VIỆN KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM
VIỆN TOÁN HỌC
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Phạm Hùng Quý

TÍNH CHẺ RA CỦA MÔĐUN ĐỐI ĐỒNG ĐIỀU ĐỊA PHƯƠNG
VÀ ỨNG DỤNG

LUẬN ÁN TIẾN SĨ TOÁN HỌC


GS. TSKH. Nguyn Tự Cờng HÀNỘI-2013

R a R M
R
H
i
a
(•)
Ext
1
R
(•, •)
0 → A → B → C → 0
0 Ext
1
R
(C, A)
H
i
a
(M) i < t
(R, m)
x ∈ b(M)
3

R
(C, A)
H
i
a
(M) i < t
t
(R, m)
x ∈ b(M)
3
b(M) = ∩
d
x
;i=1
Ann(0 : x
i
)
M/(x
1
, ,x
i−1
)M
,
x
= x
1
, , x
d
M
M a

M
a
n
)
M R
H
i
a
(M)
M a
x ∈ a M
0 → M
x
→ M → M/xM → 0.
H
i
a
(•)
··· → H
i
a
(M) → H
i
a
(M) → H
i
a
(M/xM) → H
i+1
a

m
(M) = 0 i < d M
I
M
(q) := ℓ(M/qM) − e(q; M) > 0
I
M
(q)
I
M
(q)
M H
i
m
(M) i < d
n
0
m
n
0
H
i
m
(M) = 0 i < d
n
0
= 1 H
i
m
(M)

i+1
m
(M) i < d − 1
M x
M
0 → H
i
m
(M) → H
i
m
(M/xM) → H
i+1
m
(M) → 0
i < d − 1
M/xM
x ∈ m
2
M
M/xM
H
i
m
(M/xM)
∼
=
H
i
m

t x ∈ a a M
x /∈ p p ∈ AssM, a  p
a R M
R t H
i
a
(M)
i < t
n a x a
n
H
i
a
(M/xM)
∼
=
H
i
a
(M) ⊕ H
i+1
a
(M) i < t −1
M R a
R t n
0
a
n
0
H

=
H
t−1
a
(M) ⊕ 0 :
H
t
a
(M)
a
n
0
.
N M
N q M
q M
qM M N
R
(q, M) = dim
R/m
Soc(M/qM)
Soc(N)
∼
=
0 :
N
m
∼
=
Hom

0
H
i
m
(M) = 0 i < d q
M m
2n
0
k ≤ n
0
ℓ
R

(qM :
M
m
k
)/qM

ℓ
R

(qM :
M
m
k
)/qM

=
d

R/m
Soc(H
i
m
(M)).
M
R a R t n
0
a
n
0
H
i
a
(M) = 0 i < t
a x ∈ a
n
0
0 → M/H
0
a
(M)
x
→ M → M/xM → 0
0 → H
i
a
(M) → H
i
a

i
a
(M) → H
i
a
(M/xM) → H
i+1
a
(
M) → 0
i < t − 1 x (♯)
E
i
x
Ext(H
i+1
a
(M), H
i
a
(M))
H
t
a
(
M)
∼
=
H
t

0 → 0 :
H
t−1
a
(M)
b → 0 :
H
t−1
a
(M/xM )
b → 0 :
H
t
a
(
M)
b → 0.
(♯)
t U
M
M = M/U x y (♯)
0 :
M
(x + y) = U
x+y (♯) E
i
x+y
= E
i
x

t−1
y
t U
M
M = M/U x y R x
(♯) 0 :
M
xy = U
xy (♯) E
i
xy
= yE
i
x
i < t − 1
H
t
a
(M)
∼
=
H
t
a
(M) F
t−1
x
F
t−1
xy

xy
(R, m) a b
p
1
, , p
n
ab  p
j
j ≤ n
x ab x /∈ p
j
j ≤ n
a
1
, , a
r
∈ a b
1
, , b
r
∈ b
x = a
1
b
1
+ ··· + a
r
b
r
a

