✣❸❍➴❈
■❍❯➌
❚❘×❮◆●
✣❸❍➴❈
■❙×
P❍❸▼
∗∗∗∗∗∗

❚❘❺◆
❚❍➚
❱■➏❚
❚❘■◆❍

❈❍➓
❙➮
❈❍➑◆❍
◗❯❨
❈Õ❆
▼❐❚
❙➮
❚❾✣■➎▼
P❇➆❖
▲❯❾◆
❿❱❚❍❸
❈◆❙➒❚⑩◆
❍➴❈
❖
❚❍❊❖
✣➚◆❍
❍×❰◆●
◆●❍■➊◆

❈Ù❯

❚❤ø❛
❚❤✐➯♥
❍✉➳✱
♥➠♠
✷✵✶✽

∗∗∗∗∗∗


✣❸❍➴❈
■❍❯➌
❚❘×❮◆●
✣❸❍➴❈
■❙×
P▼❍❸
∗∗∗∗∗∗

∗∗∗∗∗∗

❚❘❺◆
❚❍➚
❱■➏❚
❚❘■◆❍

❈❍➓
❙➮
❈❍➑◆❍
◗❯❨

❈Õ❆
▼❐❚
❙➮
❚❾✣■➎▼
P❇➆❖
❈❤✉②➯♥
♥❣➔♥❤✿

✣
❸❙➮
■⑨
❱Þ
▲❚❍❯❨➌❚
❙➮
▼➣
sè✿
✻✵
✹✵✻✵✹✶

▲❯❾◆
❿❱❚❍❸
❈◆❙➒❚⑩◆
❍➴❈
❖
❚❍❊❖
✣➚◆❍
❍×❰◆●
◆●❍■➊◆
❈Ù❯
❈→♥

ë❜❤÷î♥❣
❞➝♥
❦❤♦
❛❤å
❝✿

P●❙✳❚❙
P❤❛♥
➠❱❚♥ ❤✐➺♥

❚❤ø❛
❚❤✐➯♥
❍✉➳✱
♥➠♠
✷✵✶✽
✐



ổ ổ tr ự ừ r tổ số t
q ự tr tr tỹ ữủ ỗ t sỷ
ử ữ tứ ữủ ổ ố tr t ởt ổ tr
t
ồ tỹ
r t r






ữủ t ữợ sỹ ữợ ừ P P
ổ tọ ỏ t ỡ s s sỹ trồ ố ợ
t t ữợ ú ù tổ tr q tr ồ t ụ ữ t

ổ ỷ ớ ỡ qỵ ổ ồ
ồ ộ tr t tự tổ tr sốt q tr ồ
t
ổ t ỡ trữớ P ỏ t s
ồ trữớ P t tổ tr sốt õ ồ
ố ũ tổ ỡ ồ õ
trữớ P số ỵ tt số sỹ ở ú ù
tr q tr ồ t ứ q
t tỹ ổ ự tr ổ
tr ọ ỳ t sõt tổ rt ữủ sỹ õ õ ỵ ừ t
ổ ữủ tổ t ỡ
t
ồ tỹ
r t r




▼Ö❈ ▲Ö❈
❚r❛♥❣ ♣❤ö ❜➻❛

✐

▲í✐ ❝❛♠ ✤♦❛♥

✐✐


▲í✐ ❝↔♠ ì♥

✐✐✐

▼ö❝ ❧✉❝

✶

▼❐❚ ❙➮ ❑Þ ❍■➏❯ ❚❍×❮◆● ❉Ò◆●

✸

▲❮■ ◆➶■ ✣❺❯

✹

❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✼

✶✳✶

❱➔♥❤ ♣❤➙♥ ❜➟❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ ✈➔♥❤

✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✶✳✶


❱➔♥❤ ♣❤➙♥ ❜➟❝ ✈➔ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✶✳✷

✣❛ t↕♣ ①↕ ↔♥❤

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✶✳✸

❈❤✐➲✉ ❑r✉❧❧ ❝õ❛ ✈➔♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✶✳✹

❱➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✶✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✷✳✶


❍➔♠ ❍✐❧❜❡rt ✈➔ ✤❛ t❤ù❝ ❍✐❧❜❡rt

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✷✳✷

❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✷✳✸

▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ✈î✐ ❝❤➾ sè
❝❤➼♥❤ q✉② ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ ❝õ❛ ✈➔♥❤ t♦↕ ✤ë✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

❈❤÷ì♥❣ ✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣
❣✐❛♥ ①↕ ↔♥❤ P
✷✵
n

✷✳✶ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t

✷✵

✷✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn ✱ s ≤ ✺ ✳ ✳ ✳ ✳


✷✸

✷✳✸

✷✻

❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ♥✰✸ ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳ ✳

❑➳t ❧✉➟♥

✸✸
✶


❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✹

✷


ị ì ề


ị

số
số ữỡ
ổ tr trữớ õ số

R := k[x0 , ..., xn ] tự t x0 , ..., xn tr trữớ k
(M )
tr ừ Rổ M
e(A)
ố ở ừ t ở t t A
HM (t)
rt ừ ổ M
r(Z)
số q ừ t Z
(S) S
tố t t t S
M
ờ trỹ t ừ õ Md
d
d
B
r ừ B
Z(T )
ổ ừ t T tỷ t t
ừ R = k[x0 , ..., xn ]
rf
t ừ ỗ f
[a]
ố ợ t b s b a a Q
Z
Z+
Pn := Pnk





▲❮■ ◆➶■ ✣❺❯
❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ❝â t❤➸ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ t❤æ♥❣ q✉❛ ❤➔♠
❍✐❧❜❡rt✱ ❝ö t❤➸ ♥❤÷ s❛✉✿
❈❤♦ X = {P1 , ..., Ps } ❧➔ t➟♣ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn :=
Pnk ✱ ✈î✐ k ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè✳ ●å✐ ℘1 , ..., ℘s ❧➔ ❝→❝ ✐❞❡❛❧ ♥❣✉②➯♥ tè t❤✉➛♥
♥❤➜t ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝ R := k [x0 , ..., xn ] t÷ì♥❣ ù♥❣ ✈î✐ ❝→❝ ✤✐➸♠ P1 , ..., Ps ✳ ❈❤♦
ms
1
m1 , ..., ms ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ ✤➦t I := ℘m
1 ∩ ... ∩ ℘s ✳ ❚❛ ❣å✐ (X, I) ❧➔ t➟♣
✤✐➸♠ ❜➨♦ tr♦♥❣ Pn ✈➔ ❦þ ❤✐➺✉ ❧➔

Z := m1 P1 + · · · + ms Ps .
❱➔♥❤ t♦↕ ✤ë t❤✉➛♥ ♥❤➜t ❝õ❛ Z ❧➔ A := R/I ✳ ❱➔♥❤ A =
♣❤➙♥ ❜➟❝ ✈î✐ ❜ë✐ ❝õ❛ ♥â ❧➔
s

e(A) :=
i=1

t≥0

At ❧➔ ♠ët ✈➔♥❤

mi + n − 1
.
n

❍➔♠ ❍✐❧❜❡rt ❝õ❛ Z ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ HA (t) := dimk At ✱ t➠♥❣ ❝❤➦t ❝❤♦ ✤➳♥ ❦❤✐

