ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM





PHẠM THỊ LÝ



MỘT SỐ ĐẶC TRƯNG CỦA MÔ ĐUN
COHEN-MACAULAY VỚI CHIỀU >S



Chuyên ngành: ĐẠI SỐ VÀ LÝ THUYẾT SỐ
Mã số: 60.46.01.04




LUẬN VĂN THẠC SỸ TOÁN HỌC





Người hướng dẫn khoa học: TS.NGUYỄN THỊ DUNG

Thái Nguyên, năm 2014

> s
> s
(R, m) M R
dim M = d
M M
m
M
M
> s
> s
s  −1
(x
1
, . . . , x
n
) m M > s x
i
/∈ p,
p ∈ Ass
R
(M/(x
1
, . . . , x
i−1
)M) dim(R/p) > s
i = 1, . . . , n M > s

M M > s
M > s s = −1, 0, 1
M M
M
> s s = −1, 0, 1
f
> s,
s > 1
> s
m
> s M
> s
>
1
1 2
>
m
> s M
2 2
> s e(x; M) M N-dim
R
H
i
m
(M)
H
i
m
(M) p(M)
M

(R, m)
A R M R
. . .
I (R, m)
R n > 0 m
n
⊆ I ⊆ m.
√
m
n
⊆
√
I ⊆
√
m
√
I = m I m I
I m I ⊂ m
R M n > 0 m
n
M ⊆ IM.
I R M R
{a
1
, . . . , a
r
} I. dim(R/I) = 0 R/I

R
(R/I) < ∞. 

R
(I
k
M/I
k+1
M) < ∞.
k P

M,I
(k)
P

M,I
(k) = 
R
(I
k
M/I
k+1
M).
P
M,I
(n) =
n

k=0
P

M,I
(k) =

n

k=0

R
(I
k
M/I
k+1
M) = 
R
(M/I
n+1
M).
n P
M,I
(n)
M I
P
M,I
(n) I.
dim M = d x
1
, . . . , x
d
∈ m

R
(M/(x
1

, . . . , x
d
)M) < ∞.
dim M = deg(P
M,I
(n)) = min{t | ∃x
1
, , x
t
∈ m : 
R
(M/(x
1
, , x
t
)M) < ∞}.
x := x
1
, . . . , x
d
∈ m
M (M/(x))M) < ∞.
x ∈ m M x
1
, . . . , x
i
i = 1, . . . , d.
x M q = (x
1
, . . . , x

d
)
M q + Ann
R
M R,

R
(q + Ann
R
M) < ∞.
x M n = n
1
, . . . , n
d
d x(n) = x
n
1
1
, . . . , x
n
d
d
M
x
1
, . . . , x
t
m, t  d
dim(M/(x
1

, . . . , x
t
)M)  dim M − t.
x
1
, . . . , x
t
M.
x ∈ m M x
i
/∈ p,
p ∈ Ass M/(x
1
, . . . , x
i−1
)M dim R/p = d − i + 1,
i = 1, . . . , d. x ∈ m M
x /∈ p, p ∈ Ass M dim R/p = d.
x M x

M,

M m M.
I m R,
dim M = d 
R
(M/I
n+1
M) = P
M,I

(n)
n deg P
M,I
(n) = d.
e
0
, e
1
, . . . , e
d
, e
0
> 0
P
M,I
(n) = e
0

n + d
d

− e
1

n + d − 1
d − 1

+ . . . + (−1)
d
e

d
.
e
0
, . . . , e
d
M I e
i
(I, M).
e
0
M
I, e(I, M).
I M
x = x
1
. . . , x
d
M (x) I M
(x)I
n
M = I
n+1
M e(I; M) = e(x; M)
e(I; M) M I
I M
H
•
(R, m) M R
x

1
, . . . , x
t
m
M (M/(x
1
, . . . , x
t
)M) < ∞, (x
1
, . . . , x
t
)
M. 0 −→ M

−→ M −→ M

−→ 0
R x
1
, . . . , x
t
M
x
1
, . . . , x
t
M

M


x
1
, . . . , x
t
M x
2
, . . . , x
t
M/x
1
M
0 :
M
x
1
x
1
, . . . , x
t
M t = 0 (M) <
∞ e(∅; M) = (M). t > 0, (M/(x
1
, . . . , x
t
)M) < ∞
((0 :
M
x
1

