ i > 0✳
✐✐✐✮ ◆➳✉ 0 → M → M → M ” → 0 ❧➔ ❞➣② ❦❤î♣ ♥❣➢♥ ❦❤✐ ✤â ✈î✐ ♠å✐ n ≥ 0
❧✉æ♥ tç♥ t↕✐ ✤ç♥❣ ❝➜✉ ♥è✐ HIi (M ”) → HIi (M ) s❛♦ ❝❤♦ ❞➣② s❛✉ ❧➔ ❦❤î♣
0 → ΓI (M ) → ΓI (M ) → ΓI (M ”) → HI1 (M ) → HI1 (M )
→ HI1 (M ”) → HI2 (M ) → . . .

▼➺♥❤ ✤➲ ✶✳✺✳✺✳ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r✱ M ❧➔
R✲♠æ✤✉♥

❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â Hmi (M ) ❧➔ ♠æ✤✉♥ ❆rt✐♥ ✈î✐ ♠å✐ i ≥ 0✳

❚r÷î❝ ❦❤✐ ✤➳♥ ✈î✐ t➼♥❤ tr✐➺t t✐➯✉ ❝õ❛ ♠æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣
t❤æ♥❣ q✉❛ ❝❤✐➲✉ ♠æ✤✉♥ t❛ ✤➳♥ ✈î✐ ✤à♥❤ ♥❣❤➽❛ ❝❤✐➲✉ ❝õ❛ ♠æ✤✉♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✻✳
✐✮ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤✳ ❈➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ ✤ë ❞➔✐ ❝→❝ ❞➣② ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛
R ✤÷ñ❝ ❣å✐ ❧➔ ❝❤✐➲✉ ❝õ❛ R ✈➔ ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔ dim R✳
✐✐✮ ❈❤♦ M ❧➔ ♠ët R ♠æ✤✉♥✳ ❑❤✐ ✤â ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ M ❦➼ ❤✐➺✉ ❧➔ dimR M ✱ ❧➔
dim R/ Ann M ♥➳✉ M ❦❤→❝ ❦❤æ♥❣ ✈➔ ♥➳✉ M ❧➔ ♠æ✤✉♥ ❦❤æ♥❣ t❤➻ t❛ q✉② ÷î❝
dim M = −1✳

▼➺♥❤ ✤➲ ✶✳✺✳✼ ✭✣à♥❤ ❧þ tr✐➺t t✐➯✉ ❝õ❛ ●r♦t❤❡♥❞✐❡❝❦✮✳ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦
❤♦→♥ ◆♦❡t❤❡r✱ I ❧➔ ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥✳ ❑❤✐ ✤â HIi (M ) = 0 ✈î✐
♠å✐ i > dimR M ✳
✶✸


❚❛ ①➨t t❤➯♠ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ✤ë s➙✉ ❝õ❛ ♠æ✤✉♥ tr♦♥❣ ✐✤➯❛♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✽ ✭✣ë s➙✉ ❝õ❛ ♠æ✤✉♥ tr♦♥❣ ♠ët ✐✤➯❛♥✮✳ ❈❤♦ R ❧➔ ✈➔♥❤
◆♦❡t❤❡r✱

❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ s❛♦ ❝❤♦
IM = M ✳ ✣ë ❞➔✐ ❝õ❛ ♠é✐ M ✲❞➣② tè✐ ✤↕✐ tr♦♥❣ I ✤÷ñ❝ ❣å✐ ❧➔ ✤ë s➙✉ ❝õ❛
M tr♦♥❣ ✐✤➯❛♥ I ✱ ❦➼ ❤✐➺✉ ❧➔ depthI (M ) ❤♦➦❝ depth(I, M )✳ ❑❤✐ I = m ❧➔ ✐✤➯❛♥
tè✐ ✤↕✐ ❝õ❛ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ (R, m)✱ t❤➻ t❛ ✈✐➳t depth(M ) t❤❛② ❝❤♦ depthm(M )✳
I

▼➺♥❤ ✤➲ ✶✳✺✳✾✳
✐✮ ❈❤♦ M ❧➔ ♠ët ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r (A, m) ✈➔
a ∈ m ❧➔ ♠ët ♣❤➛♥ tû ❦❤æ♥❣ ❧➔ ÷î❝ ❝õ❛ ❦❤æ♥❣ tr♦♥❣ M ✳ ❑❤✐ ✤â
depth(M/aM ) = depth(M ) − 1.

