Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC SƯ PHẠM
 LÊ THỊ MAI QUỲNH

ĐẶC TRƯNG CỦA MÔĐUN COHEN–MACAULAY DÃY
QUA TÍNH CHẤT PHÂN TÍCH THAM SỐ

Chuyên ngành: Đại số và lý thuyết số
Mã số: 60.46.05
LUẬN VĂN THẠC SỸ TOÁN HỌC


| α
i
≥ 1, ∀1 ≤ i ≤ d,
d

i=1
α
i
= d + n − 1}
q(α) = (x
α
1
1
, . . . , x
α
d
d
) ∀α = (α
1
, . . . , α
d
) ∈ Λ
d,n
x
q
n
M =

α∈Λ
d,n

i = 1, . . . , t, D
0
= D
0
.
M
q M
l(M/q
n+1
M) =
t

i=0

n + d
i
d
i

l(D
i
/qD
i
)
n ≥ 0
q M
l(M/q
n+1
M) =
t

(R, m) M R−
dim M = d
www.VNMATH.com
x
1
, x
2
, . . . , x
t
∈ m
dim(M/(x
1
, . . . , x
t
)M) ≥ dim M − t.
x
1
, x
2
, . . . , x
t
M
x
1
, . . . , x
d
M
α
1
, . . . , α

(R, m) dim M = d q M
l(M/qM) < ∞
F
q,M
(n) = l(M/q
n+1
M).
www.VNMATH.com
R =

t≥0
R
t
R
0
R = R
0
[x
1
, . . . , x
r
] x
i
d
i
F
q,M
(n)
P
q,M

+· · ·+e
d
(q, M).
e
0
(q, M) q
x = {x
1
, x
2
, . . . , x
d
} e
0
(q, M) = e
(
x, M)
R M R−
x ∈ R M− 0 :
M
x = 0 xa = 0
∀a ∈ M, a = 0 x
1
, . . . , x
n
R M−
(x
1
, . . . , x
n

, . . . , x
n
M−
α
1
, . . . , α
n
{x
α
1
1
, . . . , x
α
n
n
}
M−
x
1
, . . . , x
n
M−
x
1
, . . . , x
n
M−
M R−
x
1

1
, . . . , x
d
)M
M
M M−
N M
dim N < dim M M/N x
1
, . . . , x
i
M (x
1
, . . . , x
i
)M∩N = (x
1
, . . . , x
i
)N
i = 1 x
1
M ∩ N = x
1
N x
1
N ⊆
x
1
M ∩ N x

i
)M ∩ N (1).
a ∈ (x
1
, . . . , x
i
)M ∩ N a = x
1
a
1
+ · · · + x
i
a
i
a
j
∈ M
j = 1, . . . , i a ∈ N a
i
∈ (N + (x
1
, . . . , x
i−1
)M) : x
i
x
1
, . . . , x
i
M/N−

+ c
b
j
∈ M j = 1, · · · , i − 1 c ∈ N
a − x
i
c ∈ (x
1
, . . . , x
i−1
)M ∩ N = (x
1
, . . . , x
i−1
)N
a ∈ (x
1
, · · · , x
i
)N (x
1
, . . . , x
i
)M∩N ⊆ (x
1
, . . . , x
i
)N (2)
(x
1

M 2
D
0
= H
0
m
(M) 0 M
m
D
i−1
D
i
dim D
i−1
< dim D
i
i = 1, 2, . . . , t
www.VNMATH.com
M
D : D
0
⊂ D
1
⊂ . . . ⊂ D
t
= M
M dim D
i
= d
i

.
F : M
0
⊂ M
1
⊂ . . . ⊂ M
t
= M
M
j
D
i
M
j
⊆ D
i
dim M
j
= dim D
i
.
F : M
0
⊂ M
1
⊂ . . . ⊂ M
t
= M
dim M
i

2
, . . . , x
d
}
F x
α
1
1
, . . . , x
α
d
d
F
α
1
, . . . , α
d
.
M
M
M
D : D
0
⊂ D
1
⊂ . . . ⊂ D
t
= M M
dim D
i

i
∩ N
i
= 0 dim M/N
i
= d
i
x = {x
1
, x
2
, . . . , x
d
}
x
d
i
+1
, x
d
i
+2
, . . . , x
d
∈ Ann M/N
i
D
i
∩ (x
d

i+1
, i = 0, 1, . . . , t − 1
0 :
M
x
1
⊆ 0 :
M
x
2
⊆ . . . ⊆ 0 :
M
x
d
D
i
⊆ 0 :
M
x
j
j ≥ d
i
x ∈ D
i
D
i
M x ∈ M x
j
x ∈ (x
d

M
x
j
⊆ D
i
s 0 :
M
x
j
⊆ D
s−1
t ≥ s > i 0 :
M
x
j
= 0 :
D
s
x
j
d
s
≥ d
i+1
≥ j x
j
D
s
dim 0 :
M

⊂ D
1
⊂ . . . ⊂ D
t
= M
M dim D
i
= d
i
x = (x
1
, x
2
, . . . , x
d
)
M
(1) M
(2) (x
1
, . . . , x
d
i
) M/D
i−1
i = 1, . . . , t
(3) depth M/D
i−1
= d
i