0
) < ∞, dim M
0
< dim M
1
< ··· < dim M
t
= d
M
i
/M
i−1
i = 1, 2, , t
M
F
x
= x
1
, , x
d
M M
F M
i
∩(x
d
i
+1
, , x
d
)M = 0 i = 0, 1, , t−1, d

F,M
(x) x = x
1
, , x
d
M F
I
F
(M) = ℓ(H
0
m
(M/M
0
))
+
t−1

i=0
d
i+1
−1

j=1

d
i+1
− 1
j

−

j ≤ d
i+1
− 1
H
j
m
(M/(xM + M
i
))
∼
=
H
j
m
(M/M
i
) ⊕ H
j+1
m
(M/U
M
(0))
i ≤ t − 1 j < d − 1 x M
m
3n
0
c
t−1
0 :
H

⊆ M
1
⊆ ··· ⊆ M
t
= M
x
= x
1
, , x
d
M F
m
n
, n ≫ 0 I
F,M
(x
)
I
F,M
(x
) = ℓ(H
0
m
(M/M
0
))
+
t−1

i=0

t
= M
x
= x
1
, , x
d
M F
m
n
, n ≫ 0 (x
) M
N
R
((x
), M) = dim
R/m
Soc(H
0
m
(M))
+
t−1

i=0
d
i+1

j=1


i
(M)
b(M) = ∩
d
x
;i=1
Ann(0 : x
i
)
M/(x
1
, ,x
i−1
)M
,
x
= x
1
, , x
d
M
a(M) ⊆ b(M) ⊆ a
0
(M) ∩ ··· ∩ a
d−1
(M)
(R, m)
M a(M) b(M)
0 :
M

∼
=
H
t
I
(
M)
0 :
H
t−1
I
(M/xM )
b(M)
∼
=
H
t−1
I
(M) ⊕ (0 :
H
t
I
(M)
b(M)).
x
= x
1
, , x
d
M

3
i ≤ d U
i
(M)
∼
=
U
M/(x
i+2
, ,x
d
)M
(0) 0 ≤ i ≤ d − 1
U
i
(M)
M m I udeg(I, M) M
I d e g(I, M) M
I
udeg(I, M) = d e g(I, M) +
d−1

i=0

deg(I, U
i
(M)),

deg(I, U
i

i
a
(M) i < t supp( H
i
a
(M))
i < t
a R M R
t H
i
a
(M)
supp(H
i
a
(M)) i < t Ass
R
(H
t
a
(M))
M R
M a
f
a
(M) = inf{i ∈ N
0
|H
i
a

, , a
t
)

n
1
, ,n
t
∈N
Ass M/(a
n
1
1
, , a
n
t
t
)M
R
Ext
1
R
(•, •)
0 Ext
1
R
(•, •)
Ext
1
R

a
(M) = Γ
b
(M) M
√
a =
√
b
Γ
a
(•)
R
0 → M
′
→ M → M
′′
→ 0
0 → Γ
a
(M
′
) → Γ
a
(M) → Γ
a
(M
′′
).
H
i

−→ ··· .
Γ
a
(•) I
•
0
Γ
a
(d
−1
)
−→ Γ
a
(I
0
)
Γ
a
(d
0
)
−→ Γ
a
(I
1
)
Γ
a
(d
1

i M a
H
i
a
(M) := Ker(Γ
a
(d
i
))/Im(Γ
a
(d
i−1
)).
R
0 → L → M → N → 0.
a R
0 → H
0
a
(L) → H
0
a
(M) → H
0
a
(N) → H
1
a
(L) → ···
→ H

M R i ∈ N
0
Φ
i
a
: H
i
a
(M)
∼
=
lim
−−→
n∈N
Ext
i
R
(R/a
n
, M).
˘
C a