✐✐✮ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ I ❧➔ ♠ët ✐✤➯❛♥
❝õ❛ R s❛♦ ❝❤♦ IM = M ✳ ❑❤✐ ✤â t❛ ❝â
depthI (M ) = min {n | ExtnR (R/I, M ) = 0} = min {n | HIn (M ) = 0} .

✶✹

ữỡ

ỳ t
ừ ổ ố ỗ ữỡ
r ữỡ t ổ tt R tr õ ỡ
I ừ R

ổ ổ t
t t tr ổ ữủ ợ t
oăsr

ổ ởt Rổ M ữủ ồ ổ

tỗ t ởt ổ ỳ s N ừ M s ổ
tữỡ M/N ổ rt

t ứ tr t t r ợ ổ

ợ ổ tr ợ ổ rt t sỷ M R
ổ tr ú õ t ồ ổ N M õ ổ tữỡ
M/N
= 0 tr trữớ ủ ró r N ổ ỳ s ừ
M tọ M/N ổ rt t M Rổ



ố ợ trữớ ủ M Rổ rt t t ồ N = 0
õ N ổ ỳ s ừ M tọ M/N = M/0
= M
Rổ rt M ụ Rổ

ỡ ỳ t s t ợ ổ õ ữợ ổ
ổ tữỡ rở ữ s
N ổ ừ Rổ M õ M
N M/N
M õ ExtiR(P, M ) ợ ồ i 0 ồ
Rổ ỳ s P
ự sỷ M õ tỗ t ổ ỳ s
M1 ừ M s M/M1 rt õ N M1 ổ ỳ
s ừ N t t õ N/(N M1)
= (N + M1 )/M1 ổ ừ
M/M1 N/(N M1 ) rt N
t t ự tọ M/N t (M1 + N )/N
=
M1 /(N M1 ) ổ ỳ s ừ M/N ỗ tớ õ tọ
(M/N )/((M1 + N )/N )
= M/(M1 + N )
= (M/M1 )/((M1 + N )/M1 ) ổ
rt r M/N
ữủ t sỷ N M/N õ t õ ổ
ỳ s N1 =< x1, . . . , xn > ừ N s N/N1 rt õ ỳ
tỷ y1, . . . , ym ừ M s M/(< y1, . . . , ym > +N ) rt sỷ
P1 =< y1 , . . . , ym > t M2 = N1 + P1 õ M2 ỳ s ỗ
tớ t t M/M2 rt t tứ ợ
N + P1
M
M
M

=

0,
N1 + P1
M2
N1 + P1
N + P1
rt NN ++ PP1 N M
rt
+ P1
1
1

0

t õ M/M2



tr t õ


N M
+P

= M/(< y1 , . . . , ym > +N )
1

rt t t õ

N + P1
N + (N1 + P1 ) N + M2

N
N

=
=
=
N1 + P1
N1 + P1
M2
N M2
N (N1 + P1 )
=

N/N1
N
=
N1 + N P1
(N1 + N P1 )/N1

tr õ ổ (N

N/N1
rt õ tữỡ ừ ổ rt
1 + N P1 )/N1
N + P1
ổ rt ũ ợ tr t s r
N/N1 r
N1 + P1
ữủ M/M2 rt ự tọ M

tỹ ừ Rổ P t ữủ ợ
. . . P2 P1 P0 P 0

tr õ Pi ổ tỹ õ ỳ õ t õ ự
0 Hom(P0 , M ) Hom(P1 , M ) Hom(P2 , M ) . . .