1
, . . . , x
d
)M ∩ D
i
= (x
1
, . . . , x
d
i
, x
d
i+1
, . . . , x
d
)M ∩ D
i
= (x
1
, . . . , x
d
i
)M ∩ D
i
+ (x
d
i+1
, . . . , x
d
)M ∩ D

(R, m) M R−
dim M = d x = {x
1
, x
2
, . . . , x
d
} M
q x
1
, x
2
, . . . , x
d
n, s
Λ
d,n
= {(α
1
, . . . , α
d
) ∈ Z
d
| α
i
≥ 1, ∀1 ≤ i ≤ d,
d

i=1
α

α
d
d
) q
n
M
x
β
1
1
. . . x
β
d
d
m β
i
∈ N, ∀i = 1, . . . , d
d

i=1
β
i
= n
α = (α
1
, . . . , α
d
) ∈ Λ
d,n
d

q(α)M
q
n
M =

α∈Λ
d,n
q(α)M
x = x
1
, . . . , x
d
M
x M x
s y
1
, . . . , y
s
M−
m
(y
1
, . . . , y
s
)
n
M =

α∈Λ
s,n

1
, . . . , X
s
X Z[X
1
, . . . , X
s
]−
y Z[X
1
, . . . , X
s
]−
www.VNMATH.com
S = R  M M R S = R  M
S
(a, x)(b, y) = (ab, ay + bx), ∀a, b ∈ R, ∀x, y ∈ M.
f
i
= (y
i
, 0), (i = 1, . . . , s) f = f
1
, . . . , f
s
S−
(f
1
, . . . , f
i

1
, . . . , f
i
)S : f
i+1
g = (u, x), u ∈ R, x ∈ M
gf
i+1
∈ (f
1
, . . . , f
i
)S (u, x)(y
i+1
, 0) =
i

j=1
(y
j
, 0)(u
j
, x
j
)
(uy
i+1
, xy
i+1
) = (

xy
i+1
=
i

j=1
y
j
x
j
uy
i+1
∈ (y
1
, . . . , y
i
)R
xy
i+1
∈ (y
1
, . . . , y
i
)M
u ∈ (y
1
, . . . , y
i
)R : y
i+1

i+1
⊆ (f
1
, . . . , f
i
)S i = 0, . . . , s − 1
(f
1
, . . . , f
i
)S : f
i+1
= (f
1
, . . . , f
i
)S, ∀i = 0, . . . , s − 1 f =
f
1
, . . . , f
s
S−
(f)
n
S =

α∈Λ
s,n
f(α)S, ∀n ≥ 1 (1)
www.VNMATH.com

= n C
β
= (r
β
, m
β
) ∈ R × M
t =

(r
β
, m
β
)(y
β
1
1
. . . y
β
s
s
, 0)
= (

r
β
y
β
1
1

β
1
1
. . . y
β
s
s
,

m
β
y
β

1
1
. . . y
β

s
s
)
β
i
, β

i
≥ 0,
s


t =

(r
β
y
β
1
1
. . . y
β
s
s
, 0) + (0,

m
β
y
β

1
1
. . . y
β

s
s
)
=

(r

R × (y)
n
M (2)
f(α)S = y(α)R × y(α)M, n ≥ 1, α ∈ Λ
s,n
.

α∈Λ
s,n
f(α)S =

α∈Λ
s,n
y(α)R ×

α∈Λ
s,n
y(α)M (3)
(y)
n
R × (y)
n
M =

α∈Λ
s,n
y(α)R ×

α∈Λ
s,n

1
, . . . , y
s
m (y
1
, . . . , y
s
)
n
M =

α∈Λ
s,n
(y
α
1
1
, . . . , y
α
s
s
)M
n ≥ 1
(y
1
, . . . , y
i
)
n
M =

i < s
s ≥ 2
i = s − 1
(y
1
, . . . , y
s−1
)
n
M ⊆

α∈Λ
s−1,n
(y
α
1
1
, . . . , y
α
s−1
s−1
)M n ≥ 1
y(α) = (y
α
1
1
, . . . , y
α
s−1
s−1

s−1

i=1
β
i
= n >
s−1

i=1
(α
i
− 1) i(1 ≤ i ≤ s − 1)
β
i
> α
i
y
β
1
1
. . . y
β
s−1
s−1
∈ y(α) (y
1
, . . . , y
s−1
)
n

n
M
k
x ∈ (y
1
, . . . , y
s−1
)
n
M + y
k
s
M, x ∈ (y
1
, . . . , y
s−1
)
n
M + y
k+1
s
M.
k = 0
x ∈ (y
1
, . . . , y
s−1
)
n
M+y

)
n
M.
x
x = y + y
k+1
s
a y ∈ (y
1
, . . . , y
s−1
)
n
M, a ∈ M
2
x − y ∈

α∈Λ
s,k+n
(y
α
1
1
, . . . , y
α
s
s
)M = (y
1
, . . . , y

s
M ⊆ (y
α
1
1
, . . . , y
α
s
s
)M x − y ∈
(y
α
1
1
, . . . , y
α
s
s
)M.
α
s
≥ k + 1