✤↕t ✤÷ñ❝ sè ❜ë✐ e(A)✱ t↕✐ ✤â ♥â ❞ø♥❣✳ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ Z ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ sè
♥❣✉②➯♥ ❜➨ ♥❤➜t t s❛♦ ❝❤♦ HA (t) = e(A) ✈➔ ♥â ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ r❡❣(Z)✳
❱➜♥ ✤➲ t➻♠ ❝❤➦♥ tr➯♥ ❝õ❛ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ✤➣ ✤÷ñ❝ r➜t ♥❤✐➲✉ ♥❣÷í✐ q✉❛♥
t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ◆➠♠ ✶✾✻✶✱ ❙❡❣r❡ ✭①❡♠ ❬✶✼❪✮ ✤➣ ❝❤➾ r❛ ✤÷ñ❝ ❝❤➦♥ tr➯♥ ❝õ❛ ❝❤➾ sè
❝❤➼♥❤ q✉② ❝❤♦ ❝→❝ t➟♣ ✤✐➸♠ ❜➨♦ tê♥❣ q✉→t Z = m1 P1 + · · · + ms Ps tr♦♥❣ P2 ✿

reg(Z) ≤ max m1 + m2 − 1,

m1 + · · · + ms
2

✈î✐ m1 ≥ · · · ≥ ms ✳

✣➳♥ ♥➠♠ ✶✾✾✶✱ ❈❛t❛❧✐s❛♥♦ ✭①❡♠ ❬✻❪✮ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ tr➯♥ ❝❤♦ ♠ët t➟♣ ✤✐➸♠ ❜➨♦
ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ P2 ✳ ❱➔♦ ♥➠♠ ✶✾✾✸✱ ❈❛t❛❧✐s❛♥♦✱ ❚r✉♥❣ ✈➔ ❱❛❧❧❛ ✭①❡♠ ❬✼❪✮ ✤➣
♠ð rë♥❣ ❦➳t q✉↔ ♥➔② ❝❤♦ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ð ✈à tr➼ tê♥❣ q✉→t tr♦♥❣ Pn ✿

m1 + · · · + ms + n − 2
.
n
◆➠♠ ✶✾✾✻✱ ◆✳❱✳❚r✉♥❣ ✭①❡♠ ❬✷✵❪✮ ✤➣ ❞ü ✤♦→♥ r➡♥❣ ❝❤➦♥ tr➯♥ ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦
t✉ý þ Z = m1 P1 + · · · + ms Ps tr♦♥❣ Pn ❧➔
reg(Z) ≤ max m1 + m2 − 1,

r❡❣(Z) ≤ max {Tj |j = 1, ..., n} ,
tr♦♥❣ ✤â
✹


Tj = max


q
l=1

mi l + j − 2
|Pi1 , ..., Piq ♥➡♠ tr➯♥ ♠ët ❥✲♣❤➥♥❣ ✳
j

❍✐➺♥ ♥❛②✱ ❝❤➦♥ tr♦♥❣ ❞ü ✤♦→♥ ❝õ❛ ◆✳❱✳❚r✉♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❝❤➦♥ tr➯♥ ❝õ❛ ❙❡❣r❡✳
❈❤➦♥ tr➯♥ ❙❡❣r❡ ✤➣ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣ tr♦♥❣ ♥❤✐➲✉ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t✿
tr÷í♥❣ ❤ñ♣ n = 2, 3 ✭①❡♠ ❬✾❪✱ ❬✶✵❪✮✱ ❬✶✽❪✱ ❬✶✾❪✮✱ ❝❤♦ t➟♣ ✤✐➸♠ ❦➨♣ Z = 2P1 + · · · + 2Ps
tr♦♥❣ P4 ✭①❡♠ ❬✷✵❪✮✱ ❝❤♦ t➟♣ n + 2 ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✭①❡♠ ❬✸❪✮✱ ❝❤♦
n + 3 ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✭①❡♠ ❬✷❪✮✱✳✳✳✳ ●➛♥ ✤➙②✱ ❯✳ ◆❛❣❡❧ ✈➔ ❇✳ ❚r♦❦
✤➣ ❝❤ù♥❣ ♠✐♥❤ ❝❤➦♥ tr➯♥ ❝õ❛ ❙❡❣r❡ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t ✭①❡♠ ❬✷✹❪✮✳
❇➔✐ t♦→♥ t➼♥❤ ✤÷ñ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ❧➔ ❦❤â ❤ì♥ ❜➔✐ t♦→♥ t➻♠ ❝❤➦♥ tr➯♥
❝❤♦ r❡❣(Z)✳ ◆➠♠ ✶✾✽✹✱ ❉❛✈✐s ✈➔ ●❡r❛♠✐t❛ ✭①❡♠ ❬✽❪✮ ✤➣ t➼♥❤ ✤÷ñ❝ ❝❤➾ sè ❝❤➼♥❤ q✉②
❝❤♦ t➟♣ ✤✐➸♠ ❜➨♦ Z = m1 P1 + · · · + ms Ps ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ t❤➥♥❣ ❝õ❛ Pn ✿

reg(Z) = m1 + · · · + ms − 1.
❱➔♦ ♥➠♠ ✶✾✾✸✱ ❈❛t❛❧✐s❛♥♦✱ ❚r✉♥❣ ✈➔ ❱❛❧❧❛ ✭①❡♠ ❬✼❪✮ ✤➣ t➼♥❤ ✤÷ñ❝ ❝❤➾ sè ❝❤➼♥❤ q✉②
r❡❣(Z) ❝❤♦ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ ❝♦♥❣ ❤ú✉ t✛ ❝❤✉➞♥ tr♦♥❣ Pn ✿

mi + n − 2
.
n
◆➠♠ ✷✵✶✷✱ ❚❤✐➺♥ ✭①❡♠ ❬✷✶❪✮ ❝ô♥❣ ✤➣ t➼♥❤ ✤÷ñ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣(Z) ❝❤♦ s✰✷ ✤✐➸♠
❜➨♦ s❛♦ ❝❤♦ ❝❤ó♥❣ ❦❤æ♥❣ ♥➡♠ tr➯♥ ✭s−✶✮✲♣❤➥♥❣ tr♦♥❣ Pn ✿
r❡❣(Z) = max m1 + m2 − 1,

r❡❣(Z) = max {Tj |j = 1, ..., n} ,

tr♦♥❣ ✤â

Tj = max

q
l=1

mil + j − 2
|Pi1 , ..., Piq ♥➡♠ tr➯♥ ♠ët ❥✲♣❤➥♥❣ .
j

◆➠♠ ✷✵✶✼✱ ❚❤✐➺♥ ✈➔ ❙✐♥❤ ✭①❡♠ ❬✷✸❪✮ ✤➣ t➼♥❤ ✤÷ñ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣✭Z ✮ ❝❤♦ t➟♣ s
✤✐➸♠ ❜➨♦ ✤ç♥❣ ❜ë✐ s❛♦ ❝❤♦ ❝❤ó♥❣ ❦❤æ♥❣ ♥➡♠ tr➯♥ ✭r − 1✮✲♣❤➥♥❣✱ s ≤ r + 3 tr♦♥❣ Pn ✿
r❡❣(Z) = max {Tj |j = 1, ..., n} ,
tr♦♥❣ ✤â

Tj = max

mq + j − 2
|Pi1 , ..., Piq ♥➡♠ tr➯♥ ♠ët ❥✲♣❤➥♥❣ .
j

❱î✐ ♠♦♥❣ ♠✉è♥ ✤÷ñ❝ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t❤➯♠ ✈➲ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣
✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn ✈➔ ✤÷ñ❝ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦ P●❙✳❚❙
✺


P❤❛♥ ❱➠♥ ❚❤✐➺♥✱ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿ ✧❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦✧ ✤➸
t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤ó♥❣ tæ✐ ✤➣ t➼♥❤ ✤÷ñ❝ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠
❜➨♦ ❝❤÷❛ ♥➡♠ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤ñ♣ tr➯♥✳

◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❣ç♠ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥
✈➲ t➟♣ ✤✐➸♠ ❜➨♦ ✈➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤✳
❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ sü t➼♥❤ t♦→♥ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët sè t➟♣ ✤✐➸♠ ❜➨♦✱ ❝ö t❤➸
❧➔✿ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ s ✤✐➸♠ ❜➨♦ ♣❤➙♥ ❜✐➺t tr♦♥❣ Pn ✱ s ≤ 5❀ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛
t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣ ✈➔ ♠ët sè tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t ❝❤➾ sè ❝❤➼♥❤
q✉② ❝õ❛ n + 3 ✤✐➸♠ ❜➨♦ ❦❤æ♥❣ s✉② ❜✐➳♥ tr♦♥❣ Pn ✳

✻


❈❤÷ì♥❣ ✶
▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ❦þ ❤✐➺✉ Pn ✿❂ Pnk ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ ♥✲❝❤✐➲✉ tr➯♥
tr÷í♥❣ ✤â♥❣ ✤↕✐ sè k ✱ R ✿❂ k[x0 , ..., xn ❪ ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ t❤❡♦ ❝→❝ ❜✐➳♥ x0 , x1 , ..., xn
✈î✐ ❤➺ sè tr➯♥ k ✳ ❈→❝ ✈➔♥❤ ✤÷ñ❝ ①➨t tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à
1 = 0✳
❈→❝ ✤à♥❤ ♥❣❤➽❛✱ ✤à♥❤ ❧þ✱ ♠➺♥❤ ✤➲ ❝â t❤➸ ❞➵ ❞➔♥❣ t➻♠ t❤➜② ð ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✹❪✱
❬✺❪✱ ❬✶✶❪✲❬✶✻❪✳

✶✳✶ ❱➔♥❤ ♣❤➙♥ ❜➟❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ ❝❤✐➲✉ ❑r✉❧❧
❝õ❛ ✈➔♥❤
✶✳✶✳✶ ❱➔♥❤ ♣❤➙♥ ❜➟❝ ✈➔ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳✶✳ ❱➔♥❤ S ✤÷ñ❝ ❣å✐ ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ♥➳✉
S=

Sd
d∈Z

❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♥❤â♠ ❛❜❡♥ Sd s❛♦ ❝❤♦ ✈î✐ ❜➜t ❦ý d, e t❤➻


Sd Se ⊆ Sd+e .
▼é✐ ♣❤➛♥ tû s ∈ Sd ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t ❜➟❝ d✳ ◆➳✉ Sd ❂ ✵ ✈î✐ ♠å✐
d < 0 t❤➻ S ✤÷ñ❝ ❣å✐ ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ❞÷ì♥❣✳

❱➼ ❞ö ✶✳✶✳✶✳✶✳ ❱➔♥❤ ✤❛ t❤ù❝ R ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✈➻
R=

Rd
d≥0

tr♦♥❣ ✤â

Rd =

αc0 ...cn xc00 ...xcnn , αc0 ...cn ∈ k

f ∈R|f =
c0 +···+cn =d

✈➔ Rd Re ⊆ Rd+e .
✼


ởt I ừ S ữủ ồ t t
õ s tỷ t t

ử ỡ tự f

= xc11 ...xcnn tỷ t t ừ Rc1 +ããã+cn

c1 + ã ã ã + cn f s r ởt t t
f = {gf |g R}.

Y

ởt t t ý ừ Pn

I(Y ) = {f R | f tự t t f (P ) = 0, P Y }
ởt ừ R ữủ ồ t t ừ Y tr R

ử P P

n

ởt t t ừ R

= {f R | f (P ) = 0 f t t }

ỵ ởt ừ S =

kZ

Sk

s tữỡ ữỡ

t t
t ý t t t t ak ừ ụ tở Z
S/I ợ {(S/I)k }kZ tr õ
k k

ự I ởt ừ S

I t t t {b } ởt s ừ I
ỗ tỷ t t b d ú õ ợ a = m
i=1 ri bi ởt tỷ
(k)
ừ I ri = kZ ri t ừ ri t t t t ó
r
a=
ak
kZ
(kd1 )
1
1

ợ ak r

(kdm )
m
m

b +ã ã ã+r

b

ak õ k tỷ ừ I ợ ồ k Z

ữủ ởt s t ý ừ I õ t t t ừ
tt tỷ ừ s ụ ởt s ừ I I t
t

sỷ ữủ t ú õ ợ (S/I)k (Sk + I)/I
t õ
S/I =
(S/I)k .
kZ




❇➙② ❣✐í✱ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❜✐➸✉ ❞✐➵♥ ❝õ❛ ♠é✐ ♣❤➛♥ tû t❤✉ë❝ S/I t❤➔♥❤ tê♥❣ ❝õ❛ ❝→❝
♣❤➛♥ tû t❤✉ë❝ (S/I)k ✱ k ∈ Z ❧➔ ❞✉② ♥❤➜t✿ ●✐↔ sû k∈Z sk = 0 tr♦♥❣ ✤â sk = sk + I
t❤✉ë❝ (S/I)k ✱ sk ∈ Sk ✳ ❑❤✐ ✤â k∈Z sk ∈ I ♥➯♥ sk ∈ I ✳ ❉♦ ✤â sk ❂ ✵ ✈î✐ ♠å✐ k ∈ Z✳
❱➟②

S/I =

(S/I)k ,
k∈Z

❤❛② S/I ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✈î✐ ♣❤➙♥ ❜➟❝ {(S/I)k }k∈Z ✳
◆❣÷ñ❝ ❧↕✐✱ ❝❤♦ a = k∈Z ak ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ I ✱ ak ∈ Sk ✈î✐ ♠å✐ k ∈ Z✳ ❑❤✐
✤â k∈Z ak t❤✉ë❝ S/I ✈î✐ ak = ak + I ✳ ❉♦ ✤â ak ∈ I ✈î✐ ♠å✐ k ∈ Z✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳✹✳ ❈❤♦ S =

d∈Z

Sd ❧➔ ♠ët ✈➔♥❤ ♣❤➙♥ ❜➟❝✱ ♠ët S ✲♠æ✤✉♥ ♣❤➙♥

❜➟❝ ❧➔ ♠ët S ✲♠æ✤✉♥ M ✈î✐


M=

Mn ,
n∈Z

tr♦♥❣ ✤â Mn ❧➔ ♥❤â♠ ❛❜❡♥ s❛♦ ❝❤♦ ✈î✐ ♠å✐ m, n ∈ Z t❤➻

Sn Mm ⊆ Mn+m .
❈→❝ ♣❤➛♥ tû ❝õ❛ Mn ✤÷ñ❝ ❣å✐ ❧➔ ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t ❜➟❝ n✳

❱➼ ❞ö ✶✳✶✳✶✳✹✳ R = k[x , ..., x ] ❧➔ ♠ët ✈➔♥❤ ♣❤➙♥ ❜➟❝✳ ▲ó❝ ✤â✱ R ❧➔ ♠ët R✲♠æ✤✉♥
0

n

♣❤➙♥ ❜➟❝ ✈î✐

R=

Rd ,
d≥0

tr♦♥❣ ✤â

Rd = {f ∈ R | f =

c0 +···+cn =d

αc0 ...cn xc00 ...xcnn , αc0 ...cn ∈ k}✳


✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳✺✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✈î✐ ♣❤➙♥ ❜➟❝ (A )

✈➔ E ❧➔
♠ët A✲✤↕✐ sè✳ E ✤÷ñ❝ ❣å✐ ❧➔ A✲✤↕✐ sè ♣❤➙♥ ❜➟❝ ♥➳✉ ♥❤â♠ ❝ë♥❣ ❝õ❛ E ❧➔ tê♥❣ trü❝
t✐➳♣ ❝→❝ ♥❤â♠ ❛❜❡♥ En
En
E=
n n∈Z

n≥0

s❛♦ ❝❤♦ ✈î✐ ❜➜t ❦➻ m, n t❛ ❝â✿

Am En ⊆ Em+n
✈➔

Em En ⊆ Em+n .

