dd dãy đặc trưng euler poincaré và ứng dụng vào nghiên cứu cấu trúc một số lớp mở rộng của môđun cohen macaulay

▼ơ❝ ▲ơ❝
▼ë ➤➬✉

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❈❤➢➡♥❣ ✶✳ ❈❤✉➮♥ ❜Þ

✶✵

✶✳✶✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣

✳

✳

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✶✳✷✳ ❑✐Ĩ✉ ➤❛ t❤ø❝ ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝

✶✳✸✳ ➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦

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✶✵

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✶✸

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✶✹

❈❤➢➡♥❣ ✷✳ ❞❞✲❉➲② ✈➭ ❝➳❝ ➤➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦
✷✳✶✳ ❈➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ❞❞✲❞➲②

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✷✳✷✳ ➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ❝đ❛ ♣❤ø❝ ❑♦s③✉❧

✷✳✸✳ ▲✐➟♥ ❤Ư ✈í✐ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣

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✶✻

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✶✼

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✷✺

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✸✸

❈❤➢➡♥❣ ✸✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②

✹✺

✸✳✶✳ ▲ä❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ❤Ư t❤❛♠ sè tèt

✸✳✷✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②

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✸✳✸✳ ➜➷❝ tr➢♥❣ ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②

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✹✻

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✺✹

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✺✾

❈❤➢➡♥❣ ✹✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②
✹✳✶✳ ▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②

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✹✳✷✳ ➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ✈➭ ❤➺♥❣ sè

✹✳✸✳ ➜➷❝ tr➢♥❣ t❤❛♠ sè

❑Õt ❧✉❐♥

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✳

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❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✳

✶

✼✵
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✳

IF (M )

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✼✶

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✼✽

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✽✺

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✾✸

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✾✻


▼ë ➤➬✉
◆❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ♠➠➤✉♥ t❤➠♥❣ q✉❛ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛

❤➭♠ ➤é ❞➭✐ ❝ñ❛ ♠➠➤✉♥ ♠♦❞✉❧♦ ♠ét ❤Ö t❤❛♠ sè ❧➭ ♣❤➢➡♥❣ ♣❤➳♣ ➤➲ ①✉✃t ❤✐Ö♥

tõ ❧➞✉ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳

❚õ ♥❤÷♥❣ ♥➝♠ ✺✵ ❝đ❛ t❤Õ ❦û tr➢í❝✱ ❙❡rr❡ ➤➲

❝❤Ø r❛ ❝ã t❤Ĩ ❞ï♥❣ ♣❤ø❝ ❑♦s③✉❧ ➤Ĩ tÝ♥❤ ❜é✐ ❝đ❛ ♠ét ♠➠➤✉♥ ➤è✐ ✈í✐ ♠ét ❤Ư

t❤❛♠ sè✱ tõ ➤ã ➤➢❛ r❛ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❤➭♠ ➤é ❞➭✐✱ sè ❜é✐ ✈í✐ ➤é ❞➭✐ ❝đ❛ ❝➳❝

♠➠➤✉♥ ➤å♥❣ ➤✐Ị✉ ❑♦s③✉❧✳ ❈➳❝ ♠è✐ ❧✐➟♥ ❤Ư ➤ã ➤➢ỵ❝ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣

❝➳❝ ❝➠♥❣ tr×♥❤ ❝đ❛ ❆✉s❧❛♥❞❡r✲❇✉❝❤s❜❛✉♠ ✈➭ ❝➳❝ t➳❝ ❣✐➯ ❦❤➳❝✱ ❞➱♥ ➤Õ♥ ♥❤÷♥❣

❦Õt q✉➯ ♠➭ ♥❣➭② ♥❛② trë t❤➭♥❤ ❝➡ ❜➯♥ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ➜Ó ♣❤➳t ❜✐Ĩ✉

❝❤Ý♥❤ ①➳❝✱

✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐

❑ý ❤✐Ư✉

m M
✱

R

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝ñ❛

✱ tr♦♥❣ ➤ã

x

✳

( )

❧➭ ❤➭♠ ➤é ❞➭✐✱

❑❤✐ ❞✃✉ ➤➻♥❣ t❤ø❝ ①➯② r❛✱

(M/xM ) = e(x, M ) M
✱

➤Þ❛ ♣❤➢➡♥❣✱

✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝ã ❝❤✐Ị✉

e(x, M )

➤è✐ ✈í✐ ❤Ư t❤❛♠ sè

❝❤♦

❧➭ ♠ét ✈➭♥❤ ❣✐❛♦ ❤♦➳♥✱

❧➭ ♠ét

x = x1 , . . . , x d ∈ m

(M/xM )
M

(R, m)

t❛ ❧✉➠♥ ①Ðt

M

✳

◆♦❡t❤❡r

dim M = d

✳

❑❤✐ ➤ã t❛ ❧✉➠♥ ❝ã

e(x, M )

❧➭ sè ❜é✐ ❝đ❛

♥❣❤Ü❛ ❧➭ tå♥ t➵✐

x

s❛♦

➤➢ỵ❝ ❣ä✐ ❧➭ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❈ã t❤Ĩ

♥ã✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝✃✉ tró❝ ♠➠➤✉♥ ➤➢ỵ❝ ♥❣❤✐➟♥

❝ø✉ ❦ü ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ♥❤✃t tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥✳ ◆Õ✉

(M/xM ) = e(x, M )

▼❛❝❛✉❧❛② t❛ ❝ị♥❣ ❝ã

M

✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè

❧➭ ❈♦❤❡♥✲

x

M

❝đ❛

✳

▼ë ré♥❣ ➤➬✉ t✐➟♥ t❤❡♦ ❤➢í♥❣ ♥➭② ❝đ❛ ❧í♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❦❤➳✐

♥✐Ö♠ ♠➠➤✉♥ ❇✉❝❤s❜❛✉♠ ❞♦ ❙t

s❛♦ ❝❤♦ tå♥ t➵✐ ♠ét ❤➺♥❣ sè

✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè

I(M ) = 0

✳

✱

✳

❝❦r❛❞ ✈➭ ❱♦❣❡❧ ➤➢❛ r❛✳ ➜ã ❧➭ ❝➳❝ ♠➠➤✉♥

I(M )

t❤á❛ ♠➲♥

M

(M/xM )

❝❤✃t

(M/xM ) = e(x, M ) + I(M )

◆❤➢ ✈❐② ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ❇✉❝❤s❜❛✉♠ ✈í✐

t➢➡♥❣

e(x, M ) + C

tù

C

♠➭ tå♥ t➵✐ ❤➺♥❣ sè

♥❤➢

♠➠➤✉♥

s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè

✳ ❍➺♥❣ sè

✈➭ ❝ã t➟♥ ❧➭ ❤➺♥❣ sè ❇✉❝❤s❜❛✉♠ ❝ñ❛

tÝ♥❤

M

◆➝♠ ✶✾✼✾✱ ❜❛ ♥❤➭ t ọ ờ r

ét

M

x

u
ă

M



C

ỏ t ➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭

❝đ❛

I(M )

❈➳❝ ♠➠➤✉♥ ♥❤➢ ✈❐② ❝ã r✃t ♥❤✐Ị✉

❈♦❤❡♥✲▼❛❝❛✉❧❛②

✶

x

✈➭

➤➢ỵ❝

❣ä✐

❧➭

❝➳❝

♠➠➤✉♥


❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❘â r➭♥❣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭ ❇✉❝❤s❜❛✉♠

❧➭

❝➳❝

tr➢ê♥❣

♠➠➤✉♥

❤ỵ♣

r✐➟♥❣

❝đ❛

❈♦❤❡♥✲▼❛❝❛✉❧❛②

♠➠➤✉♥

s✉②

ré♥❣

❈♦❤❡♥✲▼❛❝❛✉❧❛②

➤➢ỵ❝

♣❤➳t

tr✐Ĩ♥

s✉②

r✃t

ré♥❣✳

♥❤❛♥❤

▲ý

tr♦♥❣

t❤✉②Õt

t❤❐♣

❦û

✽✵ ✈➭ ♥❤÷♥❣ ♥➝♠ ➤➬✉ t❤❐♣ ❦û ✾✵ ❝đ❛ t❤Õ ❦û ✷✵ ❜ë✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ◆✳ ❚✳

❈➢ê♥❣✱ ❙❝❤❡♥③❡❧✱ ◆✳ ❱✳ r t

...

s

ì ọ số ý ệ

u
ă

ó ề ø♥❣ ❞ô♥❣ tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ✈➭

x(n) = xn1 1 , . . . , xnd d

❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ t❤×

✈í✐

n1 , . . . , n d

❝❦r❛❞✱ ❱♦❣❡❧✱ ▲✳ ❚✳ ❍♦❛✱ ❇r♦❞♠❛♥♥✱ ●♦t♦✱

✱ ✈í✐

n1 , . . . , n d > 0

(M/x(n)M ) = n1 . . . nd e(x, M ) + I(M )

➤đ ❧í♥ ✭➤Ĩ ♥❣➽♥ ❣ä♥ t❛ sÏ ❞ï♥❣ ❦ý ❤✐Ö✉

(M/x(n)M )

M

✳ ◆Õ✉

0

✮✱ ♥ã✐

M

❦❤➠♥❣ ♣❤➯✐ ❧➭

♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❙❤❛r♣ ➤➷t ❝➞✉ ❤á✐✿ ❤➭♠

(M/x(n)M )

r✐➟♥❣✱

❝ã ❞➵♥❣ ➤❛ t❤ø❝ t❤❡♦

❧➭ ♠ét ➤❛ t❤ø❝ t❤❡♦

n1 , . . . , n d

❦❤✐

n1 , . . . , n d

n1 , . . . , n d

n1 , . . . , n d

0

✳

❑❤✐

❦❤➠♥❣❄ ❑❤➠♥❣ ❦❤ã

ó tể tì ợ í ụ ỉ r tr➯ ❧ê✐ ❧➭ ♣❤đ ➤Þ♥❤✱ ❞➱♥ ➤Õ♥ ❝➞✉ ❤á✐ t✐Õ♣

t❤❡♦ ❧➭ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ♥➭♦ t❤× ❤➭♠

(M/x(n)M )

❝ã ❞➵♥❣ ➤❛ t❤ø❝✳

▼ét ➤✐Ị✉

❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤➢ỵ❝ ◆✳ ❚✳ ❈➢ê♥❣ ➤➢❛ r❛ tr♦♥❣ ❬✽❪ q✉❛ ❦❤➳✐ ♥✐Ư♠ ✉♣✲❞➲②✳ ❍➡♥

♥÷❛✱ tr♦♥❣ ❜➭✐ ❜➳♦ ❬✾❪ ➠♥❣ ❝ị♥❣ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t

❤➭♠

(M/x(n)M )

t❤➢➡♥❣

❝đ❛

♠ét

♠ét ❤Ư t❤❛♠ sè

❦ý ❤✐Ư✉

❧✐♥❤

✈➭♥❤

t❛

s✉②

✈➭♥❤

x

❝đ❛

●♦r❡♥st❡✐♥✱

M

s❛♦ ❝❤♦

◆✳

❚✳

❈➢ê♥❣

(M/x(n)M )

a(M ) = a0 (M )a1 (M ) . . . ad−1 (M )

❤ã❛

❧➭

❧✉➠♥ ❜Þ ❝❤➷♥ tr➟♥ ❜ë✐ ♠ét ➤❛ t❤ø❝✳

tư

❝đ❛

t❤➢➡♥❣

r❛

❧✉➠♥

♠➠➤✉♥

❝đ❛

tå♥

➤è✐

♠ét

t➵✐

➤å♥❣

✈➭♥❤

♠ét

❤Ư

➤✐Ị✉

➤Þ❛

●♦r❡♥st❡✐♥

t❤❛♠

✈í✐

➤➲

tõ

xi ∈ a(M/(xi+1 , . . . , xd )M ) i = 1, . . . , d
✱

✳

r➺♥❣

❧✉➠♥

❧➭ ✈➭♥❤

tå♥

t➵✐

ai (M ) = Ann(Hmi (M ))
Hmi (M )

♠ét

❦Õt

x = x1 , . . . , x d

sè

r❛

R

❧➭ ♠ét ➤❛ t❤ø❝✳ ❈ơ t❤Ĩ



tì

ỉ



ủ

q

tỏ

M

ủ

R



tí

t

ột ệ t số ợ

t ❚✳ ❈➢ê♥❣ ❣ä✐ ❧➭ ♠ét ❤Ö t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✱ ❤➡♥ ♥÷❛ ❦❤✐ ➤ã

d

(M/x(n)M ) =

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ),

✭

✮

i=0
❧➭ ♠ét ➤❛ t❤ø❝ ✈í✐ ♠ä✐

n1 , . . . , n d > 0

✳

❑❤➳✐ ♥✐Ö♠ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝

s❛✉ ➤ã ➤➲ ➤➢ỵ❝ ❑❛✇❛s❛❦✐ sư ❞ơ♥❣ ♥❤➢ ♠ét ❝➠♥❣ ❝ơ t❤❡♥ ❝❤èt ➤Ĩ ❣✐➯✐ ❜➭✐ t♦➳♥

✷


▼❛❝❛✉❧❛② ❤ã❛ ♠ét ➤❛ t➵♣ ➤➵✐ sè ❞♦ ❋❛❧t✐♥❣s ➤➷t r❛✱ tõ ➤ã ➤➢❛ r❛ ❝➞✉ tr➯ ❧ê✐

❦❤➻♥❣ ➤Þ♥❤ ❝❤♦ ❣✐➯ t❤✉②Õt ❝đ❛ ❙❤❛r♣ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ♣❤ø❝ ➤è✐ ♥❣➱✉✳ ❈➳❝