α∈Λ
s−1,n
(y
α
1
1
, . . . , y

s
,
i(1 ≤ i ≤ − 1) β
i
≥ α
i
(y
β
1
1
, . . . , y
β
s−1
s−1
)M ⊆
(y
α
1
1
, . . . , y
α
s−1
s−1
, y
α
s
s
)M

α∈Λ

s−1
)
n
M ⊆ (y
α
1
1
, . . . , y
α
s−1
s−1
)M
⊆ (y
α
1
1
, . . . , y
α
s−1
s−1
, y
α
s
s
)M
y ∈ (y
1
, . . . , y
s−1
)

s
s
)M = (y
1
, . . . , y
s
)
k+n
M.
www.VNMATH.com
(2) f (y
1
, . . . , y
s
)
k+n
M,
f f = y
β
1
1
. . . y
β
s
s
a
s

i=1
β

1
1
. . . y
α
s−1
s−1
a
s−1

i=1
α
i
= n, α
i
≥ 0, ∀i = 1, . . . , s − 1 a ∈ M
s−1

i=1
β
i
= k + n − β
s
≥
s−1

i=1
α
i
= n y
β

(y
1
, . . . , y
s
)
k+n
M ⊆ (y
1
, . . . , y
s−1
)
n
M + y
k+1
s
M
x − y ∈ (y
1
, . . . , y
s
)
k+n
M
x ∈ (y
1
, . . . , y
s−1
)
n
M + (y

s
M.
(y
1
, . . . , y
s−1
)
n
M =

α∈Λ
s−1,n
(y
α
1
1
, . . . , y
α
s−1
s−1
)M.
(y
1
, . . . , y
i
)
n
M =

α∈Λ

α∈Λ
i+1,k+m
(y
α
1
1
, . . . , y
α
i
i
, y
α
i+1
i+1
)M
k, m ≥ 1, i < s α = (α
1
, . . . , α
i+1
) ∈ Λ
i+1,k+m
x ∈ (y
α
1
1
, . . . , y
α
i
i
, y

1
+ · · · + α
i
x ∈ (y
α
1
1
, . . . , y
α
i
i
, y
α
i+1
i+1
)M x
s y
1
, . . . , y
s
m (y
1
, . . . , y
s
)
n
M = ∩
α∈Λ
s,n
(y

i
)
m+1
M
∀k, m ≥ 1
2.1.3(ii)
y
k
i+1
M ∩ (y
1
, . . . , y
i
)
m
M ⊆ (y
1
, . . . , y
i
, y
i+1
)
k+m
M
(y
1
, . . . , y
i
, y
i+1

i+1
=
k + m.
n
i+1
≥ k n
1
+ · · · + n
i
+ (n
i+1
− k) = m ≥ 1
y
n
1
1
. . . y
n
i
i
y
n
i+1
i+1
a = y
k
i+1
(y
n
1

n
1
1
. . . y
n
i
i
y
n
i+1
i+1
a ∈ (y
1
, . . . , y
i
)
m+1
M
y
n
1
1
. . . y
n
i
i
y
n
i+1
i+1

1
, . . . , y
i
)
m+1
M
k, m ≥ 1 1 ≤ i < s.
www.VNMATH.com
x = {x
1
, x
2
, . . . , x
d
} M
∀1 ≤ i < j ≤ d
k ≥ 1 q
i
M : x
n
j
= q
i
M + 0 :
M
x
k
j
∀n ≥ k
x

n
j
M ∩ q
i
M ⊆ x
n
j
(x
j
, q
i
)M + q
m+1
i
M
x
n
j
M ∩ q
i
M ⊆ x
n
j
M ∩ [x
n
j
(x
j
, q
i

, q
i
)M + q
m+1
i
M
x
n
j
M ∩ q
i
M ⊆ x
n
j
(x
j
, q
i
)M + q
m+1
i
M
x
n
j
M ∩ q
i
M ⊆ x
n
j

⊆
(x
j
, q
i
)M + 0 :
M
x
n
i
k  0
q
i
M : x
k
j
= q
i
M : x
k+1
j
0 :
M
x
k
j
= 0 :
M
x
k+1

+c c ∈ 0 :
M
x
k
j
x
n
j
a ∈ q
i
M
www.VNMATH.com
n ≥ k, b ∈ q
i
M : x
n+1
j
a ∈ x
j
(q
i
M : x
n+1
j
) + q
i
M + 0 :
M
x
k

M + 0 :
M
x
k
j
q
i
M : x
n
j
= q
i
M + 0 :
M
x
k
j
n ≥ k.
(R, m) M R−
M
M
M
⇒ x = x
1
, . . . , x
d
M
q
n
M =

M−
t > 1 M = M/D
t−1
x = x
1
, . . . , x
d
M
x M M
dim
R
M = d x M−
q
n
M =

α∈Λ
d,n
q(α)M, ∀n ≥ 1.
www.VNMATH.com