ỗ ổ M sỷ ử ỵ ứ õ ExtiR(P, M )
ổ tữỡ ừ ổ õ õ ụ t ỵ

t t ổ t ữủ
rtsr ữ s
ổ I t ởt Rổ M ữủ ồ ổ
I t õ tọ
SuppR(M ) V (I)
ExtiR(R/I, M ) Rổ ỳ s ợ ồ i 0
ởt t tr ổ I t ữủ ự
rss tr



▼➺♥❤ ✤➲ ✷✳✶✳✺✳ ✭❬✶✼✱ ❍➺ q✉↔ ✸✳✹❪✮ ❈❤♦ x ❧➔ ♣❤➛♥ tû t❤✉ë❝ ✐✤➯❛♥ I ❝õ❛ R ✈➔
R✲♠æ✤✉♥ M

t❤ä❛ ♠➣♥ Supp(M ) ⊆ V (I)✳ ●✐↔ sû r➡♥❣ (0 :M
✤➲✉ ❧➔ I ✲❝♦❢✐♥✐t❡ t❤➻ M ❧➔ ♠æ✤✉♥ I ✲❝♦❢✐♥✐t❡✳

x)

✈➔ M/xM

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ R✲♠æ✤✉♥ M ❧➔ ♠✐♥✐♠❛①✱ t❤➻ t❛ ❝â t❤➯♠ ♠ët ❞➜✉ ❤✐➺✉
s❛✉ ✤➙② ♥ú❛ ❝ô♥❣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ▲✳ ▼❡❧❦❡rss♦♥✳

▼➺♥❤ ✤➲ ✷✳✶✳✻✳ ✭❬✶✼✱ ▼➺♥❤ ✤➲ ✹✳✸❪✮ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥ ♠✐♥✐♠❛① s❛♦ ❝❤♦
Supp(M ) ⊆ V (I)✳ ❑❤✐ ✤â M

❧➔ I ✲❝♦❢✐♥✐t❡ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ (0 :M I) ❧➔ ♠æ✤✉♥ ❤ú✉
❤↕♥ s✐♥❤✳ ❍ì♥ ♥ú❛✱ ♥➳✉ ❝â ♠ët ♣❤➛♥ tû x ∈ I s❛♦ ❝❤♦ (0 :M x) ❧➔ I ✲❝♦❢✐♥✐t❡
t❤➻ M ❧➔ I ✲❝♦❢✐♥✐t❡✳

❚❛ ❦➳t t❤ó❝ ♠ö❝ ♥➔② ❜ð✐ ♠ët ❜ê ✤➲ ❦➽ t❤✉➟t s❛✉ ✤➙②✳

❇ê ✤➲ ✷✳✶✳✼✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ M ❧➔ R✲

♠æ✤✉♥✳ ●✐↔ sû (0 :M I) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔ ✤➦t Kn = (0 :M I n) ✈î✐ n = 1, 2, . . .✳
❑❤✐ ✤â✱ ✈î✐ ♠å✐ n ≥ 1 t❛ ❝â ❜❛♦ ❤➔♠ t❤ù❝ s❛✉ ✤➙②
Supp(Kn+2 /Kn+1 ) ⊆ Supp(Kn+1 /Kn ).

❈❤ù♥❣ ♠✐♥❤✳ ❱➻ (0 :M I) ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ♥➯♥ (0 :M I n) ❝ô♥❣ ❧➔
R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈î✐ ♠å✐ n✳ ❉♦ ✤â ✈î✐ ♠é✐ n ≥ 1✱ t❛ ❝â ✤➥♥❣ t❤ù❝
Supp(Kn+1 /Kn ) = V (AnnR Kn+1 /Kn ).

❚❛ q✉② ❜➔✐ t♦→♥ ✈➲ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤
V (AnnR (Kn+2 /Kn+1 )) ⊆ V (AnnR (Kn+1 /Kn )).

