Luận văn: ĐẶC TRƯNG CỦA MÔĐUN COHEN–MACAULAY DÃY QUA TÍNH CHẤT PHÂN TÍCH THAM SỐ pot

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

NGƯỜI HƯỚNG DẪN KHOA HỌC:
GS.TSKH NGUYỄN TỰ CƯỜNG





THÁI NGUYÊN NĂM 2008
R m M R−
dim M = d x = x
1
, . . . , x
d
M q = (x
1
, . . . , x
d
) M x
n
Λ
d,n
= {(α
1
, . . . , α
d
) ∈ Z
d
| 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
q(α)M ∀n ≥ 1

M
R−
R R
dim R ≥ 2, R
M
x x
M
2
1
2
(R, m) M R−
M
M
M
M
D : D
0
⊂ D
1
⊂ . . . ⊂ D
t
= M M
D
i
= D
i
/D
i−1
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

i=0


n + d
i
d
i

l(D
i
/qD
i
)
n ≥ 0
(R, m) M R−
dim M = d x = (x
1
, x
2
, . . . , x
d
)
x
i
∈ m , ∀i = 1, . . . , d l
R
(M/x
M) < ∞
M
(R, m) M R−
dim M = d
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
d
x
α
1

1
, . . . , x
α
d
d
M
x ∈ m x M
x ∈ p p ∈ Ass R dimR/p = d
x
1
, . . . , x
d
∈ m
x
i+1
∈ p, ∀p ∈ Ass R(M/(x
1
, . . . , x
i
)M), dim R/p = d − i
i = 0, . . . , d − 1 {x
1
, . . . , x
d
} M
M
(R, m) dim M = d q M
l(M/qM) < ∞
F
q,M

(n) = l(M/q
n+1
M).
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
(n) d
n
F
q,M
(n) = P

q,M
(n).
e
0
(q, M)(> 0), e
1
(q, M), . . . , e
d
(q, M)
P
q,M
(n) = e
0
(q, M)

n + d
d

+e
1
(q, M)

n + d − 1
d − 1

+· · ·+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
)M = M x
i
M/(x
1
, . . . , x

i−1
)M−
i = 1, . . . , n
M R−
x
1
, . . . , x
n
M−
x
1
, . . . , x
i
M− x
i+1
, . . . , x
n
M/(x
1
, . . . , x
i
)M− 1 ≤ i ≤ n − 1.
x
1
, . . . , 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
, . . . , x
t
M−
x
1
, . . . , x

t
M
I R M R−
M = IM M−
I I R− M depth R(I, M)
(R, m) R−
M depth
R
M depth M
(R, m)
M R−
depth M ≤ dim R/p ≤ dim M, ∀p ∈ Ass M
M
M = 0 M = 0 depth M = dim M. R
R−
M
∀p ∈ Ass M dim R/p = dim M
x
1
, . . . , x
d
∈ m M− M
M/(x
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
1
M ∩ N ⊆ x
1
N y ∈ x
1
M ∩ N
y ∈ x

1
M y = x
1
m m ∈ M y = x
1
m ∈ N
x
1
m + N = 0 + N M/N x
1
(m + N) = 0 m + N = 0
m ∈ N y = x
1
m ∈ x
1
N
i > 1 (x
1
, . . . , x
i
)N ⊆ (x
1
, . . . , 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−
(N + (x
1
, . . . , x
i−1
)M) :
M
x

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

F : M
0
⊂ M
1
⊂ . . . ⊂ M
t
= M
M
i
M F M
dim M
i−1
< dim M
i
i = 1, 2, . . . , t
D : D
0
⊂ D
1
⊂ . . . ⊂ D
t
= M
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
M
D : D
0
⊂ D
1
⊂ . . . ⊂ D
t
= M
M dim D
i
= d
i
D
i
=

dim(R/p
j
)≥d
i+1
N

j
i = 1, 2, . . . , t − 1
0 =
n

j=1
N
j
0 M N
j
p
j
−
j = 1, 2, . . . , n
N M dim N < dim M
D
i
N ⊆ D
i
dim N = dim 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
= d
i
x
= {x
1
, x
2
, . . . , x
d

}
M F
M
i
∩ (x
d
i
+1
, x
d
i
+2
, . . . , x
d
)M = 0
i = 1, 2, . . . , t − 1
M
x = {x
1
, x
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
= d
i
1.3.2 D
i
=

dim(R/p
j
)≥d
i+1

N
j
0 =
n

j=1
N
j
0 M
N
i
=

dim(R/p
j
)≤d
i
N
j
D
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
, x
d
i
+2
, . . . , x
d
)M ⊆
D

i
∩ N
i
= 0.
x = {x
1
, x
2
, . . . , x
d
}
M D
i
= 0 :
M
x
j
j = d
i
+ 1, . . . , 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
i
+1
, . . . , x
d
)M
∀j = d
i
+1, . . . , d x
j
x ∈ D

i
x
j
x = 0 x ∈ 0 :
M
x
j
.
0 :
M
x
j
⊆ D
i
d
i
< j < d
i+1
0 :
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
x
j
< d
s
0 :
M
x
j
⊆ D
s−1
s 0 :