❦Õt q✉➯ ➤ã t❤ó❝ ➤➮② ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❦ü ❤➡♥ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝➳❝ ❤Ư t❤❛♠ sè

♣✲❝❤✉➮♥ t➽❝ ♥➭② ❝ị♥❣ ♥❤➢ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ✈➭♥❤ ✈➭

♠➠➤✉♥✳ ❇➯♥ t❤➞♥ ❝➳❝ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ❝ã r✃t ♥❤✐Ò✉ tÝ♥❤ ❝❤✃t tèt✳ ❍➬✉

❤Õt ❝➳❝ tÝ♥❤ ❝❤✃t ♥➭② ➤Ị✉ ❞♦ ❝➳❝ ❤Ư t❤❛♠ sè ♥➭② t❤á❛ ♠➲♥ ❝➠♥❣ t❤ø❝ ✭

✮ ë

tr➟♥✳ ❱× ✈❐② tr♦♥❣ ❧✉❐♥ ➳♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ➤➷t ✈✃♥ ➤Ị ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t

❝đ❛ ❝➳❝ ❤Ư t❤❛♠ sè t❤á❛ ♠➲♥ ✭

✮ ❝ị♥❣ ♥❤➢ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ❝❤ó♥❣✳ ❈➳❝ ❤Ư

t❤❛♠ sè ♥❤➢ ✈❐② ❧➭ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❞❞✲❞➲② ➤➢ỵ❝ ➤Þ♥❤ ♥❣❤Ü❛

tr♦♥❣ ❧✉❐♥ ➳♥ ♥➭②✳

▼ét ♠ë ré♥❣ ❦❤➳❝ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② t❤❡♦ ❤➢í♥❣ ❤♦➭♥ t♦➭♥ ❦❤➳❝

❧➭ ❦❤➳✐ ♥✐Ư♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳

❚❛ ❣ä✐

M

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✭t➢➡♥❣ ø♥❣✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

s✉② ré♥❣ ❞➲②✮ ♥Õ✉ tå♥ t➵✐ ♠ét ❧ä❝ ❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝ñ❛

M

M0 ⊂ M1 ⊂ . . . ⊂ Mt = M,
s❛♦

❝❤♦

Mi /Mi−1

(M0 ) < ∞ dim M0 < dim M1 < . . . < dim Mt = d

t

ứ



s



ỗ

rộ

ớ

i = 1, 2, . . . , t

✳ ❈➳❝ ❧ä❝ ♥❤➢ ✈❐② ➤➢ỵ❝ ❣ä✐ ❧➭ ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭t➢➡♥❣ ø♥❣✱

❧ä❝

❈♦❤❡♥✲▼❛❝❛✉❧❛②

▼❛❝❛✉❧❛②

✐➤➟❛♥

✈➭

♥❣✉②➟♥

s✉②

ré♥❣✮✳

❈♦❤❡♥✲▼❛❝❛✉❧❛②

tè

❧✐➟♥

❦Õt

❝đ❛

❈❤ó

s✉②

ý

r➺♥❣

ré♥❣

♠➠➤✉♥

❧➭

t❤á❛

tr♦♥❣

❦❤➠♥❣

♠➲♥

❦❤✐

❝➳❝

tré♥

♠➠➤✉♥

❧➱♥✱

❈♦❤❡♥✲

♥❣❤Ü❛

❧➭

dim R/p = dim M

❝➳❝

❤♦➷❝

dim R/p = 0

✭tr♦♥❣ tr➢ê♥❣ ợ s rộ tì



tố





s



rộ

ết



ủ

ó



ố

ề





tù

ý

t

ổ

từ



0



ế



dim M

ột ➤✐Ĩ♠ ❦❤➳❝ ❜✐Ưt ❝➡ ❜➯♥ ❣✐÷❛ ❝➳❝ ❧í♣ ♠➠➤✉♥ ♥➭②✳ ❈✃✉ tró❝ ♠➠➤✉♥

❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ①✉✃t ❤✐Ư♥ tù ♥❤✐➟♥ tr♦♥❣ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ➜➵✐ sè ❣✐❛♦

❤♦➳♥ ✈➭♦ ❝➳❝ ❜➭✐ t♦➳♥ tổ ợ ợ t ị ĩ t tr

tr

trờ ợ tr ị ♣❤➢➡♥❣ ➤➢ỵ❝

✸


①Ðt ❜ë✐ ◆✳ ❚✳ ❈➢ê♥❣✲▲✳ ❚✳ ◆❤➭♥ ❬✶✽❪✱

❙❝❤❡♥③❡❧ ❬✸✼❪✳

❍✐Ö♥ ♥❛② ✈✐Ư❝ ♥❣❤✐➟♥

❝ø✉ ❝✃✉ tró❝ ♠➠➤✉♥ ♥➭② ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝✱

➤➷❝ ❜✐Ưt ❧➭ ❝➳❝ ứ ụ tr tổ ợ ý tết ồ tị ✭①❡♠ ❬✶✾❪✱ ❬✷✵❪✱ ✳ ✳ ✳ ✮✳

❇➟♥ ❝➵♥❤ ➤ã✱ ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②

tõ ❦❤Ý❛ ❝➵♥❤ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ❝ị♥❣ ❧➭ ♠ét ✈✃♥ ➤Ị q✉❛♥ trä♥❣ ✈➭ t❤✉ ❤ót ❝➳❝

♥❤➭ t♦➳♥ ❤ä❝✳

❈➳❝ ❝➠♥❣ tr×♥❤ t✐➟✉ ❜✐Ĩ✉ t❤❡♦ ❤➢í♥❣ ♥➭② ❝ã t❤Ĩ ❦Ĩ ➤Õ♥ ❬✶✽❪✱

❬✷✽❪✱ ❬✸✼❪✱ ❬✸✽❪✳

❝❤✃t

▼ét tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ q✉❛♥ trä♥❣ ♥❤✃t ❧➭ ➤➷❝ tr➢♥❣ tÝ♥❤

❈♦❤❡♥✲▼❛❝❛✉❧❛②

❞➲②

q✉❛

tÝ♥❤

tr✐Ưt

t✐➟✉

✈➭

tÝ♥❤

❝❤✃t

❈♦❤❡♥✲▼❛❝❛✉❧❛②

❝đ❛ ➤è✐ ♥❣➱✉ ▼❛t❧✐s ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ▼ét ♠ë ré♥❣

❦❤➳❝ ❦❤➳ tù ♥❤✐➟♥ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

s✉②

ré♥❣

▼➠➤✉♥

t➢ỵ♥❣

❞➲②

➤➢ỵ❝

◆✳

❚✳

❈♦❤❡♥✲▼❛❝❛✉❧❛②

t✐Õ♣

t❤❡♦

❝đ❛

❈➢ê♥❣

❞➲②

❝❤ó♥❣

t➠✐

✈➭

✈➭

▲✳

❚✳

◆❤➭♥

➤➢❛

❈♦❤❡♥✲▼❛❝❛✉❧❛②

tr♦♥❣

❧✉❐♥

➳♥

♥➭②✳

r❛

s✉②

tr♦♥❣

ré♥❣

❈❤ó♥❣

t➠✐

❜➭✐

❞➲②

sÏ

❜➳♦

❧➭

❝❤Ø

❬✶✽❪✳

❤❛✐

r❛

➤è✐

r➺♥❣

➤è✐ ✈í✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❧✉➠♥

tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè t❤á❛ ♠➲♥ ❝➠♥❣ t❤ø❝ ✭

✮ ë tr➟♥✳

❚õ ➤ã ❝❤ó♥❣ t➠✐ ø♥❣

❞ơ♥❣ ➤Ĩ q✉❛② ❧➵✐ ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ♥➭②✳ ▼➷❝ ❞ï ➤Þ♥❤ ♥❣❤Ü❛

❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ✈➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ❦❤➳ ❣✐è♥❣

♥❤❛✉✱

t✉② ♥❤✐➟♥ ❝ò♥❣ t➢➡♥❣ tù ♥❤➢ ➤è✐ ✈í✐ ❝➳❝ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➭

❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❝➳❝ ❦ü t❤✉❐t ❧➭♠ ✈✐Ư❝ ✈í✐ ❤❛✐ ❧í♣ ♠➠➤✉♥ ë tr➟♥ ❧➭

❦❤➳❝ ♥❤❛✉✱ ♠❛♥❣ ➤➷❝ t❤ï ❝đ❛ tõ♥❣ ❧í♣ ♠➠➤✉♥✳

▲✉❐♥ ➳♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ❜è♥ ❝❤➢➡♥❣✳

❈❤➢➡♥❣ ✶ ❧➭ ❝❤➢➡♥❣ ❝❤✉➮♥ ❜Þ✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ♥➟✉ ❧➵✐ ♥❣➽♥

❣ä♥ ♠ét sè ❦Õt q✉➯ q✉❡♥ ❜✐Õt tr♦♥❣ ➜➵✐ sè ❣✐❛♦ ❤♦➳♥ ➤Ĩ t✐Ư♥ ❝❤♦ ✈✐Ư❝ tr×♥❤

❜➭② ❝➳❝ ❦Õt q✉➯ tr♦♥❣ ❝➳❝ ❝❤➢➡♥❣ s❛✉✳ ❈ơ t❤Ĩ ❧➭ tr♦♥❣ ❚✐Õt ✶✱ ❝❤ó♥❣ t➠✐ sÏ ♥➟✉

❧➵✐ ❦❤➳✐ ♥✐Ö♠ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✱ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❞✲❞➲②✱ ❞✲❞➲②

♠➵♥❤✱ ❤Ö t❤❛♠ sè ❝❤✉➮♥ t➽❝ ✈➭ ♠ét sè ❦Õt q✉➯ ❧✐➟♥ q✉❛♥✱ ❝❤ñ ②Õ✉ tõ ❝➳❝ ❜➭✐

❜➳♦ ❬✹✻❪✱ ❬✹✸❪✳

❚r♦♥❣ ❚✐Õt ✷✱ ❝❤ó♥❣ t➠✐ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❦✐Ĩ✉ ➤❛ t❤ø❝✱ ❤Ư

t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ✈➭ ♠ét số tí t ủ ú ợ trì tr

❬✶✵❪✱ ❬✸✵❪✱ ❬✸✶❪✳ ▼ét sè ❦Õt q✉➯ ✈Ò ➤➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ❝ñ❛

✹


ứ s ợ trì tr ết

ết q ♥➭② ❝❤đ ②Õ✉ tõ ❬✶✼❪✱

❬✹✺❪✳

❈➳❝

❚r♦♥❣

❦Õt

q✉➯

❈❤➢➡♥❣

❝đ❛

✷

❧✉❐♥

❝❤ó♥❣

➳♥

t➠✐

➜ã ❧➭ ❝➳❝ ❞➲② ❝➳❝ ♣❤➬♥ tử

i = 1, 2, . . . , s

tì

ợ

ớ

trì

tệ

x1 , . . . , x s ∈ m

xn1 1 , . . . , xni i

❧➭

❞✲❞➲②

tr♦♥❣

♥✐Ö♠

❝➳❝

❈❤➢➡♥❣

❞❞✲❞➲②

tr➟♥

✷✱

♠ét

✸

tr➟♥

✹✳

♠➠➤✉♥✳

n1 , . . . , n s
ni+1
M/(xi+1 , . . . , xns s )M ✳

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

✈➭

>0

✱

▼ét

tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ❈❤➢➡♥❣ ✷ ❧➭ ➤➷❝ tr➢♥❣ tÝ♥❤ ❝❤✃t ❞❞✲❞➲② ❝đ❛ ❤Ư

t❤❛♠ sè t❤➠♥❣ q✉❛ ❤➭♠ ➤é ❞➭✐ ✈➭ sè ❜é✐✳

❈ơ t❤Ĩ ❝❤ó♥❣ t➠✐ ❝ã ➤Þ♥❤ ❧ý ✭①❡♠

➜Þ♥❤ ❧ý ✷✳✶✳✽✮✳

➜Þ♥❤ ❧ý✳ ●✐➯ sö

x = x1 , . . . , x d

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛

M✳

❈➳❝ ➤✐Ị✉ s❛✉ ❧➭

t➢➡♥❣ ➤➢➡♥❣✿
✭✐✮

x ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M ✳

✭✐✐✮ ❱í✐ ♠ä✐

n1 , . . . , nd > 0✱
d

(M/x(n)M ) =

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ).
i=0

✭✐✐✐✮ ❚å♥ t➵✐ ❝➳❝ sè ♥❣✉②➟♥

a0 , a1 , . . . , ad

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

n1 , . . . , nd > 0✱

d

(M/(xn1 1 , . . . , xnd d )M )