◆❤÷ ✈➟② t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ AnnR(Kn+1/Kn) ⊆ AnnR(Kn+2/Kn+1) ❧➔ ✤õ✳
▲➜② x ∈ AnnR(Kn+1/Kn) ♥➯♥ xKn+1 ⊆ Kn s✉② r❛ I nxKn+1 = 0✳ ❚❛
❝❤ù♥❣ ♠✐♥❤ I n+1xKn+2 ⊆ I nxKn+1✳ ❚❤➟t ✈➟② ❧➜② tò② þ ω ∈ Kn+2 s✉② r❛
✶✽

I n+2 ω = 0

❤❛② Iω ⊆ (0 :M I n+1) = Kn+1✳ ❚ø ✤â t❛ ❝â
I n+1 xω = I n x(Iω) ⊆ I n xKn+1 .

❱➻ ω ❧➔ tò② þ t❤✉ë❝ Kn+2 ♥➯♥ t❛ ❝â I n+1xKn+2 ⊆ I nxKn+1✳
◆❤÷ ✈➟② ✈î✐ ♠å✐ x ∈ AnnR(Kn+1/Kn) t❛ s✉② r❛ I n+1xKn+2
I n xKn+1 = 0 ❝â ♥❣❤➽❛ ❧➔ xKn+2 ⊆ (0 :M I n+1 ) = Kn+1 ❤❛② x
AnnR (Kn+2 /Kn+1 )✳ ❱➟② AnnR (Kn+1 /Kn ) ⊆ AnnR (Kn+2 /Kn+1 ).

⊆
∈

✷✳✷ ❈❤✐➲✉ ❤ú✉ ❤↕♥ ❜➟❝ ♠ët ✈➔ t➼♥❤ ❝❤➜t ♠✐♥✐♠❛①
❚r÷î❝ t✐➯♥ t❛ ♥❤➢❝ ❧↕✐ ♠ët t✐➯✉ ❝❤✉➞♥ ❆rt✐♥ ❝õ❛ ▼❡❧❦❡rss♦♥ ❬✶✻❪ ✈➔ ♠ët
❦➳t q✉↔ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r tr♦♥❣ ❜➔✐ ❜→♦ ❬✺❪✳
❇ê ✤➲ ✷✳✷✳✶✳ ✭❬✶✻✱ ✣à♥❤ ❧þ ✶✳✸❪✮ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥✱ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛
R✳ ❑❤✐ ✤â ♥➳✉ (0 :M I) ❧➔ ♠æ✤✉♥ ❆rt✐♥ ✈➔ M ❧➔ ♠æ✤✉♥ I ✲①♦➢♥✱ t❤➻ M ❧➔ ♠ët
♠æ✤✉♥ ❆rt✐♥✳
❇ê ✤➲ ✷✳✷✳✷✳ ✭❬✺✱ ❇ê ✤➲ ✷✳✷❪✮ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✱ M ❧➔ R✲♠æ✤✉♥
❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤æ♥❣✱ ✈➔ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ▲➜② t ❧➔ sè ♥❣✉②➯♥ ❦❤æ♥❣
➙♠ s❛♦ ❝❤♦ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ♠✐♥✐♠❛① ✈î✐ ♠å✐ i < t✳ ❑❤✐ ✤â R✲♠æ✤✉♥
HomR (R/I, HIt (M )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ✣➦❝ ❜✐➺t t➟♣ ❤ñ♣ AssR (HIt (M )) ❧➔ ❤ú✉
❤↕♥✳
❚✐➳♣ t❤❡♦ ❧➔ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❇❛❤♠❛♥♣♦✉r✲◆❛❣❤✐♣♦✉r tr♦♥❣ ❬✺❪✳
▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ✭❬✺✱ ✣à♥❤ ❧þ ✷✳✸❪✮ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✱ M
❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤æ♥❣ ✈➔ I ❧➔ ✐✤➯❛♥ ❝õ❛ R✳ ▲➜② t ❧➔ sè
♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦ HIi (M ) ❧➔ ♠✐♥✐♠❛① ✈î✐ ♠å✐ i < t✳ ❑❤✐ ✤â R✲♠æ✤✉♥
HomR (R/I, HIt (M )) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳ ✣➦❝ ❜✐➺t t➟♣ ❤ñ♣ AssR (HIt (M )) ❧➔ ❤ú✉