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳✻✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✱ R =

Rn
❧➔ ♠ët A✲✤↕✐ sè ♣❤➙♥ ❜➟❝✳ R ✤÷ñ❝ ❣å✐ ❧➔ ♠ët A✲✤↕✐ sè ♣❤➙♥ ❜➟❝ ❝❤✉➞♥ ♥➳✉ R0 = A
✈➔ R ✤÷ñ❝ s✐♥❤ ❜ð✐ ❝→❝ ♣❤➛♥ tû ❜➟❝ ✶✱ tù❝ ❧➔ R = R0 [R1 ]✳
✾

n≥0


t


f ởt tự t t ừ R
Z(f ) = {P Pn |f (P ) = 0}
ữủ ồ t ổ ừ f

T t tỷ t t ừ R
Z(T ) = {P Pn |f (P ) = 0, f T }
ữủ ồ t ổ ừ T

ởt t Y

ừ Pn ữủ ồ t số tỗ t ởt
t T tỷ t t ừ R s Y = Z(T )

ủ ừ t số t số ởt ồ tý ỵ t
số t số Pn t số

ổổ tr P

ữủ t ũ ừ
t số ữủ ồ tổổ rs
n

r ừ

B ởt 0 r ừ B
tr ừ số n s tỗ t 0 1
B ỵ r ừ B B

ããã


n tố ừ

ử k ởt trữớ ú õ k = 0 t k ởt

trữớ k õ ởt tố t {0} õ k = 0
R = k[x0 , ..., xn ú õ R = n + 1 t t õ
tố

0 = {0}

1 = x0

2 = x0 , x1

...

n+1 = x0 , x1 , ..., xn .

ợ f n+1 t õ f tự õ tỷ tỹ 0
g R \ n+1 J = x0 , x1 , ..., xn , g ừ R t õ tự g õ
tỷ tỹ c = 0
õ g c n+1 J 1 = c1 (g (g c)) J R = J
n+1 ỹ
ỡ ỳ ợ ồ k {1, 2, ..., n + 1} tố ừ R s

k

k+1 ,




t❛ ❝❤ù♥❣ ♠✐♥❤ ℘ = ℘k+1 ✿
❚❛ ❝â ℘ \ ℘k = ∅✳ ❉♦ ℘ ⊆ ℘k+1 ♥➯♥ ♠å✐ ✤❛ t❤ù❝ tr♦♥❣ ℘ \ ℘k ❝❤ù❛ ➼t ♥❤➜t ♠ët
✤ì♥ t❤ù❝ ❞↕♥❣
ik+1
xikk xk+1
...xinn , ik ≥ 1
✭✶✳✶✮
●å✐ lk ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ ❜➨ ♥❤➜t ik s❛♦ ❝❤♦ ♠ët ✤ì♥ t❤ù❝ ❝â ❞↕♥❣ ✭✶✳✶✮ ❝â ❤➺ sè
❦❤→❝ 0 tr♦♥❣ ❝→❝ ✤ì♥ t❤ù❝ tr♦♥❣ ℘ \ ℘k ✳
▲ó❝ ✤â✱ tç♥ t↕✐ ♠ët ✤❛ t❤ù❝ f ∈ ℘ \ ℘k ❝❤ù❛ ♠ët ✤ì♥ t❤ù❝ ❝â ❞↕♥❣ ✭✶✳✶✮ ✈î✐ ik = lk
i

k+1
xlkk xk+1
...xinn .

✭✶✳✷✮

❚❛ ✈✐➳t f = f1 + f2 ✱ tr♦♥❣ ✤â

cij xij ∈ ℘k ,

f1 =
ij ≥1;

✈î✐ ♠å✐

j=∈{1,...,k−1}


cij xij = xlkk h, h ∈ R.

f2 =
ij =0;

✈î✐ ♠å✐

j=∈{1,...,k−1};ik ≥lk

❉♦ f2 ❝❤ù❛ ✤ì♥ t❤ù❝ ❞↕♥❣ ✭✶✳✷✮✱ tr♦♥❣ ✤â ❜✐➳♥ xk ❝â ❜➟❝ lk ♥➯♥ ✤❛ t❤ù❝ h ❝❤ù❛ ♠ët
✤ì♥ t❤ù❝ ❦❤æ♥❣ ❝❤✐❛ ✤÷ñ❝ ❜ð✐ xi ✱ ∀i = 1, k − 1✳
❙✉② r❛ h ∈
/ ℘k+1 ✱ ❞♦ ✤â xlkk −1 h ∈
/℘
❚❛ ❝â f2 = f − f1 ∈ ℘ ✭ ✈➻ f ∈ ℘ ✈➔ f1 ∈ ℘k−1 ℘ ✮✱

f2 = xlkk h = xk (xlkk −1 h).
l −1

❉♦ xkk h ∈
/ ℘ ✈➔ ℘ ❧➔ ✐❞❡❛❧ ♥❣✉②➯♥ tè ♥➯♥ xk ∈ ℘ ✳ ❉♦ ✤â✱ ℘ ⊆ x1 , ..., xk ✱ ❤❛②
℘ = ℘k+1 ✳
❱➟② ❞✐♠R = n + 1✳
❝✮ ❈❤♦ P = (1, 0, ..., 0) ∈ Pn ❝â ✐❞❡❛❧ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ I = (x1 , ..., xn )✳ ▲ó❝
✤â✱ ❞✐♠(R/I) = 1✳ ❚❤➟t ✈➟②✱ ✈î✐ f ∈ R = k[x0 , ..., xn ]✱ t❛ ❝â t❤➸ ✈✐➳t f = h + g ✈î✐
h ∈ k[x0 ] ✈➔ ❦❤æ♥❣ ❝â ✤ì♥ t❤ù❝ ❦❤→❝ ❦❤æ♥❣ ♥➔♦ ❝õ❛ g t❤✉ë❝ k[x0 ]✳ ❳➨t →♥❤ ①↕

ϕ : R −→ k[x0 ]
✤÷ñ❝ ①→❝ ✤à♥❤ ϕ(f ) = h ✈î✐ ♠é✐ f = h + g ✳


✶✶


✰ ❱î✐ f1 = h1 + g1 , f2 = h2 + g2 ∈ R✱ t❛ ❝â

ϕ(f1 + f2 ) = h1 + h2
= ϕ(f1 ) + ϕ(f2 ),
ϕ(f1 f2 ) = h1 h2
= ϕ(f1 )ϕ(f2 ).
❱➻ ✈➟② ϕ ❧➔ ♠ët ✤ç♥❣ ❝➜✉✳
✰ ❱î✐ ♠é✐ h ∈ k[x0 ]✱ t❛ ❧➜② f = h ❧ó❝ ✤â ϕ(f ) = h ♥➯♥ ϕ ❧➔ ♠ët t♦➔♥ ❝➜✉✳

Ker(ϕ) = {f ∈ R|ϕ(f ) = 0}
= (x1 , ..., xn ).
❉♦ ✤â✱ R/(x1 , ..., xn ) ∼
= k[x0 ] ♥➯♥ ❞✐♠(R/I) = dimk[x0 ] = 1✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳✷✳ ❈❤♦ M ❧➔ ♠ët B ✲♠æ✤✉♥✱ M = ✵✳ ❑❤✐ ✤â
❆♥♥(M ) = {a ∈ B|aM = 0}
❧➔ ♠ët ✐❞❡❛❧ ❝õ❛ B ✳ ❈❤✐➲✉ ❑r✉❧❧ ❝õ❛ ♠æ✤✉♥ M ❧➔
❞✐♠M := ❞✐♠(B/Ann(M )).