=

ai n 1 . . . n i .
i=0

▼ét ❤Ö q✉➯ ❝đ❛ ➤Þ♥❤ ❧ý ♥➭② ❧➭ ♠ä✐ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ➤Ị✉ ❧➭ ❞❞✲❞➲②✳

◆❣➢ỵ❝ ❧➵✐✱ tõ ♠ét ❦Õt q✉➯ ❝đ❛ ◆✳ ❚✳ ❈➢ê♥❣ t❤× ♠ä✐ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈í✐

sè ♠ị ➤đ ❧í♥ ❧✉➠♥ ❧➭ ♠ét ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✳

t❤❛♠

sè

t❤❛♠ sè

k

❧➭

❞❞✲❞➲②

✈➭

❤Ö

x = x1 , . . . , x d

❝ñ❛ ♣❤ø❝ ❑♦s③✉❧ ❝ñ❛

t❤❛♠

❝ñ❛

M

M

sè

♣✲❝❤✉➮♥

t➽❝

❧➭

❉♦ ➤ã sù tå♥ t➵✐ ❝ñ❛ ệ

t



ớ

ỗ

ệ

t ị ĩ tr rPré

ứ ớ

x



d

(1)ik (Hi (x, M )),

χk (x, M ) =
i=k

✺


tr♦♥❣ ➤ã

Hi (x, M )

♥ã✐ r➺♥❣

χ0 (x, M ) = e(x, M )

❞➱♥

❧➭ ♠➠➤✉♥ ➤å♥❣ ➤✐Ò✉ ❑♦s③✉❧ t❤ø

✈➭

χk (x, M )

0

i

✳ ▼ét ❦Õt q✉➯ ❝đ❛ ❙❡rr❡

✈í✐ ♠ä✐

k = 0, 1, . . . , d

✱

χ1 (x, M ) = (H0 (x, M )) − χ0 (x, M ) = (M/xM ) − e(x, M )

➤Õ♥

✳

(M/x(n)M )

◆❤➢ ✈❐② tÝ♥❤ ❝❤✃t ➤❛ t❤ø❝ ❝đ❛ ❤➭♠

χ1 (x(n), M )

✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣

t❤ø❝ ❝đ❛ ❤➭♠

x

t➢➡♥❣ ➤➢➡♥❣ ✈í✐ tÝ♥❤ ➤❛

❧➭ ❞❞✲❞➲② t❤×

d−1

χ1 (x(n), M ) =

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ).
i=0

➜✐Ò✉

❣✐➯

♥➭②

◆✳

❚✳

❞➱♥

➤Õ♥

❈➢ê♥❣

χk (x(n), M )

❧➭

♠ét

❬✶❪✿

❝➞✉

♣❤➯✐

♠ét

➤❛

❤á✐

♠ë

❝❤➝♥❣

t❤ø❝

tr♦♥❣

x

♥Õ✉

t❤❡♦

❧✉❐♥

❧➭

♠ét

n1 , . . . , n d

➳♥

t✐Õ♥

❤Ư

✈í✐

sÜ

t❤❛♠

♠ä✐

❦❤♦❛

sè

❤ä❝

❝đ❛

♣✲❝❤✉➮♥

k > 0

❄

❚r➯

t➳❝

t➽❝

❧ê✐

t❤×

❝➞✉

❤á✐ ♥➭② ❝❤ó♥❣ t➠✐ ❝ã ❦Õt q✉➯ q✉❛♥ trä♥❣ t❤ø ❤❛✐ ❝đ❛ ❈❤➢➡♥❣ ✷ ✭①❡♠ ➜Þ♥❤ ❧ý

✷✳✷✳✸✮✳

x = x1 , . . . , x d

➜Þ♥❤ ❧ý✳ ❈❤♦
❞❞✲❞➲② tr➟♥

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛

M✳

●✐➯ sư

x

❧➭ ♠ét

M ✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ n1 , . . . , nd > 0✱
d−k

χk (x(n), M ) =

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )Hk1 (xi+2 ,...,xd ,M ) ).
i=0

ệ

q



tứ



tr

ờ



ị



ò ỉ r tờ ủ



ỏ

tr

k (x(n), M )



ữ

ú

t

tr trờ ợ

P ❝✉è✐ ❝đ❛ ❈❤➢➡♥❣ ✷ ➤➢ỵ❝ ❞➭♥❤ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ tÝ♥❤ t ủ

ó

ột

ệ

t



R

số





M





trì

ó

sử

tờ ợ sử ó ♣❤ø❝ ➤è✐ ♥❣➱✉✳

➤➯♠ ♠ä✐ ♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤

➤ã

❚r♦♥❣

M

❞ơ♥❣

❦❤➳✐

♥✐Ư♠

●✐➯ t❤✐Õt ♥➭② ❜➯♦

➤Ị✉ ❝ã ♠ét ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲②✱ ❜➟♥ ❝➵♥❤

❝ã ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t ❦❤➳❝ ➤ã♥❣ ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉

❝ã sư ❞ơ♥❣ ❞❞✲❞➲② ♥❤➢ tÝ♥❤ ➤ã♥❣ ❝đ❛ q✉Ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ tÝ♥❤

❝❛t❡♥❛r②✱ ✳ ✳ ✳ ❚✉② ♥❤✐➟♥✱ tr♦♥❣ t❤ù❝ tÕ ❝ã ♥❤✐Ị✉ ✈Ý ❞ơ ✈➭♥❤

R

❦❤➠♥❣ ❝ã ♣❤ø❝

➤è✐ ♥❣➱✉ ♥❤➢♥❣ ✈➱♥ ❧➭ ❝❛t❡♥❛r②✱ ❝ã q✉Ü tÝ❝❤ ❦❤➠♥❣ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧➭ ➤ã♥❣

✈➭

❝ã

❤Ư

t❤❛♠

sè

❧➭

❞❞✲❞➲②✳

❚r♦♥❣

t✐Õt

✻

❝✉è✐

❝đ❛

❝❤➢➡♥❣

♥➭②✱

❝❤ó♥❣

t➠✐

❜á


❣✐➯ t❤✐Õt

R

❝ã ♣❤ø❝ ➤è✐ ♥❣➱✉✱ ❝❤Ø ❣✐➯ sư

M

❝ã ♠ét ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈➭

♥❣❤✐➟♥ ❝ø✉ sù t❤❛② ➤ỉ✐ ❝đ❛ ❝➳❝ tÝ♥❤ ❝❤✃t ❦❤➳❝✳ ▼ét sè ❦Õt q✉➯ ❜❛♥ ➤➬✉ t❤❡♦

❤➢í♥❣ ♥➭② ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ị ó ột



trờ ợ ệt ủ

ị ý rệt ể ts ợ trì tr tết

ố ủ ♥➭②✳

❈❤➢➡♥❣ ✸ ➤➢ỵ❝ ❞➭♥❤ ❝❤♦ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳

❈❤ó♥❣ t➠✐ tr➢í❝ ❤Õt ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❧ä❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ❤Ư

t❤❛♠ sè tèt✳ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ♥➭② ➤ã♥❣ ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉

tr♦♥❣ ❝❤➢➡♥❣ ♥➭② ✈➭ ❝➯ ❝❤➢➡♥❣ s❛✉ ✈Ò ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳

❚❛ ♥ã✐ ♠ét ❧ä❝

F : M0 ⊂ M1 ⊂ . . . ⊂ Mt = M

M

❝➳❝ ♠➠➤✉♥ ❝♦♥ ❝đ❛

t❤á❛

♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ♥Õ✉

dim M0 < dim M1 < . . . < dim Mt = dim M =

d

✳

✳

❑ý ❤✐Ư✉

di = dim Mi

sè tèt ➤è✐ ✈í✐

x1 , . . . , x di

F

♥Õ✉

x = x1 , . . . , x d

▼ét ❤Ö t❤❛♠ sè

(xdi +1 , . . . , xd )M ∩ Mi = 0

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛

Mi

❧➭ ♠ét ❤Ư t❤❛♠

i = 0, 1, . . . , t

✈í✐

✳ ❑❤✐ ➤ã

✈➭ t❛ ❝ã t❤Ĩ ①Ðt ❤✐Ö✉

t

IF,M (x) = (M/xM ) −

e(x1 , . . . , xdi , Mi ).
i=0

IF,M (x)
➤➲

➤➢ỵ❝

❝ã ♥❤✐Ị✉ tÝ♥❤ ❝❤✃t t➢➡♥❣ tù ♥❤➢ ❤✐Ư✉

①Ðt

▼❛❝❛✉❧❛②

IF,M (x)

tr➢í❝

s✉②

❧✉➠♥

n1 , . . . , n d
IF,M (x)

t❤×

0

ré♥❣

❧➭

➤➞②

✈➭

♠ét

❤➭♠

♥❤✐Ị✉

sè

♥➭②

t➢➡♥❣

❦❤✐

♥❣❤✐➟♥

✈✃♥

❝ø✉

➤Ị

➞♠✱

❦❤➠♥❣

❣✐➯♠✱

➤➢➡♥❣

✈í✐

♠➠➤✉♥

❦❤➳❝

❦❤➠♥❣

❦❤✐

IM (x) = (M/xM )−e(x, M )

tr♦♥❣

①Ðt

✳ ✳ ✳ ✳

➤➵✐

sè

IF,M (x(n))

❈ị♥❣

(M/xM )

❈♦❤❡♥✲▼❛❝❛✉❧❛②✱

❝❤ó

ý

❣✐❛♦

♥❤➢

r➺♥❣

❤♦➳♥✳

♠ét

❜✃t

❈♦❤❡♥✲

❈ơ

t❤Ĩ✱

❤➭♠

t❤❡♦

➤➻♥❣

t❤ø❝

t
i=0 e(x1 , . . . , xdi , Mi )

❧➭

♠ét ♠ë ré♥❣ ➤➳♥❣ ❝❤ó ý ❝đ❛ ❜✃t ➤➻♥❣ t❤ø❝ q✉❡♥ ❜✐Õt ❣✐÷❛ ➤é ❞➭✐ ✈➭ sè ❜é✐

(M/xM )

e(x, M )

✳

❑Õt

q✉➯

❝❤Ý♥❤

❝đ❛

➤Þ♥❤ ❧ý s❛✉ ✭①❡♠ ➜Þ♥❤ ❧ý ✸✳✸✳✷✮✳

➜Þ♥❤ ❧ý✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿
✭✐✮

M

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✳

✼

❈❤➢➡♥❣

✸

❝ã

t❤Ó

tã♠

t➽t

tr♦♥❣


F

✭✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝

x = x1 , . . . , x d

tèt

x = x1 , . . . , x d

◆❤➢ ✈❐②✱ ❦❤✐

F

➤è✐ ✈í✐

F

✭✐✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝

M

. . . ⊂ Mt = M

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭ ♠ét ❤Ö t❤❛♠ sè tèt
s❛♦ ❝❤♦

IF,M (x(n)) = 0✱ ✈í✐ ♠ä✐ n1 , . . . , nd > 0✳

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè

F ✱ IF,M (x) = 0✳

➤è✐ ✈í✐

❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲②✱ tå♥ t➵✐ ♠ét ❧ä❝

s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè tèt

F : M0 ⊂ M1 ⊂

x = x1 , . . . , x d

t❛ ❧✉➠♥ ❝ã

t

(M/x(n)M ) =

n1 . . . ndi e(x1 , . . . , xdi , Mi )
i=0

❧➭

♠ét

➤➷❝

➤❛

t❤ø❝

tr➢♥❣

❝đ❛

✈í✐

♠ét

♠ä✐

❤Ư

n1 , . . . , n d > 0

✱

t❤❛♠

sè

❧➭

tr♦♥❣

❞❞✲❞➲②

q✉❛

➤ã

❤➭♠

di = dim Mi
➤é

❞➭✐

ë

✳

❚õ

❈❤➢➡♥❣

♠ét

✷✱

t❛

s✉② r❛ ♠ä✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➲② ❧✉➠♥ ❝ã ♠ét ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳

❑❤✐

①Ðt

❝➳❝

❦Õt

q✉➯

e(x1 , . . . , xdi , Mi )

tr➟♥

tr♦♥❣

tr➢ê♥❣

❤ỵ♣

✈➭♥❤

❙t❛♥❧❡②✲❘❡✐s♥❡r✱

❝➳❝

❤Ư

sè

➤➢ỵ❝ tÝ♥❤ t➢ê♥❣ ♠✐♥❤ t❤➠♥❣ q✉❛ sè ❝➳❝ ♠➷t ❝ù❝ ➤➵✐ ❝đ❛

♣❤ø❝ ➤➡♥ ❤×♥❤ t➢➡♥❣ ø♥❣ ✈í✐ ✈➭♥❤ ➤ã✳

❚r♦♥❣

❝❤➢➡♥❣

❝✉è✐

❝ï♥❣

▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳

❝❤ó♥❣

t➠✐

♥❣❤✐➟♥

ré♥❣ t❤× ❧✉➠♥ tå♥ t➵✐ ♠ét ❤➺♥❣ sè

F IF,M (x) < C
✱

❝➳❝ ❤Ö t❤❛♠ sè tèt ❝đ❛

❝❤♦

❧í♣

❝➳❝

♠➠➤✉♥

❑Õt q✉➯ ➤➬✉ t✐➟♥ ❝❤ó♥❣ t➠✐ ❝❤Ø r❛ ❧➭ ♥Õ✉

♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲② ✈➭

➤è✐ ✈í✐

❝ø✉

➤

✳

➷t

M

C

❧➭ ♠ét

❧➭ ♠ét ❧ä❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉②

s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè tèt

IF (M ) = supIF,M (x)

tr♦♥❣ ➤ã

x

x

❝đ❛

M

❝❤➵② tr➟♥

x

➤è✐ ✈í✐ ❧ä❝

IF,M (x(n)) = IF (M )