❤↕♥✳
✶✾


❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✷ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡
✈î✐ ♠å✐ i < t✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ i✳ ❚r÷í♥❣ ❤ñ♣ i = 0 ❧➔ ❤✐➸♥
♥❤✐➯♥✳
❚✐➳♣ t❤❡♦ t❛ ❣✐↔ sû ❦➳t q✉↔ ✤➣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❝→❝ ❣✐→ trà ♥❤ä ❤ì♥
i ✈î✐ i > 0✳ ❚❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ t❛ ❝â HIj (M ) ❧➔ I ✲❝♦❢✐♥✐t❡ ✈î✐ ♠å✐ j < i✳
⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✷✳✷ ✈➔ ❣✐↔ t❤✐➳t✱ t❛ ❝â HomR(R/I, HIi (M )) ❧➔ ❤ú✉ ❤↕♥
s✐♥❤✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✶✳✻ t❛ ❝â HIi (M ) ❧➔ I ✲❝♦❢✐♥✐t❡✳ ❉♦ ✤â t❛ t❤✉ ✤÷ñ❝ HIi (M )
❧➔ I ✲❝♦❢✐♥✐t❡ ✈î✐ ♠å✐ i < t✳

❇ê ✤➲ ✷✳✷✳✹✳ ✭❬✺✱ ✣à♥❤ ❧þ ✷✳✺❪✮ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✱ M ❧➔
R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤æ♥❣ ✈➔ I

❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❈❤♦ t ❧➔ sè ♥❣✉②➯♥
❦❤æ♥❣ ➙♠ s❛♦ ❝❤♦ HIi (M ) ❧➔ ♠✐♥✐♠❛① ✈î✐ ♠å✐ i < t ✈➔ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥
♠✐♥✐♠❛① ❝õ❛ HIt (M )✳ ❑❤✐ ✤â R✲♠æ✤✉♥ HomR(R/I, HIt (M )/N ) ❧➔ ❤ú✉ ❤↕♥
s✐♥❤✳ ✣➦❝ ❜✐➺t✱ s✉② r❛ t➟♣ AssR(HIt (M )/N ) ❧➔ ❤ú✉ ❤↕♥✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ t❤➜② HomR(R/I, HIt (M ) ❧➔ ❤ú✉ ❤↕♥
s✐♥❤✳ ❍ì♥ ♥ú❛✱ ✈➻ ❞➣② 0 → N → HIt (M ) ❧➔ ❞➣② ❦❤î♣ ♥➯♥ t❛ ❝â ❞➣②
0 → (0 :N I) → (0 :H (M ) I) ❝ô♥❣ ❧➔ ❞➣② ❦❤î♣ s✉② r❛ (0 :N I) ❧➔ ❤ú✉
❤↕♥ s✐♥❤✳
❱➻ N ❧➔ I ✲①♦➢♥ ✈➔ ♠✐♥✐♠❛①✱ ♥➯♥ t❛ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✶✳✻ s✉② r❛ ✤÷ñ❝
N ❧➔ I ✲❝♦❢✐♥✐t❡✳ ❚❛ ①➨t ❞➣② ❦❤î♣ s❛✉
t
I

0 → N → HIt (M ) → HIt (M )/N → 0.

❚→❝ ✤ë♥❣ ❤➔♠ tû HomR(R/I, −) t❛ ✤÷ñ❝ ❞➣② s❛✉ ❝ô♥❣ ❧➔ ❦❤î♣
HomR (R/I, HIt (M ) → HomR (R/I, HIt (M )/N ) → ExttI (R/I, N ).

❚ø ✤â s✉② r❛ HomR(R/I, HIt (M )/N ) ❧➔ ❤ú✉ ❤↕♥ s✐♥❤✳
✷✵