f-môđun suy rộng và tập iđêan nguyên tố liên kết của môđun đối đồng điều địa phương

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








F
f
f
(R, m) M R
dim M = d
f
f

l(H
i
I
(M)) < ∞, i < d
f R
f
f
x
1
, . . . , x
r
m
M x
i
/∈ p, p ∈ Ass
R
M/(x
1
, . . . , x
i−1
)M
dim R/p > 1 i = 1, . . . , r. I R
dim M/IM > 1. M I
gdepth(I; M)
M I

t
1
, ,t
n

∈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,

> 1 M;
R f
1
d 1
f
R M
R
M R p
R M 0 = x ∈ M
p = Ann
R
(x).
Q M M
M/Q = 0 a ∈ ZD(M/Q), n ∈ N a
n
(M/Q) = 0.
p =

Ann
R
(M/Q) R, Q
p M.
N R M N
Q
i
i = 1, . . . , n,
N = Q
1
∩ . . . ∩ Q
n

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

i
Ass
R
M/N
Q
i
Q
i
(x
n
) ⊆ R m
k ∈ N n
0
x
n
− x
m
∈ m
k
n, m ≥ n
0
. (x
n
) ⊆ R k ∈ N
n
0
x
n
∈ m
k

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

y
n
)

R

R
m

R.

R
m R
(z
n
) ⊆ M m
k ∈ N n
0
z
n
− z
m
∈ m
k
M
n, m ≥ n
0
.
m


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

R
M.
Ass M Supp M
Ass
R
p
(M
p
) = {qR
p
: q ∈ Ass
R
(M), q ⊆ p}.
Ass
R
M = {

p ∩

R :

p ∈ Ass

R

M}.
Ass

R


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

)
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
1
, . . . , x
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

, . . . , 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).
(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 :

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 i 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.
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

(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) → . . .
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

1
, . . . , a
n
R M
M/(a
1
, . . . , a
n
)M = 0
(a
1
, . . . , a
i−1
)M :
M
a
i
= (a
1
, . . . , a
i−1
)M i = 1, . . . , n.
I R M = IM
M I I
M I
M I depth(I, M). M = IM
depth(I, M) = ∞.
M a
1
, . . . , 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

1
1
, . . . , a
t
n
n
M
I, t
1
, . . . , t
n
.
depth(I, M) M
depth(M)  dim(M).
I R
depth(I, M) = inf{i | Ext
i
R
(R/I, M) = 0} = inf{i | H
i
I
(M) = 0}.
depth(I, M) = t.
Ass
R
(Ext
t
R
(R/I, M)) = Ass
R

(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

p
/(x
1
/1, . . . , x
n
/1)M
p
).
M
(R, m). M
M = 0 M = 0 dim M = depth M. R
R
(R, m) M R
M R p ∈ Ass M,
dim M = dim R/p. M
M H
i
m
(M) = 0,
i = dim M.
M M
p
p ∈ Spec R, M
p
= 0 depth(p, M) = depth
R
p
M
p
.

(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

(x
1
, . . . , x
i−1
)M :
M
x
i
⊆

n0
(x
1
, . . . , x
i−1
)M : m
n
.
x ∈ m f x /∈ p
p ∈ Ass(M) \ {m} x
1
, . . . , x
r
∈ m f
x
i
/∈ p, p ∈ Ass(M/(x
1
, . . . , x
i−1

)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

1
, . . . , n
r
x
1
, . . . , x
r
f i = 1, . . . , n
dim((x
1
, . . . , x
i−1
)M :
M
x
i
/(x
1
, . . . , x
i−1
)M)  0.
I ⊆ m R
M I
f t I
(Hom
R
(R/I, M)) < ∞ x ∈ I
f
t > 0
(Ext

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

dim(Ext
t
R
(R/I, M)) > 0. f I
I R. f M
I f depth(I, M), f
M I f I
∞
M M
f f f
I ⊆ R ht
M
(I) (I+Ann M)/ Ann M
R/ Ann M. dim M > 0.
U(M) = {p ∈ Supp M : dim R/p > 0}.
f
x
1
, . . . , x
t
M
p ∈ Ass(M/(x
1
, . . . , x
t
)M) dim R/p ≥ 1 dim R/p = d − t.
depth M
p
= d − dim R/p p ∈ U(M).
ht