F

M

❈♦❤❡♥✲

✈í✐ ♠ä✐

F

✳ ▲✉➠♥ tå♥ t➵✐ ♠ét ❤Ö t❤❛♠ sè

x

s❛♦

n1 , . . . , n d > 0

✳ ❉♦ ➤ã

t

(M/x(n)M ) =

n1 . . . ndi e(x1 , . . . , xdi , Mi ) + IF (M )
i=0

✈➭

x

❧➭ ♠ét ❞❞✲❞➲② tr➟♥

M

✳ ❍➺♥❣ sè

IF (M )

➤è✐ ✈í✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

s✉② ré♥❣ ❞➲② ➤ã♥❣ ✈❛✐ trß t➢➡♥❣ tù ♥❤➢ ❤➺♥❣ sè ❇✉❝❤s❜❛✉♠

♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳

✹ ❧➭ ✈✐Ö❝ tÝ♥❤ ❤➺♥❣ sè

IF (M )

I(M )

➤è✐ ✈í✐

❑Õt q✉➯ q✉❛♥ trä♥❣ t❤ø ❤❛✐ ❝ñ❛ ❈❤➢➡♥❣

t❤➠♥❣ q✉❛ ➤é ❞➭✐ ❝ñ❛ ♠ét sè ♠➠➤✉♥ ➤è✐ ➤å♥❣

➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳ ❚❛ ❝ã ➤Þ♥❤ ❧ý ✭①❡♠ ➜Þ♥❤ ❧ý ✹✳✷✳✻✮✳

✽


➜Þ♥❤ ❧ý✳ ❈❤♦ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②
▼❛❝❛✉❧❛② s✉② ré♥❣

M

✈í✐ ♠ét ❧ä❝ ❈♦❤❡♥✲

F : M0 ⊂ M1 ⊂ . . . ⊂ Mt = M ✳

➜➷t

di = dim Mi ✱

i = 0, 1, . . . , t − 1✳ ❑❤✐ ➤ã
t

di+1 −1 di+1 −1

IF (M ) =
i=0 k=di
❑Õt

q✉➯

q✉❛♥

trä♥❣

t❤ø

❜❛

j=1
❝ñ❛

k−1
j−1

❈❤➢➡♥❣

✹

(Hmj (M/Mi )).
❧➭

➤Þ♥❤

❧ý

s❛✉

✭①❡♠

➜Þ♥❤

❧ý

✹✳✸✳✷✮✳

➜Þ♥❤ ❧ý✳ ❈➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿
✭✐✮

M

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✳

✭✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝

x = x1 , . . . , xd ❝đ❛ M
✈í✐ ♠ä✐

F

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉✱ ♠ét ❤Ư t❤❛♠ sè tèt

➤è✐ ✈í✐

F

✈➭ ♠ét ❤➺♥❣ sè

C s❛♦ ❝❤♦ IF,M (x(n))

C

n1 , . . . , nd > 0✳

✭✐✐✐✮ ❚å♥ t➵✐ ♠ét ❧ä❝

F

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ s❛♦ ❝❤♦

IF (M ) < ∞✳

❚õ ➤Þ♥❤ ❧ý ♥➭② ❝❤ó♥❣ t➠✐ ❝❤ø♥❣ ♠✐♥❤ ♠ét t✐➟✉ ❝❤✉➮♥ ❤÷✉ ❤➵♥ ➤Ó ❦✐Ó♠ tr❛

tÝ♥❤ ❝❤✃t ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❞➲②✿

M

s✉② ré♥❣ ❞➲② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ♠ét ❧ä❝

♠ét ❤Ö t❤❛♠ sè tèt

x

❝đ❛

M

➤è✐ ✈í✐

F

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

F

s❛♦ ❝❤♦

✾

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝❤✐Ị✉ ✈➭

IF,M (x) = IF,M (x21 , . . . , x2d )

✳


❈❤➢➡♥❣ ✶
❈❤✉➮♥ ❜Þ

❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ♥➟✉ ❧➵✐ ♠ét sè ❦Õt q✉➯ q✉❡♥ ❜✐Õt tr♦♥❣ ➜➵✐ sè

❣✐❛♦ ❤♦➳♥ ♥❤➺♠ ❣✐ó♣ ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② râ r➭♥❣ ✈➭ ❤Ư t❤è♥❣ ❝➳❝ ❦Õt q✉➯ tr♦♥❣

❝➳❝ ❝❤➢➡♥❣ s❛✉✳

❚r♦♥❣ t♦➭♥ ❜é ❧✉❐♥ ➳♥ ♥➭② t❛ ❧✉➠♥ ①Ðt

(R, m)

❧➭ ♠ét ✈➭♥❤

❣✐❛♦ ❤♦➳♥ ❝ã ➤➡♥ ✈Þ✱ ➤Þ❛ ♣❤➢➡♥❣✱ ◆♦❡t❤❡r ✈í✐ ✐➤➟❛♥ ❝ù❝ ➤➵✐ ❞✉② ♥❤✃t

R

❧➭ ột

số

x1 , . . . , x d


ớ

ủ

ỗ

ệ

t

ó

s ré♥❣
✳

M

✳ ❚❛ ❞ï♥❣

x

✱

➤Ĩ ❦ý ❤✐Ư✉ ♠ét ❤Ư t❤❛♠

✳

▼➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣

e(x, M )
x

d

✲♠➠➤✉♥ ❤÷✉ ❤➵♥ s✐♥❤ ❝ã ❝❤✐Ị✉

m M

➤➷t

IM (x) = (M/xM ) −

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét

♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

x = x1 , . . . , x d

IM (x)

0 M
✳

♥Õ✉ tå♥ t➵✐ ♠ét ❤➺♥❣ sè

❑❤✐ ➤ã✱ ➤➷t

t❤❛♠ sè✳

sè

C

s❛♦ ❝❤♦

I(M ) = max{IM (x)}

I(M )

x

➤➢ỵ❝ ❣ä✐ ❧➭

M

❝đ❛

✱

IM (x)

tr♦♥❣ ➤ã

❤➺♥❣ sè ❇✉❝❤s❜❛✉♠

x

C

✈í✐ ♠ä✐ ❤Ư t❤❛♠ sè

❝❤➵② tr➟♥ t♦➭♥ ❜é ❝➳❝ ❤Ư

❝đ❛

M

✳

❑❤➳✐ ♥✐Ư♠ ♠➠➤✉♥

❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ➤➢ỵ❝ ◆✳ ❚✳ ❈➢ê♥❣✲❙❝❤❡♥③❡❧✲◆✳ ❱✳ ❚r✉♥❣ ➤➢❛ r❛ ✈➭

♥❣❤✐➟♥ ❝ø✉ ➤➬✉ t✐➟♥ tr♦♥❣ ❬✹✻❪✳ ❈➳❝ ❦Õt q✉➯ tr♦♥❣ t✐Õt ♥➭② ➤➢ỵ❝ trÝ❝❤ ❝❤đ ②Õ✉

tõ ❬✹✻❪✱ ❬✹✸❪✳

❙❛✉ ➤➞② ❧➭ ♠ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳

✭✶✳✶✳✶✮

M

❧➭

♠ét

♠ét ❤Ư t❤❛♠ sè

x

♠➠➤✉♥

❈♦❤❡♥✲▼❛❝❛✉❧❛②

✈➭ ♠ét ❤➺♥❣ sè

C
✶✵

0

s✉②

ré♥❣

s❛♦ ❝❤♦

❦❤✐

✈➭

❝❤Ø

IM (x(n))

❦❤✐

C

tå♥

t➵✐

✈í✐ ♠ä✐


n1 , . . . , n d > 0

C

✳ ❍➡♥ ♥÷❛✱ ❤➺♥❣ sè

✈í✐ ❤➺♥❣ sè ❇✉❝❤s❜❛✉♠

✭✶✳✶✳✷✮

M

♥❤á ♥❤✃t t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ♥➭② trï♥❣

I(M )

✳

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝➳❝ ♠➠➤✉♥

➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣

Hmi (M )

❝ã ➤é ❞➭✐ ❤÷✉ ❤➵♥ ✈í✐ ♠ä✐

i=d

✳ ➜➠✐ ❦❤✐

♥❣➢ê✐ t❛ ❝ò♥❣ ❣ä✐ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❧➭ ♠➠➤✉♥ ❝ã ố ồ

ề ị ữ r số ❇✉❝❤s❜❛✉♠ ➤➢ỵ❝ tÝ♥❤ q✉❛ ➤é ❞➭✐

❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ♥➭② q✉❛ ❝➠♥❣ t❤ø❝

✭✶✳✶✳✸✮

d−1

d−1
j

I(M ) =
j=0
✭✶✳✶✳✹✮

◆Õ✉

M

▼❛❝❛✉❧❛② ✈➭

Supp M

(Hmj (M )).

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ t❤×

dim Mp + dim R/p = d

❧➭ ❝❛t❡♥❛r②✳

✈í✐ ♠ä✐

Mp

❧➭ ❈♦❤❡♥✲

p ∈ Supp M \ {m}

ữ

ề ợ ũ ú ế

R

➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛

♠ét ✈➭♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳

●✐➯ sư

❧➭ ♠ét

❝đ❛

M

M

❧➭ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❚❛ ❣ä✐ ❤Ö t❤❛♠ sè

❤Ö t❤❛♠ sè ❝❤✉➮♥ t➽❝

♥Õ✉

x

IM (x) = I(M )

✳ ◆❤➢ ✈❐②✱ ♠ä✐ ❤Ö t❤❛♠ sè

✈í✐ sè ♠ị ➤đ ❧í♥ ➤Ị✉ ❧➭ ♠ét ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝✳ ❱❛✐ trß ❝đ❛ ❤Ư t❤❛♠

sè ❝❤✉➮♥ t➽❝ tr♦♥❣ ❧ý t❤✉②Õt ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ❝ò♥❣ t➢➡♥❣ tù

♥❤➢ ✈❛✐ trß ❝đ❛ ❝➳❝ ❞➲② ❝❤Ý♥❤ q✉✐ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳

❍➬✉ ❤Õt ❝➳❝ tÝ♥❤ ❝❤✃t tèt ❝ñ❛ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ➤Ị✉ s✉② r❛

tõ ✈✐Ư❝ tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝ tr➟♥ ♠➠➤✉♥ ➤ã✳

➜Ĩ ♣❤➳t ❜✐Ĩ✉ ❝➳❝

tÝ♥❤ ❝❤✃t ❝đ❛ ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝✱ ❝❤ó♥❣ t➠✐ ♥❤➽❝ ❧➵✐ ❦❤➳✐ ♥✐Ư♠ ❞✲❞➲② ❝đ❛

❍✉♥❡❦❡✳

✭✶✳✶✳✺✮
♠ä✐

▼ét ❞➲②

x = x1 , . . . , x s ∈ m

i = 1, . . . , s

s

✈➭

j

i

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét

❞✲❞➲②

tr➟♥

M

♥Õ✉ ✈í✐

✱

(x1 , . . . , xi−1 )M : xj = (x1 , . . . , xi−1 )M : xi xj .
x

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét

♠ä✐

M

❞✲❞➲② ♠➵♥❤

n1 , . . . , n s > 0

♥Õ✉

✳

x

tr➟♥

❚❛ ♥ã✐ ❞➲②

x

M

xn1 1 , . . . , xns s

♥Õ✉

❧➭ ♠ét

M

❞✲❞➲② ♠➵♥❤ ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥

❧➭ ❞✲❞➲② ♠➵♥❤ ✈í✐ ♠ä✐ t❤ø tù ❝đ❛

✶✶

❧➭ ❞✲❞➲② tr➟♥

x1 , . . . , x s

✳

✈í✐

tr➟♥


▼ét tr♦♥❣ ♥❤÷♥❣ tÝ♥❤ ❝❤✃t q✉❛♥ trä♥❣ ❝đ❛ ❞✲❞➲② ♠➵♥❤ ❧➭ ✈✐Ư❝ ❣✐Õt ❝❤Õt ❝➳❝

♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ ✭①❡♠ ❬✷✷✱ ❚❤❡♦r❡♠ ✶✳✶✹❪ ❤♦➷❝ ❬✶✵✱ ▲❡♠♠❛