❱➼ ❞ö ✶✳✶✳✸✳✷✳ ❈❤♦ R = k[x , ..., x ❪✱ ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ ✈➔♥❤ R ❧➔ n + 1✳ ❚❛ ❝â
0

n

❆♥♥(R) = 0 ♥➯♥ ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ k ✲♠æ✤✉♥ R ❧➔


❞✐♠R = ❞✐♠(k/Ann(R)) = ❞✐♠k = 0.

✶✳✶✳✹ ❱➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳✶✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ✈➔ M

❧➔ ♠ët A✲♠æ✤✉♥✳ ▼ët ♣❤➛♥ tû
a ∈ A ✤÷ñ❝ ❣å✐ ❧➔ M ✲❝❤➼♥❤ q✉② ♥➳✉ ax = 0✱ ∀ 0 = x ∈ M ✱ tù❝ ❧➔ a ❦❤æ♥❣ ❧➔ ÷î❝
❝õ❛ 0 tr➯♥ M ✳

❱➼ ❞ö ✶✳✶✳✹✳✶✳ M = A = k[x]✱ k ❧➔ ♠ët tr÷í♥❣✳ ▲ó❝ ✤â✱ x ❧➔ ❝❤➼♥❤ q✉② tr➯♥ A✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳✷✳ ▼ët ❞➣② a , ..., a ❝→❝ ♣❤➛♥ tû ❝õ❛ A ❧➔ ♠ët M ✲❞➣② ✭❤❛② ♠ët
1

r

M ✲❞➣② ❝❤➼♥❤ q✉②✮ ♥➳✉ t❤♦↔ ✷ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✶✷


✐✮ a1 ❧➔ M ✲❝❤➼♥❤ q✉②✱ a2 ❧➔ M/(a1 M )✲❝❤➼♥❤ q✉②✱✳✳✳✱ ar ❧➔ M/((a1 , ..., ar−1 )M )✲
❝❤➼♥❤ q✉②✳
✐✐✮ M/((a1 , ..., ar )M ) = 0✳

▲÷✉ þ ✶✳✶✳✹✳✶✳ ◆➳✉ a , ..., a

❧➔ ♠ët M ✲❞➣② t❤➻ at11 , ..., atrr ❝ô♥❣ ✈➟②✱ ✈î✐ ❜➜t ❦ý ❝→❝
sè ♥❣✉②➯♥ ❞÷ì♥❣ ti ✳ ❚✉② ♥❤✐➯♥✱ ♥➳✉ a1 , ..., ar ❧➔ ♠ët M ✲❞➣② t❤➻ ❦❤æ♥❣ ❝â ♥❣❤➽❛ ❤♦→♥
✈à ❝õ❛ a1 , ..., ar ❧➔ M ✲❞➣②✳
1


❱➼ ❞ö ✶✳✶✳✹✳✷✳ ✶✮ x , ..., x

r

tr♦♥❣ ✈➔♥❤ ✤❛ t❤ù❝ R = A[x1 , ..., xr ] ❧➔ R✲❝❤➼♥❤ q✉②✳
✷✮ ❈❤♦ A = k[x, y, z]✱ k ❧➔ ♠ët tr÷í♥❣✳ ▲ó❝ ✤â✱ x, y(1 − x), z(1 − x) ❧➔ ♠ët A✲❞➣②
♥❤÷♥❣ y(1 − x), z(1 − x), x ❦❤æ♥❣ ❧➔ A✲❞➣②✳
1

r

❈❤ó þ r➡♥❣ z(1 − x) ❦❤æ♥❣ ❝❤➼♥❤ q✉② tr➯♥ A/(y(1 − x)) ✈➻

z(1 − x)y = zy − zxy = zy − zy = 0 ✭ ❞♦ y = yx)
tr➯♥ A/((y(1 − x))) ♥➯♥ ♥â ❧➔ ÷î❝ ❝õ❛ ✵✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳✸✳ ▼ët M ✲❞➣② x , ..., x
1

n

❧➔ ❝ü❝ ✤↕✐ ♥➳✉ x1 , ..., xn , xn+1 ❦❤æ♥❣ ❧➔

♠ët M ✲❞➣②✱ ✈î✐ ♠å✐ xn+1 ∈ R✳

▲÷✉ þ ✶✳✶✳✹✳✷✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r✱ M ❧➔ ♠ët A✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈➔

I ❧➔ ♠ët ✐❞❡❛❧ ❝õ❛ A ✈î✐ IM = M ✳ ❱î✐ ❜➜t ❦ý M ✲❞➣② ❝❤➼♥❤ q✉② x1 , ..., xm ✱ t❛ ❝â
(x1 , ..., xi )M = (x1 , ..., xi+1 )M ✱ ✈î✐ ♠å✐ i = 0, ..., m − 1✳ ❱➻ M ❧➔ ♠æ✤✉♥ ◆♦❡t❤❡r

♥➯♥ M ✲❞➣② ❝❤➼♥❤ q✉② x1 , ..., xm ✱ ✈î✐ xi ∈ I ❝â t❤➸ ♠ð rë♥❣ t❤➔♥❤ ♠ët ❞➣② ❝ü❝ ✤↕✐✱
tù❝ ❧➔ t❤➔♥❤ ♠ët M ✲❞➣② ❝❤➼♥❤ q✉② x1 , ..., xn tr♦♥❣ I(n ≥ m) s❛♦ ❝❤♦ ❜➜t ❦ý a ∈ I
❧➔ ÷î❝ ❝õ❛ ❦❤æ♥❣ tr♦♥❣ M/((x1 , ..., xn )M )✳

▼➺♥❤ ✤➲ ✶✳✶✳✹✳✶✳ ❚➜t ❝↔ ❝→❝ M ✲❞➣② ❝ü❝ ✤↕✐ ❝â ❝ò♥❣ ✤ë ❞➔✐ ♥➳✉ M ❧➔ ❤ú✉ ❤↕♥ s✐♥❤
✈➔ A ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳✹✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✱ t❛ ❣å✐ ✤ë ❞➔✐ tr♦♥❣ ♠➺♥❤ ✤➲

tr➯♥ ❧➔ ✤ë s➙✉ ❝õ❛ M ✈➔ ❦þ ❤✐➺✉ ❧➔ depth(M )✳ ◆➳✉ t❛ ✤❛♥❣ ♥â✐ ✈➲ M ✲❞➣② tr♦♥❣ ♠ët
✐❞❡❛❧ ❝ü❝ ✤↕✐ I ❝õ❛ A✱ t❤➻ t❛ ❦➼ ❤✐➺✉ ❞❡♣t❤(I, A)✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳✺✳ ❈❤♦ A ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ M ❧➔ ♠ët A✲♠æ✤✉♥

❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â M ✤÷ñ❝ ❣å✐ ❧➔ ♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✭✈✐➳t t➢t ❧➔ ❈▼✮ ♥➳✉
M = 0 ❤♦➦❝ ❞❡♣t❤(M ) = dimM ✳
◆➳✉ A✲♠æ✤✉♥ A ❧➔ ♠ët ♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ t❛ ♥â✐ A ❧➔ ♠ët ✈➔♥❤ ❈♦❤❡♥✲
▼❛❝❛✉❧❛②✳

❱➼ ❞ö ✶✳✶✳✹✳✸✳ ✶✮ ❇➜t ❦ý ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r 0✲❝❤✐➲✉ ✤➲✉ ❧➔
♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ✤➦❝ ❜✐➺t ✈➔♥❤ 0✲❝❤✐➲✉ ◆♦❡t❤❡r ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳
✶✸