)/(x
2
, . . . , x
t
)(0 :
M
x
1
)) < ∞,
(x
2
, . . . , x
t
) 0 :
M
x
1
.
e(x
2
, . . . , x
t
; M/x
1
M) e(x
2
, . . . , x
t
; 0 :
M

x
1
)
e(x
1
, . . . , x
t
; M) = e(x
2
, . . . , x
t
; M/x
1
M) − e(x
2
, . . . , x
t
; 0 :
M
x
1
)
M (x
1
, . . . , x
t
).
0  e(x
1
, . . . , x

t
; M)  (M/(x
1
, . . . , x
t
)M).
i x
n
i
M = 0, n
e(x
1
, . . . , x
t
; M) = 0.
0 −→ M
n
−→ . . . −→ M
1
−→ M
0
−→ 0
R x
1
, . . . , x
t
M
i
,
i = 0, . . . , n.

n

i=0
(−1)
i
e(x
1
, . . . , x
t
; M
i
) = 0.
x
1
, . . . , x
t
M. e(x
1
, . . . , x
t
; M) = 0
t > dim M.
(n
1
, . . . , n
t
) t
e(x
n
1

1
, . . . , x
n
t
t
; M) = n
1
. . . n
t
e(x
1
, . . . , x
t
; M).
I R M R
M I, H
i
I
(M),
H
i
I
(M) = R
i
(Γ
I
(M)),
R
i
(Γ

I
(M)) i I Γ
I
(−)
M.
I R. δ
0 → L
f
→ M
g
→ N → 0
R
0 → H
0
I
(L)
H
0
I
(f)
→ H
0
I
(M)
H
0
I
(g)
→ H
0

I
(N)
→ H
1
I
(L)
H
1
I
(f)
→ H
1
I
(M)
H
1
I
(g)
→ H
1
I
(N) → . . .
→ H
i
I
(L)
H
i
I
(f)

→ H
i
I
(M)
H
i
I
(g)
→ H
i
I
(N) → H
i+1
I
(L) → . . .
H
i
I
(M) = 0 i > d = dim M (R, m)
0 = M R H
d
m
(M) = 0.
R H
d
I
(M) R H
i
m
(M)

i ∈ N
0
.
R M M = 0
x ∈ R x M
Rad(Ann
R
M) p
M p
M R M
M = N
1
+ . . . + N
n
p
i
N
i
.
M = 0 M
p
i
N
i
i = 1, . . . , n
M
{p
1
, . . . , p
n

}
M M
Att
R
M N
i
, i = 1, . . . , n
M
M R M = 0
Att
R
M = ∅
R Ann(M) Att
R
M.
0 −→ M

−→ M −→ M

−→ 0 R
Att
R
M

⊆ Att
R
M ⊆ Att
R
M


∪ Att
R
M

.
m R

R,
0 m
t
, t = 0, 1, 2, . . . .

R

R r ∈ R
r.
M 0 {m
t
M},
t = 0, 1, 2, . . .

M

R
a = (a
1
, a
2
, . . .) ∈


R a
n
∈ R/m
n
x = (x
1
, x
2
, . . .) ∈

M x
n
∈ M/m
n
M, n.
ax = (a
1
x
1
, a
2
x
2
, . . .) ∈

M.
A R A Att
R
A
A


R A R

R A
R

R
Att
R
A = {

p ∩ R :

p ∈ Att

R
A}.
R
N R Att
R
(D(N)) = Ass
R
(N).
A R Ass
R
(D(A)) = Att
R
(A).
p
0

⊆ p
1
⊆ . . . ⊆ p
n
p
i
= p
i+1
, i = 1, . . . , n
R dim R
R M dim M
n n
Supp M M Supp M = V (Ann
R
M)
dim M = dim R/ Ann
R
M = sup
p∈Ass M
dim(R/p).
A N-dim
R
A,
A = 0, N-dim
R
A = −1.
A = 0, d ≥ 0, N-dim
R
A = d
N-dim

R
A < d A
0
⊆ A
1
⊆ . . .
A, n
0
N-dim
R
(A
n+1
/A
n
) < d,
n > n
0
.
R M
N-dim
R
M = 0.
M dim M = 0 M = 0 
R
(M) < ∞.
N-dim
R
A = 0 A = 0 
R
(A) < ∞