M
x
j
= D
i
M
D : D
0
⊂ D
1
⊂ . . . ⊂ D
t
= M D
i
/D
i−1
i = 1, 2, . . . , t
D : D
0
⊂ 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
i = 1, . . . , t
x = {x
1
, x
2
, . . . , x
d
}
M (x
1
, . . . , x

d
)M ∩ D
i
=
(x
1
, . . . , x
d
i
)D
i
i = 1, . . . , t − 1
D
i
M dim D
i
< M M
(x
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
i
= (x
1
, . . . , x
d
i
)M ∩ D
i
(x
1
, . . . , x
d

i
) M 1.2.11
(x
1
, . . . , x
d
i
)M ∩ D
i
= (x
1
, . . . , x
d
i
)D
i
.
(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
i
= d + n − 1}
α = (α
1
, . . . , α
d
) ∈ Λ
d,n
q(α) = (x
α
1
1
, . . . , x

α
d
d
)
q
n
M ⊆

α∈Λ
d,n
q(α)M
q(α)M = (x
α
1
1
, . . . , x
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

i=1
β
i
>
d

i=1
(α
i
− 1)
β
i
> α

i
x
β
1
1
. . . x
β
d
d
m ∈ q(α)M
q
n
M ⊆

α∈Λ
d,n
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
(y
α
1
1
, . . . , y
α
s
s
)M
n ≥ 1
y = (y
1
, . . . , y
s

) y(α) = (y
α
1
1
, . . . , y
α
s
s
).
R
Z[X
1
, . . . , X
s
] y
X = X
1
, . . . , X
s
X Z[X
1
, . . . , X
s
]−
y Z[X
1
, . . . , X
s
]−
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
)S : f
i+1
= (f
1
, . . . , f
i
)S, i = 0, . . . , s − 1.
(f
1
, . . . , f
i
)S : f
i+1
⊇ (f
1

, . . . , f
i
)S
(f
1
, . . . , f
i
)S : f
i+1
⊆ (f
1
, . . . , f
i
)S, i = 0, . . . , s − 1
g ∈ (f
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
) = (
i

j=1
y
j
u
j
,
i

j=1
y
j
x
j

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

)M
i = 0, . . . , s−1 (u, x) ∈ (f
1
, . . . , f
i
)S g ∈ (f
1
, . . . , f
i
)S
(f
1
, . . . , f
i
)S : f
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)
(f)
n
S = (y)
n
R × (y)
n
M.
t =

C
β
f
β
1
1
. . . f

β
s
s
∈ (f)
n
S β
i
≥ 0, i =
1, . . . , s,
s

i=1
β
i
= 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
. . . y
β
s
s
,

m
β
y
β
1
1
. . . y
β
s
s

) ∈ (y)
n
R × (y)
n
M.
t ∈ (y)
n
R×(y)
n
M t = (

r
β
y
β
1
1
. . . y
β
s
s
,

m
β
y
β

1
1

. . . y
β

s
s
)
β
i
, β

i
≥ 0,
s

i=1
β
i
= n,
s

i=1
β

i
= n, r ∈ R, m ∈ M f
β
i
i
=
(y

β
i
i
, 0), f
β

i
i
= (y
β

i
i
, 0), 1 ≤ i ≤ 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
β
, 0)(f
β
1
1
. . . f
β
s
s
) +

(0, m
β
)(f
β


1
1
. . . f
β

i
i
) ∈ (f)
n
S.
(f)
n
S = (y)
n
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
y(α)M.
(y)
n
M =

α∈Λ
s,n
y(α)M, n ≥ 1
(y
1
, . . . , y
s
)
n
M =


α∈Λ
s,n
(y
α
1
1
, . . . , y
α
s
s
)M
n ≥ 1.
s y
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 =

α∈Λ
i,n
(y
α
1
1
, . . . , y
α
i
i
)M n ≥ 1 i < s
y
k
i+1
M ∩ (y
1

, . . . , y
i
)
m
M ⊆ (y
1
, . . . , y
i
, y
i+1
)
k+m
M ∀k, m ≥ 1
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
) y
n
= (y
1
, . . . , y
s−1
)
n
y
β
1
1
. . . y
β
s−1
s−1
β

i
∈ N, ∀i = 1, . . . , s − 1
s−1

i=1
β
i
= n α = (α
1
, . . . , α
s−1
) ∈ Λ
s−1,n
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
M ⊆

α∈Λ
s−1,n
(y
α
1
1
, . . . , y
α
s−1
s−1
)M.
x ∈

α∈Λ
s−1,n

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

1
, . . . , y
s−1
)
n
M + y
k+1
s
M) = (y
1
, . . . , y
s−1
)
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
)
k+n
M, ∀k, n ≥ 1
(y
1
, . . . , y
s
)
k+n
M ⊆ (y
1
, . . . , y
s−1
)
n
M + y
k+1

s
M
(1) α = (α
1
, . . . , α
s
) ∈ Λ
s,k+n
α
s
≤ k, y
k
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
α
s−1
s−1
)M ⊆ (y
α
1
1
, . . . , y
α
s
s
)M
β = β
1
, . . . , β
s−1
∈ Λ
s−1,n

s−1

i=1
β
i
= s + n − 2 ≥
s−1

i=1
α
i
= s + n + k − 1 − α
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
(y
α
1
1
, . . . , y
α
s−1
s−1
)M ⊆ (y
α
1
1
, . . . , y
α
s
s
)M