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

)M dim R/p > 1 i = 1, . . . , r.
x ∈ m M x /∈ p,
p ∈ Ass
R
M dim R/p > 1.
f
f
2,
x
1
, . . . , x
r
m
x
1
, . . . , x
r
M x
1
/1, . . . , x
r
/1
M
p
p ∈ Supp M x
1
, . . . , x
r
dim R/p > 1
x

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

dim(Ext
i
R
(R/I; M))  1, i < r.
I M r
x
1
, . . . , x
r
∈ I
(Ext
r
R
(R/I; M))
p
∼
=
Hom(R/I; M/(x
1
, . . . , x
r
)M)
p
,
p ∈ Supp M dim R/p > 1.
R (M/IM)  1
I M r,
r  1.
M I dim(M/IM) > 1.
M I

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

(M)) dim R/p > 1}.
(R, m) M
R dim M = d.
m M M
p
M
Supp M M,
I R dim(M/IM) > 1. 1.4.3
M
f
M f
f
f f f
2 f (x, y)
M x /∈ p, p ∈ Ass M dim R/p = 2.
x /∈ p, p ∈ Ass M dim R/p > 1. x
y y /∈ p,
p ∈ Ass(M/xM) dim R/p = 2 − 1 = 1. y
M/xM. (x, y)
M.
3 f
(x, y, z) R. R (0) ∈ Ass R
3 x, y, z x, y, z = 0
(x, y, z) R, R
f
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
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

f M
> 1 M
dim M > 1.
T (M) = {p ∈ Supp M : dim R/p > 1}.
f
(x
1
, . . . , x
s
) M p ∈ Ass(M/(x
1
, . . . , x
s
)M)
dim R/p  2, dim R/p = d − s.
depth M
p
= d − dim R/p, p ∈ T(M).
ht
M
(p) = ht
M
(q) + ht
M
(p/q), p, q ∈ T (M) ∪ {m}
p ⊇ q, M
p
p ∈ T (M) dim R/p = d,
p ∈ min T (M).
⇒ (x

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 ∈ T (M). dim R/p = d−r. dim M/pM = d−r.
(x
1
, . . . , x
r
) M p.
(x
1
, . . . , x
r
) M.
(x
1
/1, . . . , x
r
/1) M
p
depth(M
p
) ≥ r.
depth(M
p
) + dim R/p = d.
⇒ (x
1
, . . . , x
s
) M
s s = 0 p ∈ Ass(M), dim R/p = d − r  2.
depth(M

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

1
M − (s − 1) = d − 1 − s + 1 = d − s.
⇒ p ∈ T (M), p ∈ Supp M dim R/p > 1.
M
p
p ∈ min T (M),
depth M
p
= 0. dim R/p = d. p, q ∈ T (M) ∪ {m}
q ⊆ p p = q p = m M
p
Supp M
p
ht
M
(p) = dim M
p
= dim(R
p
/qR
p
) + ht
M
p
(qR
p
) = ht(p/q) + ht
M
(q).
p = m. q


∈ 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

+ dim R/q = d = ht
M
(m).
⇒ M
p
depth M
p
= dim M
p
.
p ∈ T (M) dim M
p
= ht(p) = d − dim R/p
depth M
p
= d − dim R/p.
x = (x
1
, . . . , x
d
) M n
1
, . . . , n
d
> 0
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).
M
C x
I(x
n
1
1
, . . . , x
n
d
d
; M)  C, x M M
(H
i
m

(M)) < ∞, i  d−1,
R
M f
M
p
p ∈ Supp M \ {m}
dim R/p = d, p ∈ min T(M).
M
p
p ∈ T (M) dim R/p = d,
p ∈ min T (M).
⇔ ⇒ ⇒
R Spec R
T (M) M
M f
⇔ ¨u ⇔
f
m

M M

M m M

M f M f
R M
f

M f
(x
1

, . . . , 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

i−1
)

M)  1,
i = 1, . . . , d (x
1
, . . . , x
d
)
M, M f

p ∈ Supp

M dim

R/

p > 1.

M

p

p ∈ Spec

R p :=

p ∩ R. R
k(p) ⊗


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
+ dim k(p) ⊗

R

p
,
depth

R

p

M

p
= depth
R
p
M
p
+ depth k(p) ⊗

R

p

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