✷✳✾❪✮✳

✭✶✳✶✳✻✮

❈❤♦

x1 , . . . , x s

❧➭ ♠ét ❞✲❞➲② ♠➵♥❤ tr➟♥

M

✳ ❑❤✐ ➤ã

j
(xr+1 , . . . , xs )H(x
(M ) = 0
1 ,...,xr )

r = 1, . . . , s

✈➭

♠ét ❞✲❞➲② ♠➵♥❤ tr➟♥

M

✈í✐ ♠ä✐

j
✳ ◆ã✐ r✐➟♥❣✱ ♥Õ✉ ♠ét ❤Ư t❤❛♠ sè

x1 , . . . , x d

❧➭

t❤×

xi Hmj (M/(x1 , . . . , xk )M ) = 0
✈í✐ ♠ä✐

i = 1, . . . , d, j + k < i

✳

❚❛ ❝ã ♠ét sè ➤➷❝ tr➢♥❣ ❝đ❛ ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝✿ ●✐➯ sư

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛

✭✶✳✶✳✼✮
✭✐✮

M

x = x1 , . . . , x d

✳

❈➳❝ ♠Ö♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿

x

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝❤✉➮♥ t➽❝✳

✭✐✐✮ ❚å♥ t➵✐ ❤➺♥❣ sè

C

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

n1 , . . . , n d > 0

✱

(M/x(n)M ) = n1 . . . nd e(x, M ) + C
❤❛② t➢➡♥❣ ➤➢➡♥❣

IM (x(n)) = C

✳

✭✐✐✐✮

IM (x21 , . . . , x2d ) = IM (x)

✭✐✈✮

x

✳

❧➭ ♠ét ❞✲❞➲② ♠➵♥❤ ❦❤➠♥❣ ➤✐Ò✉ ❦✐Ư♥ tr➟♥

❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t ❦❤✐

s✉②

ré♥❣✱

❣✐➯

(x1 , . . . , xd )R

sư

✳

M

❝ã

♠ét

❤Ư

M

M

✳

❦❤➠♥❣ ♥❤✃t t❤✐Õt ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

t❤❛♠

sè

x1 , . . . , x d

❧➭

❞✲❞➲②✳

➤

➷t

q =

❑Õt q✉➯ s❛✉ ❝ñ❛ ◆✳ ❱✳ ❚r✉♥❣ ❬✹✷✱ ❚❤❡♦r❡♠ ✹✳✶❪ ➤➢ỵ❝ ❞ï♥❣

❦❤✐ ❝❤ó♥❣ t➠✐ ①Ðt ❤➭♠ ❍✐❧❜❡rt✲❙❛♠✉❡❧ ❝ñ❛ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉②

ré♥❣ ❞➲② tr♦♥❣ ❈❤➢➡♥❣ ✹✳

✭✶✳✶✳✽✮

d

(M/q

n+1

M) =
i=0

✶✷

n+i
ed−i (q, M )
i


tr♦♥❣ ➤ã

ed (q, M ) = (0 :M x1 /(0 :M x1 ) ∩ qM )

✱

ei (q, M ) = ((x1 , . . . , xd−i )M : xd−i+1 /((x1 , . . . , xd−i )M : xd−i+1 )∩qM )
− ((x1 , . . . , xd−i−1 )M : xd−i /((x1 , . . . , xd−i−1 )M : xd−i ) ∩ qM )
✈í✐

0
✈➭

e0 (q, M ) = (M/qM )
− ((x1 , . . . , xd−1 )M : xd /((x1 , . . . , xd−1 )M : xd ) ∩ qM ).
✶✳✷
❑ý

❑✐Ĩ✉ ➤❛ t❤ø❝ ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝
❤✐Ö✉

ai (M ) = Ann Hmi (M )

➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣ t❤ø

i
d

❝đ❛

M

❧➭

✳ ➜➷t

✐➤➟❛♥

❧✐♥❤

❤ã❛

tư

❝đ❛

♠➠➤✉♥

➤è✐

a(M ) = a0 (M )a1 (M ) . . . ad−1 (M )

➤å♥❣

✈➭

∞

Ann(x1 , . . . , xi−1 )M : xi /(x1 , . . . , xi−1 )M

b(M ) =
x i=1 t=0
✈í✐

x

❝❤➵② tr➟♥ t♦➭♥ ❜é ❝➳❝ ❤Ư t❤❛♠ sè ❝đ❛

✭✶✳✷✳✶✮

✳ ❚õ ❬✹✾✱ ❙❛t③ ✷✳✹✳✺❪ t❛ ❝ã

a(M ) ⊆ b(M ) ⊆ a0 (M ) ∩ a1 (M ) ∩ . . . ∩ ad−1 (M )

▼ét

t➽❝

M

♥Õ✉

✳

❤Ö

t❤❛♠

sè

x = x1 , . . . , x d

➤➢ỵ❝

xi ∈ a(M/(xi+1 , . . . , xd )M )

✈í✐

❣ä✐

♠ä✐

❧➭ ➯♥❤ ➤å♥❣ ❝✃✉ ❝đ❛ ♠ét ✈➭♥❤ ●♦r❡♥st❡✐♥✱

dim R/a(M ) < d

✱

❧➭

♠ét

❤Ö

t❤❛♠

sè

♣✲❝❤✉➮♥

i = d, d − 1, . . . , 1

✳

❑❤✐

R

❙❝❤❡♥③❡❧ ➤➲ ❝❤Ø r❛ tr♦♥❣ ❬✹✾❪ ❧➭

❞♦ ➤ã ❧✉➠♥ tå♥ t➵✐ ♠ét ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ❝đ❛

M

tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✳ ➜è✐ ✈í✐ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✱ ♠ét ❦Õt q✉➯ ❝ñ❛ ◆✳ ❚✳

❈➢ê♥❣ ❬✶✵❪ ♥ã✐ r➺♥❣ ❤➭♠ ➤é ❞➭✐

(M/x(n)M )

❧➭ ♠ét ➤❛ t❤ø❝ t❤❡♦

n1 , . . . , n d

❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝

✭✶✳✷✳✷✮

(M/x(n)M ) = n1 . . . nd e(x, M )
d−1

+

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M )
i=0

✶✸


✈í✐ ♠ä✐

n1 , . . . , n d > 0

✳

❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✱ ➤è✐ ✈í✐ ♠ét ❤Ư t❤❛♠ sè

❦ú✱ ❤➭♠

➤❛

t❤ø❝✱

IM (x(n)) = (M/x(n)M ) − n1 . . . nd e(x, M )

t✉②

♥❤✐➟♥

❤➭♠

♥➭②

❧✉➠♥

❜Þ

❝❤➷♥

tr➟♥

❜ë✐

♠ét

x = x1 , . . . , x d

❜✃t

❝ã t❤Ĩ ❦❤➠♥❣ ❧➭ ♠ét

➤❛

t❤ø❝

✈➭

❜❐❝

♥❤á

x

✳

M

✳

♥❤✃t ❝đ❛ ❝➳❝ ➤❛ t❤ø❝ ❝❤➷♥ tr➟♥ ➤ã ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈✐Ư❝ ❝❤ä♥ ❤Ö t❤❛♠ sè

p(M )

❚❛ ❦ý ❤✐Ö✉ ❜❐❝ ➤❛ t❤ø❝ ♥❤á ♥❤✃t ➤ã ❜ë✐

◆❤➢ ✈❐②✱ ♥Õ✉ ❦ý ❤✐Ư✉ ❜❐❝ ❝đ❛ ➤❛ t❤ø❝

0

❧➭

✈➭ ❣ä✐ ❧➭ ❦✐Ĩ✉ ➤❛ t❤ø❝ ❝đ❛

−∞

t❤×

M

✭t➢➡♥❣ ø♥❣✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✮ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐

ø♥❣✱

p(M )

0

✮✳

❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

p(M ) = −∞

✭t➢➡♥❣

▲✐➟♥ ❤Ư ✈í✐ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱

◆✳ ❚✳

❈➢ê♥❣ ➤➲ ❝❤Ø r❛ r➺♥❣

✭✶✳✷✳✸✮

p(M )

dim R/a(M )

✳

❉✃✉ ➤➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐

R

❧➭ ✈➭♥❤ t❤➢➡♥❣

❝đ❛ ♠ét ✈➭♥❤ ●♦r❡♥st❡✐♥✳

❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ t❛ ❝ã

✭✶✳✷✳✹✮

p(M )

IM (x(n)) =

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ),
i=0

✈í✐ ♠ä✐

✶✳✸

n1 , . . . , n d > 0

✳

➜➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦

❳Ðt ♠ét ❤Ư t❤❛♠ sè

❝đ❛ ♣❤ø❝ ❑♦s③✉❧

x = x1 , . . . , x d

K(x, M )

❝ñ❛

M

✳ ➜➷❝ tr rPré

k

ợ ị ĩ ở

d



(1)ik (Hi (x, M )).

k (x, M ) =
i=k
❚❛ ❧✉➠♥ ❝ã

χ0 (x, M ) = e(x, M )

✈➭

χk (x, M )

0

✈í✐ ♠ä✐

k = 0, 1, . . . , d

❉♦ ➤ã

χ1 (x, M ) = (H0 (x, M )) − χ0 (x, M ) = (M/xM ) − e(x, M ) = IM (x).
➜é s➞✉ ❝đ❛

M

➤➢ỵ❝ ➤➷❝ tr➢♥❣ q✉❛ sù tr✐Ưt t✐➟✉ ❝đ❛

✶✹

χk (x, M )

♥❤➢ s❛✉

✳

✭✶✳✸✳✷✮

depth(M ) = max{k : Hd−k+1 (x, M ) = 0}
= max{k : χd−k+1 (x, M ) = 0}
= max{k : χj (x, M ) = 0

✈í✐ ♠ä✐

❑Õt q✉➯ s❛✉ ❝ñ❛ ◆✳ ❚✳ ❈➢ê♥❣ ✈➭ ❱✳ ❚✳ ❑❤➠✐ ❬✶✼✱

χk (x, M )

j > d − k}.

❈♦r♦❧❧❛r② ✷✳✷❪ ❧✐➟♥ ❤Ö ❝➳❝

✈➭ sè ❜é✐ ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤å♥❣ ➤✐Ị✉ ❑♦s③✉❧ ➤ã♥❣ ✈❛✐ trß q✉❛♥ trä♥❣

tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ❝❤ó♥❣ t➠✐ ✈Ị tÝ♥❤ ➤❛ t❤ø❝ ❝ñ❛ ❤➭♠

✭✶✳✸✳✸✮

χk (x(n), M )

✳

d−k

χk (x, M ) =

e(x1 , . . . , xi , (0 : xi+1 )Hk−1 (xi+2 ,...,xd ,M ) ).
i=0

❚r➢ê♥❣ ❤ỵ♣

k=1

❝❤Ý♥❤ ❧➭ ❝➠♥❣ t❤ø❝ q✉❡♥ t❤✉é❝ ❝ñ❛ ❆✉s❧❛♥❞❡r✲❇✉❝❤s❜❛✉♠

❬✷✱ ❈♦r♦❧❧❛r② ✹✳✸❪✳

✭✶✳✸✳✹✮

d−1

IM (x) = χ1 (x, M ) =

e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ).
i=0

❳Ðt

χk (x(n), M )

χk (x(n), M )
❚✉② ♥❤✐➟♥✱

❦❤➠♥❣

❤➭♠

♥❤➢

❧➭

♠ét

♠ét

➤❛

χk (x(n), M )

❤➭♠

t❤ø❝✱

t❤❡♦

❝➯

n1 , . . . , n d > 0

✳

tr♦♥❣

tr➢ê♥❣

❤ỵ♣

◆ã✐

❝❤✉♥❣✱

n1 , . . . , n d

0

✳

❦❤➠♥❣ ❣✐➯♠ ✈➭ ❜Þ ❝❤➷♥ tr➟♥ ❜ë✐ ❝➳❝ ➤❛ t❤ø❝✳

❇❐❝ ♥❤á ♥❤✃t ❝ñ❛ ❝➳❝ ➤❛ t❤ø❝ ♥➭② ❦❤➠♥❣ ♣❤ơ t❤✉é❝ ✈✐Ư❝ ❝❤ä♥ ❤Ư t❤❛♠ sè ✈➭

➤➢ỵ❝ ❦ý ❤✐Ư✉ ❧➭

❦✐Ĩ✉ ➤❛ t❤ø❝ ❝đ❛

pk (M )

✳

M

❚❛ ❝ã✱

p0 (M ) = d

✳

✶✺

❧➭ ❝❤✐Ò✉ ✈➭

p1 (M ) = p(M )

❧➭


❈❤➢➡♥❣ ✷
❞❞✲❉➲② ✈➭ ❝➳❝ ➤➷❝ tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ
❜❐❝ ❝❛♦

❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❞❞✲❞➲② ✈➭ ♥❣❤✐➟♥ ❝ø✉ ♠ét

sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ❝➳❝ ❞➲② ♥➭②✳

◆ã✐ ♠ét í tì

ột ị ĩ ❦❤➳❝ ❝đ❛ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ tr♦♥❣ ❬✶✵❪ t❤➠♥❣ q✉❛