✷✮ ◆➳✉ k ❧➔ ♠ët tr÷í♥❣ t❤➻ R = k[X1 , X2 ]/(X12 , X1 , X2 ) ❦❤æ♥❣ ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳
❚❤➟t ✈➟②✱ ♥➳✉ ℘ ❧➔ ✐❞❡❛❧ tr♦♥❣ R s✐♥❤ ❜ð✐ ❝→❝ ↔♥❤ ❝õ❛ X1 , X2 t❤➻ ❞✐♠R℘ = 1 ♥❤÷♥❣
℘R℘ ❝❤➾ ❣ç♠ ❝→❝ ÷î❝ ❝õ❛ 0✳

✶✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦
✶✳✷✳✶ ❍➔♠ ❍✐❧❜❡rt ✈➔ ✤❛ t❤ù❝ ❍✐❧❜❡rt


✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳✶✳ ▼ët ✤❛ t❤ù❝ sè ❧➔ ♠ët ✤❛ t❤ù❝ P (z) ∈ Q[z] s❛♦ ❝❤♦ P (n) ∈ Z
✈î✐ ♠å✐ n ✤õ ❧î♥✱ n ∈ Z✳

❱➼ ❞ö ✶✳✷✳✶✳✶✳ ❛✮ ▼å✐ ✤❛ t❤ù❝ ❝â ❤➺ sè ♥❣✉②➯♥ ✤➲✉ ❧➔ ✤❛ t❤ù❝ sè✳
❜✮ ❱î✐ ♠é✐ i ∈ N ✤❛ t❤ù❝ s❛✉ ❧➔ ✤❛ t❤ù❝ sè

x+i
,
i

Pi (x) =
tr♦♥❣ ✤â

(x + i)(x + i − 1)...(x + 1)
x+i
=
, i ∈ N∗ ,
i
i!
✈➔ q✉② ÷î❝

x+0
= 1.
0

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳✷✳ ❈❤♦ M

❧➔ ♠ët R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤✱ M =
t∈Z Mt ✱ Mt ❧➔ ❝→❝ ♥❤â♠ ❛❜❡♥✳ ❍➔♠ ❍✐❧❜❡rt ❝õ❛ M ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔


HM (t) = dimk Mt , t ∈ Z.

❱➼ ❞ö ✶✳✷✳✶✳✷✳ ❱î✐ ♠é✐ n ∈ N t❛ ❝â

HR (t) = dimk Rt =

n+t
, t ∈ N.
n

✭✶✳✸✮

❚❤➟t ✈➟②✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ tr➯♥ n✳
❱î✐ n = 0 t❤➻ R = k[x0 ❪✱ ❞♦ ✤â dimk Rt = 1✱ t ∈ N✳ ❱➟② ✭✶✳✸✮ ✤ó♥❣ ✈î✐ n = 0✳
●✐↔ sû ✭✶✳✸✮ ✤ó♥❣ ✈î✐ n = k ✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✸✮ ✤ó♥❣ ✈î✐ n = k + 1✿ ❚❛
❝â R = k[x0 , ..., xk+1 ]✱ ❦❤✐ ✤â dimk Rt ✤ó♥❣ ❜➡♥❣ sè ❤↕♥❣ tû tr♦♥❣ ❦❤❛✐ tr✐➸♥
(x0 + x1 + · · · + xk+1 ✮t ✳ ❚❛ ❝â

(x0 + x1 + · · · + xk+1 )t = ((x0 + x1 + · · · + xk ) + xk+1 )t
t

t
(x0 + x1 + · · · + xk )i xt−i
k+1 .
i

=
i=0


✶✹


❚❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❤➻ sè ❤↕♥❣ tû tr♦♥❣ ❦❤❛✐ tr✐➸♥ (x0 + x1 + · · · + xk )i ❜➡♥❣
k+i
✳ ❉♦ ✤â sè ❤↕♥❣ tû ❝õ❛ ✭x0 + x1 + · · · + xk+1 ✮t ❜➡♥❣
k
t

k+i
t+k+1
=
,
k
k+1

i=0

k, t ∈ N.

✣à♥❤ ❧þ ✶✳✷✳✶✳✶✳ ❈❤♦ M

❧➔ ♠ët R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝â ❞✉②

♥❤➜t ♠ët ✤❛ t❤ù❝ sè PM (z) ∈ Q[z] s❛♦ ❝❤♦ HM (t) = PM (t) ✈î✐ ♠å✐ sè ♥❣✉②➯♥ t ✤õ
❧î♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳✸✳ ✣❛ t❤ù❝ P

M


①→❝ ✤à♥❤ tr♦♥❣ ✤à♥❤ ❧þ tr➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ✤❛ t❤ù❝

❍✐❧❜❡rt ❝õ❛ M ✳

✣à♥❤ ❧þ ✶✳✷✳✶✳✷✳ ❈❤♦ M = 0 ❧➔ ♠ët R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤ ❝â ❝❤✐➲✉ ❧➔
d✱ ❧ó❝ ✤â ✤❛ t❤ù❝ ❍✐❧❜❡rt PM (t) ❝â ❜➟❝ d − 1 ✈➔ ✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣
d−1

t+d−i−1
d−i−1

(−1)i ei

PM (t) =
i=0

✈î✐ e0 , ..., ed−1 ∈ Z✳

✶✳✷✳✷ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳✶✳ ❈❤♦ X = {P , ..., P } ❧➔ t➟♣ ❤ñ♣ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣
1

s

P ✳ ▼é✐ ✤✐➸♠ Pi ∈ P ✈î✐ i = 1, ..., s❀ t❛ ①→❝ ✤à♥❤ ♠ët t➟♣
n

n


℘i = {f ∈ R|f t❤✉➛♥ ♥❤➜t , f (Pi ) = 0}.
✣➙② ❧➔ ♠ët ✐❞❡❛❧ ♥❣✉②➯♥ tè t❤✉➛♥ ♥❤➜t ❝õ❛ R✱ ❣å✐ ❧➔ ✐❞❡❛❧ ♥❣✉②➯♥ tè t❤✉➛♥ ♥❤➜t ①→❝
✤à♥❤ ❜ð✐ Pi ∈ Pn ✳
❈❤♦ m1 , ..., ms ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✣➦t✿
ms
1
I := ℘m
1 ∩ ... ∩ ℘s ,

❚❛ ❣å✐ (X, I) ❧➔ t➟♣ ✤✐➸♠ ❜➨♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn ✈➔ ❦þ ❤✐➺✉ t➟♣ ✤✐➸♠ ❜➨♦
❧➔

Z := m1 P1 + · · · + ms Ps .
✶✺


m1 = m2 = ã ã ã = ms = m t Z ữủ ồ t ỗ ở tr Pn
m1 = m2 = ã ã ã = ms = 2 t Z ữủ ồ t tr Pn
A := R/I ữủ ồ t ở t t ừ Z ú õ A
số A = t0 At (A) = 1
t rt

HA (t) = dimk At .
ữớ t ự ữủ r rt HA (t) t t õ t
mi + n 1
õ õ ứ
ữủ số ở e0 = si=1
n


ố t t s H

A

(t) = e0 ữủ ồ số

q ừ t Z ỵ r(Z)