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

)M, ∀α ∈ Λ
s,k+n
x − y ∈

α∈Λ
s,k+n
(y
α
1
1
, . . . , y
α
s
s
)M = (y
1
, . . . , y
s
)
k+n
M.
(2) f (y
1
, . . . , y
s
)
k+n
M,
f f = y
β

1
1
. . . y
β
s
s
a
s

i=1
β
i
= k + n β
i
≥
0, ∀i = 1, . . . , s a ∈ M β
s
≥ k + 1
f ∈ y
k+1
s
M (y
1
, . . . , y
s
)
k+n
M ⊆ y
k+1
s

M β
s
≤ k
(y
1
, . . . , y
s−1
)
n
M y
α
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
1
1
. . . y
β
s
s
a ∈ (y
1
, . . . , y
s−1
)
n
M
f ∈ (y
1
, . . . , y
s−1
)
n

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

k+1
s
M
x ∈ (y
1
, . . . , y
s−1
)
n
M + y
k+1
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
i,n
(y
α
1
1
, . . . , y
α
i
i
)M n ≥ 1, i ≤ s
x ∈ y
k
i+1
M ∩ (y
1
, . . . , y
i
)
m

M
x ∈ (y
1
, . . . , y
i
, y
i+1
)
k+m
M =

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
α
i+1
i+1
)M x ∈ y
k
i+1
M α
i+1
≥
k + 1 x ∈ (y
1
, . . . , y
i
)
m
M x
y
β
1

1
. . . y
β
i
i
a a ∈ M β
1
+ · · · + β
i
= m > α
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
α
1
1
, . . . , y
α
s
s
)M
n ≥ 1 1 ≤ i < s
y
k
i+1
M ∩ (y
1
, . . . , y
i
)
m
M ⊆ y
k

i+1
(y
1
, . . . , y
i
, y
i+1
)M + (y
1
, . . . , 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
)
k+m
M ⊆ y
k
i+1
(y
1
, . . . , y
i
, y
i+1
)M + (y
1
, . . . , y
i
)
m+1
M
∀k, m ≥ 1
a ∈ M (n

1
, . . . , n
i
, n
i+1
) ∈ Z
i+1
n
1
+· · ·+n
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
1
. . . y
n
i
i
y
n
i+1
−k
i+1
)a ∈ y
k
i+1
(y
1
, . . . , y
i+1
)M
n
i+1

< k n
i+1
≤ k − 1 n
1
+· · ·+n
i
= k+m−n
i+1
≥ m+1
y
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
a ∈ y
k
i+1
(y
1
, . . . , y
i+1
)M + (y
1
, . . . , y
i
)
m+1
M.
(y
1
, . . . , y
i
, y
i+1

)
k+m
M ⊆ y
k
i+1
(y
1
, . . . , y
i+1
)M + (y
1
, . . . , y
i
)
m+1
M
k, m ≥ 1 1 ≤ i < s.
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, ∀n ≥ 1.
n ≥ 1
x
n
j
M ∩ q
i
M ⊆ x
n
j

(x
j
, q
i
)M + q
m
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
M ∩ [x
n
j
(x
j
, q
i
)M + q
m
i
M]
= x
n
j
(x
j
, q
i
)M + x
n
j
M ∩ q
m
i
M.
2.1.4
x
n

j
M ∩ q
m
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 + q
m+1
i

M
x
n
j
M ∩ q
i
M ⊆ x
n
j
(x
j
, q
i
)M
x
n
j
(q
i
M : x
n
j
) ⊆ x
n
j
M ∩ q
i
M ⊆ x
n
j

(x
j
, q
i
)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
j
q
i
M : x
n
j
⊆ (x
j
, q
i
)M + 0 :
M
x
k
j
∀n ≥ k a ∈ q
i
M : x
n
j
a = x

j
b+x
1
b
1
+· · ·+ x
i
b
i
+c c ∈ 0 :
M
x
k
j
x
n
j
a ∈ q
i
M
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
j
∀n ≥ k
q
i
M : x
n
j
= x
j
(q
i
M : x
n+1
j
) + q
i
M + 0 :
M
x
k
j
= x

j
(q
i
M : x
n
j
) + q
i
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 =

α∈Λ
d,n
q(α)M q
x q(α) = (x
α
1
1
, . . . , x
α
d
d
) (α
1
, . . . , α
d
) = α ∈ Λ
d,n
D : D
0
⊂ D
1

⊂, . . . , D
t
= M M
t D M x
t = 1 M R− x
1
, . . . , x
d
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.