Att
R
A = {m}.
0 −→ A

−→ A −→ A

−→ 0
R
N-dim
R
A = max{N-dim
R
A

, N-dim
R
A

}.
N-dim
R
A  dim R/ Ann
R
A = max{dim R/p : p ∈ Att
R
A}
A N-dim
R
A < dim R/ Ann

R
A.
N-dim

R
A = dim

R/ Ann

R
A = max{dim

R/

p :

p ∈ Att

R
A}.
(R, m) A R A

R
N-dim
R
A = N-dim

R
A.
N-dim A N-dim

R
A N-dim

R
A.
(R, m) M R
dim M = d.
N-dim H
d
m
(M) = d.
N-dim H
i
m
(M)  i i  d − 1.
x ∈ m A N-dim(0 :
A
x) = N-dim A−1.
> s
(R, m) M
R dim M = d 1
M
> s
> s
s = −1, 0, 1
f f
2
> s
e(x; M) M N-dim
R

H
i
m
(M)
H
i
m
(M) p(M) M.
R M
x ∈ m
M x /∈ p p ∈ Ass(M)
x
1
, . . . , x
n
∈ m M x
i
/∈ p
p ∈ Ass
R
M/(x
1
, . . . , x
i−1
)M.
x ∈ m f x /∈ p
p ∈ Ass(M) \{m} x
1
, . . . , x
n

m
f x
i
/∈ p, p ∈ Ass(M/(x
1
, . . . , x
i−1
)M) \ {m}
i = 1, . . . , n
x ∈ m M
x /∈ p, p ∈ Ass
R
M dim R/p > 1.
x
1
, . . . , x
n
m M
x
i
/∈ p, p ∈ Ass
R
M/(x
1
, . . . , x
i−1
)M dim R/p > 1
i = 1, . . . , n.
f
f

f
x
1
, . . . , x
n
M f
x
1
, . . . , x
n
M
x
1
, . . . , x
n
M f
a
1
, . . . , a
n
x
a
1
1
, . . . , x
a
n
n
M
f

x
1
, . . . , x
n
∈ p f
M x
1
/1, . . . , x
n
/1 M
p
p ∈ Supp M \{m} p ∈ Supp M dim R/p > 1
x
i
/1 x
i
R
p
, i = 1, . . . , n.
x
1
, . . . , x
n
m M
M i = 1, . . . , n
(x
1
, . . . , x
i−1
)M :

M
x
i
= (x
1
, . . . , x
i−1
)M.
(x
1
, . . . , x
i−1
)M :
M
x
i
⊆

t0
(x
1
, . . . , x
i−1
)M :
M
m
t
).
x
1

, . . . , x
n
M f
i = 1, . . . , n
dim((x
1
, . . . , x
i−1
)M :
M
x
i
/(x
1
, . . . , x
i−1
)M) = −1,
dim((x
1
, . . . , x
i−1
)M :
M
x
i
/(x
1
, . . . , x
i−1
)M)  0)).

f
R M

R
(H
i
m
(M)) < ∞ i < d
> s
x
= x
1
, . . . , x
d
M
I(x; M) = 
R
(M/(x)M) − e(x; M) I(M) = sup
x
I(x; M),
x M.
M
I(M) I(x; M)  I(M)
x M
x M C
x
I(x
n
1
, . . . , x

n
d
; M)  C
x
n
x M I(x
2
1
, . . . , x
2
d
; M) = I(x; M)
x
M x M
I(x
n
1
1
, . . . , x
n
d
d
; M) = I(x; M) n
1
, . . . , n
d
 1
M f
M M = 0
M = 0 M

M M = 0 M f
f f
M f M = 0
M f
f
f f
f 3 f
f f
M m
M f
> s
N-dim
R
(H
i
m
(M))  1 i < d.
p(M)  1.
x = x
1
, . . . , x
d
M k ∈ {1, . . . , d}
I(x
2
1
, . . . , x
2
k−1
, x

k
, x
2
k+1
, . . . , x
2
d
; M) = I(x
1
, . . . , x
d
; M),
x M C
x
n
n
1
, . . . , n
d
> 0
I(x
n
1
1
, . . . , x
n
d
d
; M)  n
k

C
x
.
M f
R M
f
> s
s  −1 x
1
, . . . , x
n
m. i = 1, . . . , n, M
i
= M/(x
1
, . . . , x
i−1
)M
x
1
, . . . , x
n
M > s x
i
/∈ p
p ∈ Ass(M
i
) dim R/p > s i.
x
1