ử P P m Z

Z = mP õ r(Z) = m 1 t
ồ t t P t A = R/m õ rt
HA (t) k At t t õ t số ở
n

+

e0 =

m+n1
.
n

õ

HA (t) = dimk At
= dimk Rt dimk [m ]t
n+t
dimk [m ]t .
=

n
ợ t m 2 t k [m ]t = 0 s r

HA (t) =

n+t
m+n1
<
= e0
n
n


ợ t = m 1 t

HA (m 1) = dimk Am1
n+m1
=
t ử
n
m+n1
=
= e0 .
n
r(Z) = m 1



❱✐➺❝ t➼♥❤ ❝❤➾ sè ❝❤➼♥❤ q✉② r❡❣✭Z ✮ tr♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t ❧➔ ❦❤æ♥❣ ❞➵ ❞➔♥❣✱
✈➜♥ ✤➲ ♥➔② ❝á♥ ❧➔ ♠ët ❜➔✐ t♦→♥ ♠ð✳


✶✳✷✳✸ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ✈î✐
❝❤➾ sè ❝❤➼♥❤ q✉② ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ ❝õ❛ ✈➔♥❤ t♦↕
✤ë✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✶✳ ●✐↔ sû C ✈➔ D ❧➔ ❤❛✐ ♣❤↕♠ trò ❝❤♦ tr÷î❝✳❚❛ ♥â✐ r➡♥❣ Φ ❧➔ ♠ët

❤➔♠ tû ❤✐➺♣ ❜✐➳♥ tø C ✈➔♦ D ♥➳✉ ♠é✐ ✈➟t E ❝õ❛ C t÷ì♥❣ ù♥❣ ✈î✐ ♠é✐ ✈➟t Φ(E) ❝õ❛ D
✈➔ ♠é✐ ❝➜✉ ①↕ f : E −→ F ❝õ❛ C t÷ì♥❣ ù♥❣ ✈î✐ ♠é✐ ❝➜✉ ①↕ Φ(f ) : Φ(E) −→ Φ(F )
❝õ❛ D s❛♦ ❝❤♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷ñ❝ t❤♦↔ ♠➣♥ ✈î✐ ♠å✐ ✈➟t ❝õ❛ C ✈➔ ♠å✐ sì ✤ç

E

f

✲

g

F

✲

H

tr♦♥❣ C ✿

Φ(id(E)) = idΦ(E) ✈➔ Φ(f og) = Φ(f )oΦ(g).

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✷✳ ❈❤♦ F : M od(R) −→ M od(R) ❧➔ ❤➔♠ tû ❤✐➺♣ ❜✐➳♥✱ F ✤÷ñ❝
❣å✐ ❧➔ ❤➔♠ tû ❦❤î♣ tr→✐ ✭ ♣❤↔✐ ✮ ♥➳✉ ✈î✐ ♠é✐ ❞➣② ❦❤î♣ ♥❣➢♥


0

A

✲

α✲

β

B

✲

C

0

✲

t❤➻ ❞➣② t÷ì♥❣ ù♥❣ s❛✉ ❧➔ ❦❤î♣✳

0



F (A)

✲


F (A)

F (α)

✲

F (α)

F (B)

✲

F (B)

F (β)

✲

F (β)

F (C)

✲

F (C)

✲

0





.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✸✳ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥✳ ▼ët ❞➣② ❦❤î♣ ❝→❝ R✲♠æ✤✉♥
0

✲

M

ε✲

E1

d0✲

E2

d1✲

...

✤÷ñ❝ ❣å✐ ❧➔ ❣✐↔✐ t❤ù❝ ♥ë✐ ①↕ ❝õ❛ M ♥➳✉ ❝→❝ Ei ❧➔ ❝→❝ ♠æ✤✉♥ ♥æ✐ ①↕✱ ✈î✐ ♠å✐ i =
0, 1, 2, ... ✳
✶✼



✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✹✳ ❈❤♦ ♠ët ✐❞❡❛❧ I ⊆ R ✈➔ ♠ët R✲♠æ✤✉♥ M ✱ t➟♣
ΓI (M ) := ∪n (O :M I n )
❧➔ t➟♣ ❝→❝ ♣❤➛♥ tû ❝õ❛ M ✤÷ñ❝ ❧✐♥❤ ❤♦→ ❜ð✐ ❜➟❝ ❝õ❛ I ✳ ❈❤ó þ r➡♥❣ ΓI (M ) ❧➔
♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❱î✐ ♠é✐ ✤ç♥❣ ❝➜✉ ❝õ❛ ❝→❝ R✲♠æ✤✉♥ f : M → N ✱ t❛ ❝â
f (ΓI (M )) ⊆ ΓI (N ) ✈➔ ✈➻ ✈➟② t❛ ❝â →♥❤ ①↕ ΓI (f ) : ΓI (M ) → ΓI (N ) ①→❝ ✤à♥❤ ❜ð✐
ΓI (f )(a) = f (a)✱ ✈î✐ a ∈ ΓI (M )✳
❍➔♠ tû ΓI (❴) : C (R) → C (R) ✤÷ñ❝ ❝❤♦ ♥❤÷ s❛✉ ❧➔ ❤➔♠ tû ❤✐➺♣ ❜✐➳♥✳
✐✮ ❱î✐ ♠é✐ ✈➟t A ❝õ❛ C (R) ❝❤♦ t÷ì♥❣ ù♥❣ ✈î✐ ♠é✐ ✈➟t ΓI (A) ❝õ❛ C (R)✳
✐✐✮ ❱î✐ ♠é✐ ❝➜✉ ①↕ f : A → B ❝õ❛ C (R) ❝❤♦ t÷ì♥❣ ù♥❣ ✈î✐ ❝➜✉ ①↕

ΓI (f ) : ΓI (A) → ΓI (B).
❍➔♠ tû ❤✐➺♣ ❜✐➳♥ ΓI (❴) tr➯♥ ♣❤↕♠ trò ❝→❝ R✲♠æ✤✉♥ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ tû I ✲t♦rs✐♦♥✳

▼➺♥❤ ✤➲ ✶✳✷✳✸✳✶✳ Γ (❴) ❧➔ ♠ët ❤➔♠ tû ❦❤î♣ tr→✐✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✺✳ ❱î✐ i ∈ N✱ ❤➔♠ tû ✤è✐ ✤ç♥❣ ✤✐➲✉ ✤à❛ ♣❤÷ì♥❣ t❤ù i ✈î✐ ❣✐→ ❧➔
I

I ✱ HIi (❴)✱ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ ♠ët ❤➔♠ tû ❞➝♥ ①✉➜t ♣❤↔✐ t❤ù i ❝õ❛ ΓI (❴)✳
❈❤♦ ♠ët R✲♠æ✤✉♥ M ✱ →♣ ❞ö♥❣ ❤➔♠ tû HIi (❴) ❝❤♦ M t❛ ❝â HIi (M ) ✤÷ñ❝ ❣å✐ ❧➔
♠æ✤✉♥ ✤è✐ ✤ç♥❣ ✤✐➲✉ t❤ù i ❝õ❛ M ✈î✐ ❣✐→ ❧➔ I ✿ ❈❤♦ ♠ët R✲♠æ✤✉♥ M ✈➔ I ❧➔ ♠ët
❣✐↔✐ t❤ù❝ ♥ë✐ ①↕ ❝õ❛ M ✳
I:0

✲

E0

d0 ✲

E1


d1 ✲

...

dn−1
✲

En

dn ✲

...

⑩♣ ❞ö♥❣ ΓI (❴) ✈➔♦ I t❛ ✤÷ñ❝ ♣❤ù❝ ✿

ΓI (I) : 0

✲

ΓI (E 0 )

ΓI (d0 )

✲

ΓI (E 1 )

ΓI (d1 )


✲

...