❦❤➳✐ ♥✐Ư♠ ❞✲❞➲② ✈➭ ➤Þ♥❤ ♥❣❤Ü❛ ❝❤♦ ❞➲② ❝ã ➤é ❞➭✐ tï② ý✳ ❑❤✐ ♠ét ❤Ö t❤❛♠ sè

x1 , . . . , x d

❝đ❛

M

❧➭ ♣✲❝❤✉➮♥ t➽❝ t❤× ❤➭♠ ➤é ❞➭✐

♠ét ➤❛ t❤ø❝ r✃t ➤➷❝ ❜✐Öt t❤❡♦

n1 , . . . , n d

tr➢♥❣ ❝đ❛ ❝➳❝ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳

t➢➡♥❣

➤➢➡♥❣

✈í✐

♠ét ➤❛ t❤ø❝ t❤❡♦

➤➷❝

tr➢♥❣

n1 , . . . , n d

(M/(xn1 1 , . . . , xnd d )M )

❧➭

✳ ❈❤ó♥❣ t➠✐ sÏ ❝❤Ø r❛ ➤➞② ❧➭ ♠ét ➤➷❝

◆❤×♥ tõ ❣ã❝ ➤é ♣❤ø❝ ❑♦s③✉❧✱ ➤✐Ò✉ ♥➭②

❊✉❧❡r✲P♦✐♥❝❛rÐ

❜❐❝

♠ét

χ1 (xn1 1 , . . . , xnd d , M )

❧➭

✳ ❉♦ ➤ã ❞➱♥ ➤Õ♥ ♠ét ❝➞✉ ❤á✐ tù ♥❤✐➟♥ ❧➭✿ ❝➳❝ ➤➷❝

tr➢♥❣ ❊✉❧❡r✲P♦✐♥❝❛rÐ ❜❐❝ ❝❛♦ ❤➡♥

χk (xn1 1 , . . . , xnd d , M )

❝ã ❧➭ ➤❛ t❤ø❝ ❦❤➠♥❣❄

❈➞✉ ❤á✐ ♥➭② ➤➢ỵ❝ ➤➷t r❛ tr♦♥❣ ❧✉❐♥ ➳♥ t✐Õ♥ sÜ ❦❤♦❛ ❤ä❝ ❝ñ❛ t➳❝ ❣✐➯ ◆✳ ❚✳ ❈➢ê♥❣

❧➭ ①✉✃t ♣❤➳t ➤✐Ĩ♠ ➤➬✉ t✐➟♥ ❝đ❛ t✃t ❝➯ ❝➳❝ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ❝❤ó♥❣ t➠✐ ✈Ị ❞❞✲❞➲②

tr♦♥❣ ❝❤➢➡♥❣ ♥➭②✳

❈➞✉ tr➯ ❧ê✐ ➤➬② ủ ỏ ợ trì tr

ết ❚r♦♥❣ t✐Õt ❝✉è✐✱ ❝❤ó♥❣ t➠✐ ♥❣❤✐➟♥ ❝ø✉ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ♠➠➤✉♥ ❝ã

♠ét ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲②✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤ã ❝❤ó♥❣ t➠✐ ➤➲ ♥❤❐♥ ➤➢ỵ❝ ♠ét

sè ❦Õt q✉➯ ❧✐➟♥ q✉❛♥ ➤Õ♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ➤Þ❛ ó ột

trờ ợ ệt ủ



ị ý rệt ❦✐Ĩ✉ ❋❛❧t✐♥❣s✳ ❈❤➢➡♥❣ ✷ ➤➢ỵ❝ ✈✐Õt ❞ù❛

tr➟♥ ❝➳❝ ❜➭✐ ❜➳♦ ❬✼❪✱ ❬✶✷❪ ✈➭ ❬✶✸❪✳

✶✻


✷✳✶

❈➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❞❞✲❞➲②

➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✶✳

x

♥ã✐

❧➭

❈❤♦

❞❞✲❞➲②

♠ét

♠ét

tr➟♥

❞➲②

M

♥Õ✉

❝➳❝

x

i = 1, . . . , s − 1 n1 , . . . , ns > 0
✱

✱

♣❤➬♥

❧➭

♠ét

❞➲②

tö

x = x1 , . . . , x s ∈ m

✳

❞✲❞➲②

♠➵♥❤

x1 , . . . , x i

❧➭

tr➟♥

♠ét

M

✈➭

❞✲❞➲②

❚❛

✈í✐

♠ä✐

♠➵♥❤

tr➟♥

n

i+1
M/(xi+1
, . . . , xns s )M

✳

i

❈❤ó ý ✷✳✶✳✷✳

✭ ✮ ❞❞✲❞➲② ♣❤ơ t❤✉é❝ t❤ø tù ❝ñ❛ ❞➲②✳

▼ét ❞➲② ❧➭ ❞❞✲❞➲② t❤❡♦

♠ä✐ t❤ø tù ❝ñ❛ ❝➳❝ ♣❤➬♥ tö tr♦♥❣ ❞➲② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❞➲② ➤ã ❧➭ ♠ét ❞✲❞➲② ♠➵♥❤

❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥✳

ii

✭

✮ ▼ä✐ ♣❤➬♥ ❤Ư t❤❛♠ sè ❝đ❛ ♠ét ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣ ✈í✐ sè

♠ị ➤đ ❧í♥ ➤Ị✉ ❧➭ ♠ét ❞❞✲❞➲②✳

iii

✭

✮

x1 , . . . , x s

◆Õ✉

M/(xi+1 , . . . , xs )M
iv

✭

✮

◆Õ✉

x1 , . . . , x s

❧➭

♠ét

❞❞✲❞➲②

i = 1, 2, . . . , s

❧➭

❞❞✲❞➲②

♠ét

✱ ❝ò♥❣ ❧➭ ❞❞✲❞➲② tr➟♥

v x1 , . . . , x s
✮

tr➟♥

M

✈➭

M

✈í✐ ♠ä✐

n1 , . . . , n s > 0
✭

tr➟♥

❧➭ ❞❞✲❞➲② tr➟♥

x1 , . . . , xs−1

❇ỉ ➤Ị ✷✳✶✳✸✳ ❈❤♦ ❞➲②

M

tr➟♥

M

x = x1 , . . . , x s

x1 , . . . , x i

t❤×

♠ä✐

❧➭

❞❞✲❞➲②

tr➟♥

✳

M

❞➲②

xn1 1 , . . . , xns s

✈í✐

✳

❦❤✐ ✈➭ ❝❤Ø ❦❤✐

❧➭ ❞❞✲❞➲② tr➟♥

t❤×

x1 , . . . , x s

M/xns M

✈í✐ ♠ä✐

❝➳❝ ♣❤➬♥ tư tr♦♥❣

❧➭ ♠ét ❞✲❞➲② ♠➵♥❤

n>0

✳

m✳

❈➳❝ ❦❤➻♥❣ ➤Þ♥❤

s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿
✭i✮

x ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M ✳

✭ii✮ ❱í✐ ♠ä✐

1

i

k

j

s ✈➭ n1 , . . . , ns > 0✱ t❛ ❝ã

n

n

j+1
i−1
(xn1 1 , . . . , xi−1
, xj+1
, . . . , xns s )M : xni i xnk k

n

n

j+1

i−1
= (xn1 1 , . . . , xi−1
, xj+1
, . . . , xns s )M : xnk k .

✭iii✮ ❱í✐ ♠ä✐

1
n

i

j

s✱ n1 , . . . , ns > 0✱ t❛ ❝ã

n

n

j+1
i−1
(xn1 1 , . . . , xi−1
, xj+1
, . . . , xns s )M : xni i xj j

n

n

n

j+1
i−1
= (xn1 1 , . . . , xi−1
, xj+1
, . . . , xns s )M : xj j .

✶✼


ứ

ii iii











ợ s r từ ị ĩ ❞❞✲❞➲②✳

✮✿ ❤✐Ó♥ ♥❤✐➟♥ ❦❤✐ ❧✃②

iii ⇒ ii

✭

i ⇔ ii

✭ ✮

✭

✮✿

①Ðt

1

i

k=j

k

j

✳

s n1 , . . . , n s > 0
✱

✳

❉ï♥❣ ➜Þ♥❤ ❧ý ●✐❛♦

❑r✉❧❧✱ tõ ❣✐➯ t❤✐Õt t❛ ❝ã

n

n

j+1
i−1
(xn1 1 , . . . , xi−1
,xj+1
, . . . , xns s )M : xni i xnk k

n

n

n

n

i−1
k+1
(xn1 1 , . . . , xi−1
, xk+1
, . . . , xns s )M : xni i xnk k

=
nk+1 ,...,nj

i−1
k+1
(xn1 1 , . . . , xi−1
, xk+1
, . . . , xns s )M : xnk k

=

nk+1 ,...,nj
nj+1
ni−1
=(xn1 1 , . . . , xi−1
, xj+1
, . . . , xns s )M

: xnk k .

❍Ö q✉➯ ✷✳✶✳✹✳ ▼ä✐ ❞➲② ❝♦♥ ❝đ❛ ♠ét ❞❞✲❞➲② ✭❣✐÷ ♥❣✉②➟♥ t❤ø tù✮ ❝ị♥❣ ❧➭ ♠ét
❞❞✲❞➲②✳

❑ý ❤✐Ö✉ ❞➲②

x1 , . . . , xi−1 , xi+1 , . . . , xs

❜ë✐

x1 , . . . , xi , . . . , x s

✳ ❚❛ ❝ã ♠Ư♥❤

➤Ị s❛✉✳

x = x1 , . . . , x s

▼Ư♥❤ ➤Ị ✷✳✶✳✺✳ ❈❤♦

i = 1, 2, . . . , s✱ ❞➲② x1 , . . . , xi , . . . , xs
❈❤ø♥❣ ♠✐♥❤✳

i=s

❧➭

❚❛ ❝❤ø♥❣ ♠✐♥❤ ♠Ư♥❤ ➤Ị ❜➺♥❣ q✉✐ ♥➵♣ t❤❡♦

❳Ðt tr➢ê♥❣ ❤ỵ♣

M/xns s M

❞❞✲❞➲②

✈í✐ ♠ä✐

tr➟♥

s>2

ns > 0

✳

x1 , . . . , xi , . . . , x s

❧➭

x2 , . . . , xi , . . . , x s

❧➭ ❞❞✲❞➲② tr➟♥

♥➵♣✱

x2 , . . . , xi , . . . , x s

1
✳

♠➵♥❤

❈❤ó

tr➟♥

ý

✳ ◆Õ✉

v

✳

✱

✮✱

✱

x1 , . . . , xi , . . . , xs−1

✶✽

s = 1, 2

❧➭ ❞❞✲❞➲②

x1 , . . . , xi , . . . , xs−1
t❛

❝➬♥

❝❤ø♥❣

♠✐♥❤

✳

❱×

♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt q✉✐

M/(xn1 1 , xi )M

✳

M

♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt

M/xi M

✳ ❱× ✈❐②✱ ➤Ĩ ❝❤ø♥❣

❧➭ ❞❞✲❞➲② tr➟♥

❧➭ ❞✲❞➲② ♠➵♥❤ tr➟♥

❤♦➷❝

Mi = M/xi M

➜➷t

M/xn1 1 M n1 > 0

x1 , . . . , xs−1

i=1

x1 , . . . , xs−1

❉♦

✷✳✶✳✷✭

M/xi M

s

❧➭ ❞❞✲❞➲②✱ ❞♦ ➤ã ❧➭ ❞✲❞➲② ♠➵♥❤ tr➟♥

▼➷t ❦❤➳❝✱ tõ ❍Ư q✉➯ ✷✳✶✳✹✱

q✉✐ ♥➵♣✱

❚õ

❱í✐ ♠ä✐

❉♦ ➤ã tr➢ê♥❣ ❤ỵ♣

♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt q✉✐ ♥➵♣✱

M/(xi , xns s )M




M

M/xi M

ột tr

tì ị ợ s r❛ tõ ➤Þ♥❤ ♥❣❤Ü❛✳

❧✉➠♥ ➤ó♥❣✳

tr➟♥

❧➭ ♠ét ❞❞✲❞➲② tr➟♥


♠✐♥❤

x1 , . . . , xi , . . . , x s

❧➭ ❞✲❞➲② ♠➵♥❤ tr➟♥

0 :Mi xns s = 0 :Mi xn1 1 xns s , ∀n1 , ns > 0,
❚❛

❝➬♥

❝❤ø♥❣

a ∈ xi M : xn1 1
s✉② r❛

✳ ❑❤✐ ➤ã tå♥ t➵✐

b∈M

❉➱♥ ➤Õ♥✱

xs2ns a − xns s xi c = 0

❧➭

♠➵♥❤

tr➟♥

M

✳

❱×

❤❛②

❤❛②

❦❤✐

s

❳Ðt

✳ ❉♦ ➤ã✱

s❛♦ ❝❤♦

tư

❜✃t

s
a ∈ xi M : x2n
= xi M : xns s
s
✳

✳

❦×

b ∈ xn1 1 M :

xns s b = xn1 1 c

xi M : xn1 1 ⊆ xi M : xns s

✈❐②

♣❤➬♥

✳ ❚õ ➤ã

xns s a − xi c ∈ 0 :M xn1 1 ⊆ 0 :M xns s

x1 , . . . , x s

❞♦

❚õ

✳

➤➞②

s✉②

r❛

✈➭ ♠Ư♥❤ ➤Ị ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳

3✱ ❞➲② x = x1 , . . . , xs ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M