, . . . , x
n
M
> −1, 0, 1 M f
M
x > s
> s + 1 M > s
M > s + 1.
a R M x
1
, . . . , x
n
> s
n  1 dim M/aM  s.
M > s a
> s
x
1
, . . . , x
n
M > s
i = 1, . . . , n
dim((x
1
, . . . , x
i−1
)M :
M
x
i

)/(x
1
, . . . , x
i−1
)M  s.
x
1
/1, . . . , x
n
/1 M
p
p ∈ Supp M x
1
, . . . , x
n
dim R/p > s.
> s
M,
M > s
M M > s. R
> s
> s
> −1, 0, 1
> −1, 0, 1
> s
> s + 1 > s
> s + 1
s + 1
> s x s + 1
x > s

R dim R/q = dim R
q ∈ min(Ass R) M
dim R/p = dim M,
p ∈ min(Ass M) p ⊂ q R
p = p
0
⊂ . . . ⊂ p
n
= q p
i
= p
i+1
i
p q i
p
i
p
i+1
R
p, q R p ⊂ q,
p q
Supp M p, q ∈ Supp M
p ⊂ q, p
q R R
dim R/p + ht p = dim R
p R, Supp M R/ Ann
R
M
M Supp M
dim R/p + dim M

p
= dim M p ∈ Supp M.
> s
M
> s M
dim M = d > s
M > s.
x
1
, . . . , x
r
M
p ∈ (Ass(M/(x
1
, . . . , x
r
)M))
>s
, dim R/p = d − r.
p ∈ (Supp(M))
>s
depth M
p
+ dim R/p = d.
p ∈ (Supp(M))
>s
dim M = dim M
p
+ dim R/p
dim M

p
= depth M
p
.
dim M
p
= dim M
q
+ dim R
p
/qR
p
p, q ∈ (Supp(M))
>s
∪ {m}
q ⊆ p, M
p
p ∈ (Supp(M))
>s
dim R/p = d p ∈ (Min(Supp(M)))
>s
.
R
Spec(R) (Supp(M))
>s
⇔ M > s
M
p
p ∈ (Supp(M))
>s

dim R/p = dim M
p ∈ (min(Supp(M)))
>s
.
> s

M > s M
> s R
> s
> s
> s
> s e(x; M)
M N-dim
R
H
i
m
(M)
H
i
m
(M) p(M) M
x = x
1
, . . . , x
d
M
n
1
, . . . , n

d
I(x
n
1
1
, . . . , x
n
d
d
; M) = (M/(x
n
1
1
, . . . , x
n
d
d
)M − n
1
. . . n
d
e(x; M)
n
1
, . . . , n
d
n
1
, . . . , n
d

n
1
, . . . , n
d
I(x
n
1
1
, . . . , x
n
d
d
; M) x
M p(M)
M
0 −∞ M
p(M) = −∞ M
p(M)  0,
M
(M) = {p ∈ Spec R | M
p
}
NC(M)
R M
dim M = dim R/p p ∈ min Ass M
p(M) = dim( (M)).
x ∈ m x /∈ p p ∈ (
d

i=1

Att(H
i
m
(M))) \
{m}. dim(0 :
M
x)  0.
x f M.
(Ass M)
i
⊆ Att(H
i
m
(M))
(Ass M)
i
= {p ∈ Ass M|dim R/p = i}. Ass M ⊆
(
d

i=1
Att(H
i
m
(M))). x /∈ p p ∈ Ass M \{m}
x f M.
R M R M
R
n
0

∈ N

n0
0 :
M
m
n
= 0 :
M
m
n
0
. M

= 0 :
M
m
n
0
m
n
0
M

= 0. M

⊇ mM

⊇ m
2

M

⊇ . . . ⊇ m
n
0
M

= 0.
(M

) < ∞. f
0 :
M
x ⊆

n0
0 :
M
m
n
= 0 :
M
m
n
0
. (0 :
M
x)  (0 :
M
m

n
0
) < ∞.
0 :
M
x = 0 dim(0 :
M
x) = 0 0 :
M
x = 0
dim(0 :
M
x) = −1. dim(0 :
M
x)  0.
A R x R
x /∈ p, p ∈ Att
R
A \ {m} (A/xA) < ∞.