ΓI (dn−1 )

✲

ΓI (E n )

❑❤✐ ✤â✱

HI0 (M ) := ΓI (M )
✈➔

HIi (M ) := Ker(ΓI (di ) Im(ΓI (di−1 )) ✈î✐ ✐ ❃ ✵✳

▲÷✉ þ ✶✳✷✳✸✳✶✳
• HIi (❴) ❧➔ ❤➔♠ tû ❤✐➺♣ ❜✐➳♥✳
• ◆➳✉ E ❧➔ ♠ët R✲♠æ✤✉♥ ♥ë✐ ①↕ t❤➻ HIi (E) = 0✱ ✈î✐ ♠å✐ i > 0✳
✶✽

ΓI (dn )

✲

...


✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✻✳ ❈❤♦ R ❧➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝✱ M =


n∈N

Mn ❧➔ R✲♠æ✤✉♥ ♣❤➙♥

❜➟❝✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❡♥❞ ❝õ❛ M ❧➔

end(M ) := sup{n ∈ Z|Mn = 0} ♥➳✉ s✉♣ ♥➔② tç♥ t↕✐
✈➔ end(M ) = ∞ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❦❤→❝ .

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳✼✳ ❈❤♦ R =

Rn ❧➔ ✤↕✐ sè ♣❤➙♥ ❜➟❝ ❝❤✉➞♥ ❞÷ì♥❣✱ M ❧➔
♠ët R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤✳ ❱î✐ l ∈ N✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❝❤➾ sè ❝❤➼♥❤ q✉②
❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ r❡❣✭M ✮ ❝õ❛ M ♥❤÷ s❛✉
n∈N

reg(M ) : = sup{end(HRi + (M )) + i|i ∈ N}
= sup{end(HRi + (M )) + i|0 ≤ i ≤ dimM }.
❇➙② ❣✐í t❛ ①➨t✱ ✈➔♥❤ t♦↕ ✤ë A = R/I ❝õ❛ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ Z = m1 P1 + · · · +
ms Ps ✳ ❚❛ ❝â ❝❤➾ sè ❝❤➼♥❤ q✉② ❈❛st❡❧♥✉♦✈♦✲▼✉♠❢♦r❞ r❡❣✭A✮✳ ◆❣÷í✐ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤
✤÷ñ❝ r➡♥❣

reg(A) = reg(Z),
✈î✐ r❡❣✭Z ✮ ❧➔ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ Z ✳ ❱➻ ❧þ ❞♦ ♥➔② t❛ ❝â t❤➸ ✈✐➳t r❡❣✭A✮
t❤❛② ❝❤♦ r❡❣✭Z ✮ ✈➔ ♥❣÷ñ❝ ❧↕✐✳

✶✾



❈❤÷ì♥❣ ✷
❈❍➓ ❙➮ ❈❍➑◆❍ ◗❯❨ ❈Õ❆ ▼❐❚ ❙➮
❚❾P ✣■➎▼ ❇➆❖ ❚❘❖◆● ❑❍➷◆●
●■❆◆ ❳❸ ❷◆❍ PN
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ t✐➳♣ tö❝ ❞ò♥❣ ❦þ ❤✐➺✉ Pn ✿❂ Pnk ❧➔ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤
n✲❝❤✐➲✉ tr➯♥ tr÷í♥❣ ✤â♥❣ ✤↕✐ sè k ✈➔ R := k[x0 , ..., xn ] ❧➔ ✈➔♥❤ ✤❛ t❤ù❝ t❤❡♦ ❝→❝ ❜✐➳♥
x0 , x1 , ..., xn ✈î✐ ❤➺ sè tr➯♥ k ✳ ✣✐➲✉ ♥➔② ♣❤ò ❤ñ♣ ✈î✐ ♥❤ú♥❣ q✉② ÷î❝ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦ ❬✷❪✱ ❬✸❪✱ ❬✶✽❪✱ ❬✶✾❪✱ ❬✷✶❪✲❬✷✸❪ ✤÷ñ❝ sû ❞ö♥❣ tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✳

✷✳✶ ❈❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠ tr➯♥ ✷
✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t
❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ s➩ ÷î❝ ❧÷ñ♥❣ ❝❤➾ sè ❝❤➼♥❤ q✉② ❝õ❛ t➟♣ ✤✐➸♠ ❜➨♦ ♥➡♠
tr➯♥ ✷ ✤÷í♥❣ t❤➥♥❣ ♣❤➙♥ ❜✐➺t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ①↕ ↔♥❤ Pn ✳ ❈❤ó♥❣ tæ✐ s➩ ❝➛♥ ❞ò♥❣ ❝→❝
❦➳t q✉↔ s❛✉ tr♦♥❣ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ❝õ❛ ♠➻♥❤✳
❈❤♦ Z = m1 P1 + · · · + ms Ps ❧➔ ♠ët t➟♣ ✤✐➸♠ ❜➨♦ ❜➜t ❦ý tr♦♥❣ Pn ✳ ✣➦t

T (Z) = max{Tj (Z), j = 1, ..., n},
tr♦♥❣ ✤â

Tj (Z) = max

q
l=1

mil + j − 2
|Pi1 , ..., Piq ♥➡♠ tr➯♥ ♠ët ❥✲♣❤➥♥❣ .
j

❇ê ✤➲ ✷✳✶✳✶✳ ✭❬✷✶❪✱ ▲❡♠♠❛ ✸✳✸✮ ❈❤♦ X = {P , ..., P } ❧➔ ❝→❝ ✤✐➸♠ ♣❤➙♥ ❜✐➺t tr♦♥❣
1


P ✈➔ m1 , ..., ms ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✣➦t I = ℘
n

s
m1
1

s
∩ ... ∩ ℘m
s ✳

mi

m

◆➳✉ Y = {Pi1 , ..., Pir } ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ X ✈➔ J = ℘i1 1 ∩ ... ∩ ℘ir ir ✱ t❤➻
r❡❣(R/J) ≤ r❡❣(R/I)✳
◆➳✉ t❛ ❣å✐ Z = m1 P1 + · · · + ms Ps ❧➔ t➟♣ ✤✐➸♠ ❜➨♦ ①→❝ ✤à♥❤ ❜ð✐ ✐❞❡❛❧ I ✈➔
U = mi1 Pi1 + · · · + mir Pir ❧➔ t➟♣ ✤✐➸♠ ❜➨♦ ①→❝ ✤à♥❤ ❜ð✐ ✐❞❡❛❧ J ✱ t❤➻ t❛ ❝â

reg(U ) ≤ reg(Z).
✷✵


q Z

= m1 P1 + ã ã ã + ms Ps ởt t
tr P ữủ ự tr ởt rổ t t
= Pr õ t

rổ t t ữ ổ r Pr ự
P1 := P1 , ..., Ps := Ps Z = m1 P1 + ã ã ã + ms Ps ữ ởt t tr
Pr õ ởt số ổ t s rZ t tr Pr t
n

r(Z) t
tr Pn

ỵ r Z = m P
1

1

+ ã ã ã + ms Ps ởt t

tý ỵ tr P õ
3

reg(Z) max {T1 (Z), T2 (Z), T3 (Z)} .
ỹ t q tr ú tổ ự ữủ t q s

ỵ Z = m P

+ ã ã ã + ms Ps ởt t tr Pn
P1 , ..., Ps tr ữớ t t l1 l2 t
1

1

T (Z) 1 reg(Z) T (Z).

ự sỷ Pi1 , ..., Pir tr s

T1 (Z) = mi1 + ã ã ã + mir 1.
ồ Y = mi1 Pi1 + ã ã ã + mir Pir t ờ t õ

reg(Z) reg(Y ) = T1 (Z).



t P1 , ..., Ps tr ữớ t l1 , l2 tỗ t ởt
ự q ỵ t õ

reg(Z) max{T1 (Z), T2 (Z), T3 (Z)}.



õ

T3 (Z) = max

q
l=1

mil + 1
|Pi1 , ..., Piq tr ởt t t
3



.