x1 , . . . , xi , . . . , xs ❧➭ ❞❞✲❞➲② tr➟♥ M/xni i M

❈❤ø♥❣ ♠✐♥❤✳

✳

xn1 1 a = xi b

c∈M

xi M : xn1 1 xns s ⊆ xi M : xs2ns = xi M : xns s

❍Ư q✉➯ ✷✳✶✳✻✳ ❱í✐

xi M : xns s = xi M : xn1 1 xsns

❤❛②

s❛♦ ❝❤♦

✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ tå♥ t➵✐

xn1 1 xns s a = xi xns s b = xn1 1 xi c

❞✲❞➲②

t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤

xi M : xn1 1 xns s ⊆ xi M : xns s

♠✐♥❤

xi ⊆ xn1 1 M : xns s

M/xi M

✈í✐ ♠ä✐

❦❤✐ ✈➭ ❝❤Ø

ni > 0✱ i = 1, . . . , s✳

➜✐Ị✉ ❦✐Ư♥ ❝➬♥ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ▼Ư♥❤ ➤Ò ✷✳✶✳✺✳ ❚❛ ❝❤ø♥❣

x1 , . . . , xs−1

♠✐♥❤ ➤✐Ị✉ ❦✐Ư♥ ➤đ✳ ❉♦

v

♥➟♥ t❤❡♦ ❈❤ó ý ✷✳✶✳✷✭

❧➭ ❞❞✲❞➲② tr➟♥

✮✱ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤

❝➳❝❤ ❦❤➳❝✱ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ✈í✐ ♠ä✐

x

✈í✐ ♠ä✐

❧➭ ❞✲❞➲② ♠➵♥❤ tr➟♥

n1 , . . . , n s > 0 1

i

✱

n

n

M/xns s M

ns > 0

M

j

s

✳ ◆ã✐

✱

n

n

i−1
i−1
(xn1 1 , . . . , xi−1
)M : xni i xj j = (xn1 1 , . . . , xi−1
)M : xj j .

◆Õ✉

i>1

♠➠➤✉♥

tå♥ t➵✐

t❤× ➤✐Ị✉ ♥➭② ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥ ❞♦ ❣✐➯ t❤✐Õt

M/xn1 1 M
k

t❤á❛ ♠➲♥

❧➭ ❞❞✲❞➲② tr➟♥

➤➢➡♥❣✱

✈í✐

♠ä✐

1

xnk k M :

✳

k

s

♥➟♥

tr➢ê♥❣

k = 1, j

n
xn1 1 xj j ✳

♥❣❤✐➟♥

❑❤✐ ➤ã✱

❞♦

✳

❉♦

3

♥➟♥

x1 , . . . , xk , . . . , x s

n

xnk k M : xn1 1 xj j =

❝ø✉

s

❤❛② t➢➡♥❣

❉ï♥❣ ➜Þ♥❤ ❧ý ●✐❛♦ ❑r✉❧❧ t❛ s✉② r❛

nk

❦❤✐

i = 1

n

n

0 :M xn1 1 xj j =

❚r➢í❝

✳

❤ỵ♣

0 :M/xnk k M xj j = 0 :M/xnk k M xn1 1 xj j ,

= xnk k M :

n

✈➭

❳Ðt

❧➭ ❞❞✲❞➲② tr➟♥

n

M/xnk k M
n
xj j

n1 > 0

x2 , . . . , x s

n

xnk k M : xj j = 0 :M xj j .
nk

tr➢ê♥❣

❤ỵ♣

➤➷❝

❜✐Ưt

❧➭

❝➳❝

❤Ư

t❤❛♠

sè

❧➭

❞❞✲❞➲②✱

❝❤ó♥❣ t➠✐ ♥➟✉ ♠ét tÝ♥❤ ❝❤✃t ➤➢ỵ❝ ❞ï♥❣ ❦❤✐ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣

➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✱ ❞♦ ❝❤ø♥❣ ♠✐♥❤ ❝ị♥❣ ❣✐è♥❣ ♥❤➢ tr♦♥❣ ❬✷✷✱ ❚❤❡♦r❡♠ ✷✳✸❪ ❝❤♦

tr➢ê♥❣ ❤ỵ♣ ị ợ ú t ♥➟✉ ë ➤➞②✳

✶✾


❇ỉ ➤Ị ✷✳✶✳✼✳ ▼ä✐ ❞❞✲❞➲②

x = x1 , . . . , xs tr➟♥ M

➤Ò✉ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ➤➡♥

n1 , . . . , ns ✱ m1 , . . . , ms > 0 t❛ ❝ã

t❤ø❝✱ ❝ơ t❤Ĩ ✈í✐ ♠ä✐

ms
1
(xn1 1 +m1 , . . . , xsns +ms )M : (xm
1 . . . xs ) =
s
i
(xn1 1 , . . . , xni i , . . . , xns s )M : xm
i + x(n)M.

i=1
➜Þ♥❤ ❧ý t✐Õ♣ t❤❡♦ ❧➭ ♠ét ➤➷❝ tr➢♥❣ ❝❤♦ tÝ♥❤ ❝❤✃t ❞❞✲❞➲② ❝đ❛ ♠ét ❤Ư t❤❛♠

sè t❤➠♥❣ q✉❛ ❤➭♠ ➤é ❞➭✐✳

➜➞② ❧➭ ♠ét ❦Õt q✉➯ ❝❤Ý♥❤ ❝ñ❛ ❈❤➢➡♥❣ ✷✱ ❤➬✉ ❤Õt

❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ø♥❣ ❞ơ♥❣ ❝đ❛ ❞❞✲❞➲② ➤Ị✉ ①✉✃t ♣❤➳t tõ ❦Õt q✉➯ ♥➭②✳

➜Þ♥❤ ❧ý ✷✳✶✳✽✳ ❈❤♦

x = x1 , . . . , x d

M✳

❧➭ ♠ét ❤Ö t❤❛♠ sè ❝đ❛

❈➳❝ ♠Ư♥❤ ➤Ị

s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣✿
✭✐✮

x ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M ✳

✭✐✐✮ ❱í✐ ♠ä✐

n1 , . . . , nd > 0✱

(M/x(n)M ) = n1 . . . nd e(x, M )
d−1

+

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ).
i=0

✭✐✐✐✮ ❚å♥ t➵✐ ❝➳❝ sè

a0 , . . . , a d

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

n1 , . . . , nd > 0✱

d

(M/x(n)M ) =

n 1 . . . n i ai .

i=0

◆ã✐ r✐➟♥❣✱

(M/x(n)M ) ❧➭ ♠ét ➤❛ t❤ø❝ t❤❡♦ n1 , . . . , nd ♥Õ✉ x ❧➭ ♠ét ❞❞✲❞➲②✳

❈❤ø♥❣ ♠✐♥❤✳

i ⇒ ii

✭ ✮

✭

n

✮✿

❚❛

❝ã

i+1
xd ∈ Ann (0 : xi+1
)M/(xni+2 ,...,xnd−1 )M
i+2

0, 1, . . . , d − 2 n1 , . . . , nd > 0
✱

1.3.4

✭

❞♦

x

❧➭

❞❞✲❞➲②

d−1
xnd d , xn1 1 , . . . , xd−1

t❛ ➤➢ỵ❝

✈í✐

♠ä✐

M

tr➟♥

✳

❉♦

➤ã

➳♣

n

✮ ❝❤♦ ❤Ư t❤❛♠ sè

(M/x(n)M ) = e(x(n), M ) +
✷✵

i =

d−1

(0 : xd )M/(xn1 ,...,xnd−1 )M .
1

d−1

❞ô♥❣


◆❤❐♥ ①Ðt ❧➭ sè ❤➵♥❣ t❤ø ❤❛✐ ë ✈Õ ♣❤➯✐ ❦❤➠♥❣ ♣❤ô t❤✉é❝

nd

✱ ❞➱♥ ➤Õ♥

n

d−1
(M/x(n)M ) =e(x(n), M ) + (M/(xn1 1 , . . . , xd−1
, xd )M )

n

d−1
− e(xn1 1 , . . . , xd−1
, xd , M )

n

d−1
, 0 :M xd )
=e(x(n), M ) + e(xn1 1 , . . . , xd−1

n

d−1
+ (M/(xn1 1 , . . . , xd−1
, xd )M )

n

d−1
− e(xn1 1 , . . . , xd−1
, M/xd M ).

M = M/xd M

➜➷t

❉ï♥❣ q✉✐ ♥➵♣ t❤❡♦

✳

❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✺✱

d

x1 , . . . , xd−1

❧➭ ❞❞✲❞➲② tr➟♥

M

✳

t❛ s✉② r❛

n

n

d−1
d−1
(M/(xn1 1 , . . . ,xd−1
, xd )M ) − e(xn1 1 , . . . , xd−1
, M/xd M )

n

n

d−1
d−1
= (M /(xn1 1 , . . . , xd−1
)M ) − e(xn1 1 , . . . , xd−1
,M )

d−2

e(xn1 1 , . . . , xni i , (0 : xi+1 )M /(xi+2 ,...,xd−1 )M )

=
i=0
d−2

e(xn1 1 , . . . , xni i , (0 : xi+1 )M/(xi+2 ,...,xd )M ).

=
i=0
❱❐②

(M/x(n)M ) = n1 . . . nd e(x, M )
d−1

n1 . . . ni e(x1 , . . . , xi , (0 : xi+1 )M/(xi+2 ,...,xd )M ).

+

i=0

ii ⇒ iii

✭

✮

✭

✮✿ ❍✐Ó♥ ♥❤✐➟♥✳

iii ⇒ i

✭

✮

✭ ✮✿ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❦❤➻♥❣ ➤Þ♥❤ ❜➺♥❣ q✉✐ ♥➵♣ t❤❡♦

(M/xn1 1 M ) = a1 n1 + a0
e(xn1 1 , M ) + (0 :M xn1 1 )

✳

0 :M xn1 1 = 0 :M x1
❣✐ê ①Ðt tr➢ê♥❣ ❤ỵ♣

✈í✐

d=2

✈í✐

❚õ

♠ä✐

n1 > 0

♠ä✐

✳

➤➞②

s✉②

n1 > 0

✳ ❚❛ ❝ã

r❛

❤❛②

▼➷t

a0 =

x1

❧➭

d

✳ ❳Ðt

❦❤➳❝✱

d=1

✳ ❚❛ ❝ã

(M/xn1 1 M ) =

(0 :M xn1 1 )

♠ét

✳

❞❞✲❞➲②

tr➟♥

❉♦

M

✳

➤ã✱

❇➞②

(M/(xn1 1 , xn2 2 )M ) = n1 n2 a2 + n1 a1 + a0

✳

1.3.4

❚õ ✭

✮ t❛ ❝ã

(M/(xn1 1 , xn2 2 )M ) = n1 n2 e(x1 , x2 , M )+n1 e(x1 , 0 :M xn2 2 )+ ((0 : xn1 1 )M/xn2 2 M ).
✷✶


❈è

➤Þ♥❤

tÝ♥❤

n2 > 0

✳

❝❤✃t

❈❤ó

◆♦❡t❤❡r

(0 : x1 )M/xn2 2 M
x 2 , x1

◆ã✐

x 1 , x2

❝❤ø♥❣ ♠✐♥❤

❞➲②

✳

✈➭

((0 : xn1 1 )M/xn2 2 M )

❧➭

M/x2n2 M

❝ñ❛

n1 e(x1 , 0 :M xn2 2 )

ý

✳

❉➱♥

✳

❦❤➳❝✱

x1

❧➭

❞❞✲❞➲②

❧➭ ♠ét ❞✲❞➲② ♠➵♥❤ tr➟♥

➤ỉ✐

n1

✈í✐

0

❞♦

n2 a2 + a1 = n2 e(x1 , x2 , M ) +

➤Õ♥

a0 = ((0 : xn1 1 )M/xn2 2 M )
❝➳❝❤

❦❤➠♥❣

M

❉♦ ➤ã✱

tr➟♥

➳

✳

(0 : xn1 1 )M/xn2 2 M =

M/xn2 2 M

✳

❚❛

♣ ❞ơ♥❣ t✐Õ♣ tơ❝ ✭

❝ß♥

♣❤➯✐

1.3.4

✮ ❝❤♦

t❛ ❝ã

(M/(xn1 1 , xn2 2 )M ) = n1 n2 e(x1 , x2 , M )+n2 e(x2 , 0 :M xn1 1 )+ ((0 : xn2 2 )M/xn1 1 M ).
❉Ơ

