ĐẠI HỌC THÁI NGUYÊN
ĐẠI HỌC SƯ PHẠM
ĐOÀN THỊ THU THẢO F-MÔĐUN SUY RỘNG VÀ TẬP IĐÊAN
NGUYÊN TỐ LIÊN KẾT CỦA MÔĐUN ĐỐI
ĐỒNG ĐIỀU ĐỊA PHƯƠNG CHUYÊN NGÀNH: ĐẠI SỐ VÀ LÝ THUYẾT SỐ class="bi x8 y16 w2 h4"
class="bi x8 y17 w2 h5"
F
f
f
(R, m) M R
dim M = d
f
f

∈N
Ass(M/(x
t
1
1
, . . . , x
t
n
n
)M) x
1
, . . . , x
n
M M
I gdepth(I, M) = r r i
Supp(H
i
I
(M)) Ass(H
r
I
(M))
f
f
f
1
1.4 1
f
2 f
f M,

p
i
N = 0 N = 0 N
p
i
Q
i
i = 1, . . . , n
Q
j
⊆
n

i=1;i=j
Q
i
.
N
{p
1
, . . . , p
n
}
N
M/N Ass
R
M/N Q
i
, i = 1, . . . , n
N p

n ≥ n
0
.
(x
n
), (y
n
)
(x
n
− y
n
)

R
(x
n
) + (y
n
) = (x
n
+ y
n
)
(x
n
)(y
n
) = (x
n

R

M.
p M M
R/p.
p Ann(x),
0 = x ∈ M. p ∈ Ass
R
(M). M = 0
Ass
R
(M) = ∅. ZD(M) M
M.
R
0 −→ M

−→ M −→ M

−→ 0.
Ass M

⊆ Ass M ⊆ Ass
R
M

∪ Ass M

.
M R Ass M
Ass M ⊆ Supp M V (Ann M) = Supp

M =

p∈Ass
R
M
Ass

R

M/p

M.
(R, m) M R
dim M = d,
x := (x
1
, . . . , x
d
) ∈ m
M (M/(x)M) < ∞.
x ∈ m M (x
1
, . . . , x
i
)
i = 1, . . . , d.
x M n = (n
1
, . . . , n
d

) ∈ m M x
i
/∈ p,
p ∈ Ass M/(x
1
, . . . , x
i−1
)M dim R/p = d − i + 1.
x ∈ m M x /∈ p,
p ∈ Ass M dim R/p = d.
x
M x

M,

M m M.
I m R. M R

R
(M/I
n+1
M) = P
M,I
(n) n
deg P
M,I
(n) = d. e
0
, e
1

(I, M).
e
0
M
I. e(I, M).
(x
1
, . . . , x
t
) R
M (M/(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 :

, . . . , 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

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.
0 −→ L −→ M −→ N −→ 0 R
δ
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

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

n
∈ R M
a
i
/∈ p p ∈ Ass
R
M/(a
1
, . . . , a
i−1
)M.
M (R, m)
a
1
, . . . , a
n
∈ m
M/(a
1
, . . . , a
n
)M = 0 M
M m
M depth M.
a
1
, . . . , a
n
M I a
t

(H
t
I
(M)).
p ∈ Supp(M/IM) \ {m}, x
1
, . . . , x
r
M
Ext
n
R
p
(R
p
/IR
p
, M
p
)
∼
=
Hom
R
p
(R
p
/IR
p
, M

(a
1
, . . . , a
r
) M M
M/(a
1
, . . . , a
r
)M
x = (x
1
, . . . , x
d
) M
I(x; M) = (M/xM) − e(x; M),
M I(x; M) = 0.

R m R depth R = depth

R
R

R
x
1
, . . . , x
r
m
M i = 1, . . . , r

)M) \ {m},
i = 1, . . . , r
p ∈ Supp M \ {m} x
1
, . . . , x
r
∈ p x
1
, . . . , x
r
x
1
/1, . . . , x
r
/1 M
p
x
1
, . . . , x
r
∈ p f x
n
1
1
, . . . , x
n
r
r
f
n

i
R
(R/I, M)) < ∞, i < t.
I f t.
x
1
, . . . , x
t
∈ I f p ∈ Spec(R) \ {m}
Ext
n
R
(R/I, M)
p
∼
=
Hom
R
(R/I, M/(x
1
, . . . , x
t
)M)
p
.
x
1
, . . . , x
t
f M I

M
(p) = ht
M
(q) + ht(p/q), p, q ∈ U(M) ∪ {m} p ⊇ q, M
p
p ∈ U(M) dim R/p = d,
p ∈ min U(M).
f

M M

M f M f
R M
f

M f
f
x
1
, . . . , x
r
m M x
i
/∈ p,
p ∈ Ass
R
M/(x
1
, . . . , x
i−1

i
/1, i = 1, . . . , r x
i
R
p
.
r  d − 2 M
r M
x
1
, . . . , x
r
M x
n
1
1
, . . . , x
n
r
r
M n
1
, . . . , n
r
.
x ∈ m. x
M dim(0 :
M
x)  1.
r

x
1
, . . . , x
r
M I
y ∈ I x
1
, . . . , x
r
, y
M.
M I
M I
gdepth(I; M)
f f
depth(I; M)  depth(I; M)  gdepth(I; M).
x
1
∈ I M
gdepth(I; M) = gdepth(I; M/x
1
M) + 1.
gdepth(I; M) = min{gdepth(p; M) | p ∈ V (I)}.
gdepth(I; M) = min{i | dim(Ext
i
R
(R/I; M)) > 1}
= min{i| ∃ p ∈ Supp(H
i
I

0
⊂ p
1
⊂ . . . ⊂ 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.
R dim R  2

1
, . . . , x
s
)
M p ∈ Ass(M/(x
1
, . . . , x
s
)M) dim R/p > 1
dim R/p < d − s. y ∈ p
(x
1
, . . . , x
s
, y) M. M f
(x
1
, . . . , x
s
, y) M.
y /∈ p p ∈ Ass(M/(x
1
, . . . , x
s
)M).
⇒ s
x M M f
p ∈ Ass(M/xM) dim R/p > 1 x ∈ p
dim R/p > d − 1 M
f

p
) = 0 = r dim R/p = d.
s > 0. p ∈ Supp M/x
1
M, dim R/p  2 x
1
M
p
. depth(M/x
1
M)
p
= depth(M
p
) − 1.
depth(M
p
/x
1
M
p
) = (d − 1) − dim R/p
(x
2
, . . . , x
s
) M/x
1
M
dim R/p = dim M/x


∈ min T(M) q

⊆ q. M
q
q

R
q
∈ Ass M
q
,
ht
M
(q) + ht(m/q) = ht
M
(q

) + ht(q/q

) + dim R/q
= dim(R
q
/q

R
p
) + dim R/q
= dim M
q

n
d
d
; M) = (M/(x
n
1
1
, . . . , x
n
d
d
)M) − n
1
. . . n
d
e(x; M).
M
C x
I(x
n
1
1
, . . . , x
n
d
d
; M)  C, x M M
(H
i
m

, . . . , x
d
) M
dim((x
1
, . . . , x
i−1
)M :
M
x
i
/(x
1
, . . . , x
i−1
)M)
= dim((x
1
, . . . , x
i−1
)

M :

M
x
i
/(x
1
, . . . , x

R

p
R
p
−→

R

p
dim(k(p) ⊗

R

p
) = depth(k(p) ⊗

R

p
).
dim

R

p

M

p

,
dim

R

p

M

p
− depth

R

p

M

p
= dim
R
p
M
p
− depth
R
p
M
p
= 0