❞➭♥❣

t❤✃②

n2

❦❤➠♥❣ ♣❤ơ t❤✉é❝

✈í✐

n2 > 0

♠ä✐

n1 a1 + a0 = ((0 : xn2 2 )M/xn1 1 M ) + n2 e(x2 , 0 :M x1n1 )

r➺♥❣

✳

❚õ

❝➳❝

0 :M xn1 1 = 0 :M x1
❍➡♥

♥÷❛✱

n > 0

➤Ĩ

e(x2 , 0 :M xn1 1 ) = 0

✱ ❞♦ ➤ã

❦Õt

✈➭

q✉➯

tr➟♥✱

(0 :M xn1 1 ) < ∞
xn2 (0 :M x1 ) = 0

d>2

ớ ỗ

xn1 1 M : xn2 2 = xn1 1 M : x2

➤Þ♥❤

❧ý

0 :M xn2 2 = 0 :M x1
❞♦

❤❛②

●✐❛♦

✈í✐

❑r✉❧❧

♠ä✐

e(x2 , 0 :M xn1 1 ) = 0

✳

0 :M xn1 1 = 0 :M xn2 2

0 :M xn1 1 xn2 2 = 0 :M x2n+n2 = 0 :M x2
❳Ðt

❞ï♥❣

✈➭

✳ ❱❐②

x 1 , x2

ni > 0 i = 1, . . . , d
✱

✱ ➤➷t

s✉②

r❛

n 1 , n2 > 0

✳

❱×

✳

t❛

❚õ

❧➭ ❞❞✲❞➲② tr➟♥

Mi = M/xni i M

✈❐②

➤➞②

M

tå♥

t➵✐

s✉②

r❛

✳

✳ ❚❛ ❝ã

d−1

(Mi /(xn1 1 , . . . , xni i , . . . , xnd d )Mi )

= (M/x(n)M ) =

bj n 1 . . . n i . . . n j
j=0

tr♦♥❣ ➤ã

bj =




aj

ai−1 + ni ai


nj aj

❚õ ❣✐➯ t❤✐Õt q✉✐ ♥➵♣ s✉② r❛

➤ã✱ tõ ❍Ö q✉➯ ✷✳✶✳✻ s✉② r❛

♥Õ✉

j < i − 1,

♥Õ✉

j = i − 1,

♥Õ✉

j > i − 1.

x1 , . . . , xi , . . . , x d

x1 , . . . , x d

❧➭ ❞❞✲❞➲② tr➟♥

❧➭ ♠ét ❞❞✲❞➲② tr➟♥

M

M/xni i M

✳

❉♦

✳

1.2.2

❚õ ✭

✮ t❛ ❝ã ♥❣❛② ❤Ö q✉➯ s❛✉ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✽✳

❍Ư q✉➯ ✷✳✶✳✾✳ ▼ä✐ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ➤Ò✉ ❧➭ ♠ét ❞❞✲❞➲② tr➟♥
◆❤➢ ✈❐②✱ ♥Õ✉ ♠ét ❤Ö t❤❛♠ sè

❞➭✐

(M/x(n)M )

x1 , . . . , x d

❧➭ ♠ét ➤❛ t❤ø❝ t❤❡♦

❧➭ ♠ét ❞❞✲❞➲② tr➟♥

n1 , . . . , n d

✷✷

✳ ●ä✐

M

M✳

t❤× ❤➭♠ ➤é

0 = i0 < i1 < . . . <


it = d

❧➭ ❝➳❝ ❝❤Ø sè s❛♦ ❝❤♦

t

aik n1 . . . nik

(M/x(n)M ) =
k=0

➤

ai1 , . . . , ait = 0

✈í✐

✳ ❍Ư q✉➯ s❛✉ ♥❤❐♥ ➤➢ỵ❝ ❧❐♣ tø❝ tõ

Þ♥❤ ❧ý ✷✳✶✳✽✳

❍Ư q✉➯ ✷✳✶✳✶✵✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ♥❤➢ tr➟♥ ✈➭ ♠ä✐ ❤♦➳♥ ✈Þ
♠➲♥

σ ∈ Sn

t❤á❛

σ({ik + 1, . . . , ik+1 }) = {ik + 1, . . . , ik+1 }✱ k = 0, 1, . . . , t − 1✱

❞➲②

xσ(1) , . . . , xσ(d) ❧➭ ♠ét ❞❞✲❞➲② tr➟♥ M ✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❞❞✲❞➲② ❝ã t❤Ĩ ❤♦➳♥ ✈Þ
tr♦♥❣ tõ♥❣ ➤♦➵♥✳

p(M )

◆❤➽❝ ❧➵✐✱ ❦✐Ĩ✉ ➤❛ t❤ø❝

❝đ❛

M

❧➭ ❜❐❝ ♥❤á ♥❤✃t ❝đ❛ ➤❛ t❤ø❝ ❝❤➷♥ tr➟♥

(M/x(n)M ) − n1 . . . nd e(x, M )

p(M ) = it−1

✳ ❉♦ ➤ã

❤✐Ư✉

t✐Õ♣ t❤❡♦ ❝❤ó♥❣ t➠✐ sÏ ❝❤Ø r❛ r➺♥❣

p(M )

✳ ❚r♦♥❣ ❦Õt q✉➯

❝ã t❤Ĩ tÝ♥❤ ➤➢ỵ❝ t❤➠♥❣ q✉❛ ✐➤➟❛♥ ❧✐♥❤

❤ã❛ tư ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉ ➤Þ❛ ♣❤➢➡♥❣✳

M

❍Ư q✉➯ ✷✳✶✳✶✶✳ ●✐➯ sư
➤ã

❝ã ♠ét ❤Ư t❤❛♠ sè

x1 , . . . , x d

p(M ) = dim R/a(M )✳

❈❤ø♥❣ ♠✐♥❤✳

i

d

✱

❞♦

❚õ

tÝ♥❤

1.2.3

✭

✮

❝❤✃t

x 1 , . . . , x i , . . . , x d , xi

1.1.6

✭

✮ t❛ ❝ã

t❛

❤♦➳♥

❧➭

❧✉➠♥

✈Þ

p(M )

dim R/a(M )

➤♦➵♥

❞❞✲❞➲②

❝ã

tõ♥❣

❞❞✲❞➲②

xi Hmj (M ) = 0

❚õ ➤ã s✉② r❛

tr➟♥

✈í✐ ♠ä✐

M

✱

✳

❝đ❛

❞♦

➤ã

❧➭

✈➭

❧➭

tr➟♥

M

t❤×

✈➭

dim R/a(M )

xnd d ∈ a(M )

n

i+1
xni i ∈ a(M/(xi+1
, . . . , xnd d )M )

❍Ö q✉➯ ✷✳✶✳✶✷✳ ❈❤♦
❧➭ ♠ét ❞❞✲❞➲② tr➟♥
♠ä✐

ni

x1 , . . . , x d

❞✲❞➲②

❱í✐

❍Ư

♠➵♥❤

✱ ❞➱♥ ➤Õ♥

xdp(M )+1 , . . . , xdd ∈ a(M )

❞❞✲❞➲②

tr♦♥❣

j = 0, 1, . . . , d−1

❚õ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ❤Ư q✉➯ tr➟♥ t❛ s✉② r❛ ♥Õ✉

sè

❧➭ ♠ét ❞❞✲❞➲②✳ ❑❤✐

✈í✐ ♠ä✐

ni

✈í✐

i

nd

i✱ i = 1, . . . , d✳

✷✸

tr➟♥

✷✳✶✳✶✵✱

M

✳

❚õ

xdi ∈ a(M )

✳

✳

❧➭ ♠ét ❤Ư t❤❛♠

d

✈➭

tỉ♥❣

q✉➳t

✳ ❉♦ ➤ã

❧➭ ♠ét ❤Ư t❤❛♠ sè ❝đ❛

M ✱ ❦❤✐ ➤ã xn1 1 , . . . , xnd d

q✉➯

p(M )

x1 , . . . , x d

♠ä✐

p(M ) <

M ✳ ●✐➯ sö x1 , . . . , xd

❧➭ ♠ét ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ✈í✐


◆❤➢ ✈❐②✱ t❤❡♦ ♠ét ♥❣❤Ü❛ ♥➭♦ ➤ã t❐♣ ❝➳❝ ❤Ö t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈➭ t❐♣ ❝➳❝

❤Ö t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ❧➭ ❤❛✐ t❐♣ ①✃♣ ①Ø ♥❤❛✉✳

❍Ö q✉➯ ❧➭ sù tå♥ t➵✐ ❝đ❛ ❤Ư

t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ ❧➭ t➢➡♥❣ ➤➢➡♥❣✳ ❚r♦♥❣ t❤ù❝ tÕ✱

✈✐Ư❝ ❦✐Ĩ♠ tr❛ ♠ét ❤Ö t❤❛♠ sè ❝ã ♣❤➯✐ ❧➭ ♣✲❝❤✉➮♥ t➽❝ ❤❛② ❦❤➠♥❣ t❤➢ê♥❣ ❦❤➠♥❣

❞Ơ ✈× ♣❤➯✐ tÝ♥❤ ♠ét ❧♦➵t t❐♣ ✐➤➟❛♥ ❧✐♥❤ ❤ã❛ tư ❝đ❛ ❝➳❝ ♠➠➤✉♥ ➤è✐ ➤å♥❣ ➤✐Ị✉

➤Þ❛ ♣❤➢➡♥❣✱ ♠ét ✈✐Ư❝ ♥ã✐ ❝❤✉♥❣ ❧➭ r✃t ❦❤ã✳ ❚r♦♥❣ ♥❤✐Ị✉ trờ ợ

ị ý

ệ q t ột ❝➳❝❤ ➤➡♥ ❣✐➯♥ ❤➡♥ ➤Ĩ ❧➭♠ ✈✐Ư❝ ♥➭②✳

➤

Þ♥❤ ❧ý ✷✳✶✳✽ ❝❤Ø r❛ r➺♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉②

ré♥❣ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ✈➭ ❤Ö t❤❛♠ sè ❝❤✉➮♥ t➽❝ ❤♦➭♥ t♦➭♥ trï♥❣ ♥❤❛✉✳ ◆❤➢

✈❐②✱ ❝ã t❤Ĩ ❝♦✐ ❦❤➳✐ ♥✐Ư♠ ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ❧➭ ♠ét ♠ë ré♥❣ ❝đ❛ ❤Ư t❤❛♠

sè ❝❤✉➮♥ t➽❝ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ♠➠➤✉♥ ❦❤➠♥❣ ❧➭ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s✉② ré♥❣✳ ❚❛

❝ã✿

❞✲❞➲② ♠➵♥❤ ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥

♣✲❝❤✉➮♥ t➽❝

⇒

⇒

❞❞✲❞➲②

⇒

❞✲❞➲② ♠➵♥❤✱ ✈➭ ❤Ư t❤❛♠ sè

❞❞✲❞➲②✱ t✉② ♥❤✐➟♥ ❝❤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ➤ó♥❣✳ P❤➬♥

❝✉è✐ ❝đ❛ t✐Õt ♥➭② ➤➢ỵ❝ ❞➭♥❤ ➤Ĩ ♣❤➞♥ ❜✐Ưt ❦❤➳✐ ♥✐Ư♠ ❞❞✲❞➲② ✈í✐ ❝➳❝ ❦❤➳✐ ♥✐Ư♠

❞✲❞➲② ♠➵♥❤✱ ❞✲❞➲② ♠➵♥❤ ❦❤➠♥❣ ➤✐Ị✉ ❦✐Ư♥ ✈➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝ t❤➠♥❣

q✉❛ ♠ét sè ✈Ý ❞ô✳ ❱Ý ❞ô ➤➬✉ t✐➟♥ ❝❤Ø r❛ r➺♥❣ ❞✲❞➲② ♠➵♥❤ ♥ã✐ ❝❤✉♥❣ ❦❤➠♥❣ ❧➭

❞❞✲❞➲②✳

❱Ý ❞ô ✷✳✶✳✶✸✳ ❳Ðt ✈➭♥❤
sè tr➟♥ tr➢ê♥❣

k✳

❤Ư t❤❛♠ sè ❝đ❛

R = k[[X, Y ]]

➤➷t M

M✳

= (X, Y )2 ✳

Y M :X
X, Y 2

✈í✐ ♠ä✐

dim M = 2

✈➭

X, Y 2

❧➭ ♠ét

0 :M X m Y 2n = 0 :M Y 2n

n, m > 0✳

❉♦ ➤ã

X, Y 2

✈➭

❧➭ ♠ét ❞✲❞➲②

M ✳ ▼➷t ❦❤➳❝ t❛ ❝ị♥❣ ❝ã
2

♥➟♥

❚❛ ❝ã

❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝

X m M : Y 2n = X m M : Y 2
♠➵♥❤ tr➟♥

❝➳❝ ỗ ũ từ ì tứ ớ ệ

m

=

(XY 2 , Y 3 )

♥Õ✉

m = 1,

(Y 2 )

♥Õ✉

m > 1,

❦❤➠♥❣ ❧➭ ♠ét ❞❞✲❞➲② tr➟♥

M✳

❚r♦♥❣ ✈Ý ❞ơ s❛✉ t❛ sÏ ①Ðt ♠ét ❤Ư t❤❛♠ sè ❧➭ ❞❞✲❞➲② ♥❤➢♥❣ ❦❤➠♥❣ ❧➭ ❞✲❞➲②

♠➵♥❤ ❤♦➳♥ ✈Þ ➤➢ỵ❝ ✈➭ ❦❤➠♥❣ ❧➭ ❤Ư t❤❛♠ sè ♣✲❝❤✉➮♥ t➽❝✳

✷✹