Tập Iđêan nguyên tố gắn kết và tính chất dịch chuyển địa phương

ĐẠI HỌC THÁI NGUYÊN
ĐẠI HỌC SƯ PHẠM


NGUYỄN QUANG BẠO





TẬP IĐÊAN NGUYÊN TỐ GẮN KẾT VÀ
TÍNH CHẤT DỊCH CHUYỂN ĐỊA PHƯƠNG











2012
1
N R− M
N =

r
i=1

Q
i
, Q
i
p
i
−
R− M
x M x ∈ R. M
M = S
1
+ S
2
+ · · · + S
n
,
S
i
p
i
i = 1, , n. p
i
S
i
M
M
{p
1
, . . . , p
n

} M
M M,
Att M.
R A R
A A
A
0
⊇ A
1
⊇ ⊇ A
n
⊇
A k ∈ N A
k
= A
n
n ≥ k
R R
R
R A R
A
A
R M
R
M
0
⊆ M
1
⊆ · · · ⊆ M

n
= M M
0 = M
0
⊂ M
1
⊂ . . . ⊂ M
n
= M
M M
i+1
/M
i
i =
0, 1, . . . , n − 1 M
i+1
/M
i
0
M 
R
(M)
0 = M
0
⊂ M
1
⊂ . . . ⊂ M
n
= M,
M

i
= M
i+1
i = 0, 1, . . . , n − 1.
R M M
M
M 
R
(M)
M
M
R R
R R
R
p
0
 p
1
 . . .  p
n
p
0
, p
1
, . . . , p
n
R R
R
p = p
0

 p
1
. . .  p
n
p ht p
R dim R
R
dim(R) = Sup{ht p|p ∈ Spec(R)}.
R dim R
R
M = 0 R
M dim M dim R/ Ann M M
dim M = −1
R = 0 R
dim R = 0.
(R, m) A R
(A) < ∞ A
(A) = n < ∞
m
n
A = 0.
M
. . . −→ P
2
−→ P
1
−→ P
0
−→ M −→ 0
P

i
M, N R− n ≥ 0
n Hom(−, N)
M n M N
Ext
n
R
(M, N). . . . −→ P
2
u
2
−→ P
1
u
1
−→ P
0

−→ M −→ 0
M, Hom(−, N)
0 −→ Hom(P
0
, N)
u
∗
1
−→ Hom(P
1
, N)
u

∗
2
−→ Hom(P
2
, N) −→ . . .
Ext
n
R
(M, N) = Ker u
∗
n+1
/ Im u
∗
n
n
M
M, N Ext
n
R
(M, N)
n.
Ext
S R
S
−1
(Ext
n
R
(M, N))
∼

=
Ext
n
S
−1
R
(S
−1
M, S
−1
N),
S
−1
(Ext
n
R
(M, N))
p
∼
=
Ext
n
R
p
(M
p
, N
p
)
p R.

f

: (R

, m

) −→ (R, m)
p ∈ Spec(R), p

= f
−1
(p) ∈ Spec(R

). f
f

: R

p

−→ R
p
,
f

(r

/s

) = f(r


)/f(s

) r

∈ R

s

∈ R

\ p

.
M R j
R
p
Ext
j
R

p

(M
p
, R

p

)

Ext
j
R

p

(M
p
, R

p

)
∼
=
(Ext
j
R

(M, R

))
p
R
p
I
I R. R− N
Γ
I
(N) =


n≥0
(0 :
N
I
n
). f : N −→ N

R− f
∗
: Γ
I
(N) −→ Γ
I
(N

) f
∗
(x) =
f(x). Γ
I
(−) R−
R− Γ
I
(−) I−
N
0 −→ N −→ E
0
−→ E
1

−→ E
2
−→ . . .
E
i
N R− I R.
n I− Γ
I
(−) N
n N H
n
I
(N).
0 −→ N −→ E
0
u
0
−→ E
1
u
1
−→ E
2
. . .
N, Γ
I
(−)
0 −→ Γ(E
0
)

u
∗
0
−→ Γ(E
1
)
u
∗
1
−→ Γ(E
2
) −→ . . .
H
n
I
(N) = Ker u
∗
n
/ Im u
∗
n−1
n
N
N R−
H
0
I
(N)
∼
=

Γ
I
(N).
N = N/Γ
I
(N) H
n
I
(N)
∼
=
H
n
I
(N) n ≥ 1.
0 −→ N

−→ N −→ N

−→ 0
n H
n
I
(N

) −→ H
n+1
I
(N


)
0 −→ Γ
I
(N

) −→ Γ
I
(N) −→ Γ
I
(N

) −→ H
1
I
(N

)
−→ H
1
I
(N) −→ H
1
I
(N

) −→ H
2
I
(N


) −→ . . .
S R S
−1
(H
n
I
(N))
∼
=
H
n
S
−1
I
(S
−1
N)
(R, m) 0 = M
R d
Σ = {N

: N

M dim N

< d}
N G = M/N.
dim G = d;
G d
Ass G = {p ∈ Ass(M) : dim R/p = d}

H
d
m
(G)
∼
=
H
d
m
(M).
R

R M

R

R H
i
IR

(M

)
∼
=
H
i
I
(M


) i ≥ 0
R

R
R

R
M R R

H
i
I
(M) ⊗
R
R

∼
=
H
i
IR

(M ⊗
R
R

) i ≥ 0
M R
H
i

I
(M) = 0, i > dim M.
M R
dim M = d. H
d
m
(M) = 0.
0 = a ∈ R M−
am = 0 m = 0 m ∈ M.
a
1
, . . . , a
n
R M− a
i
M/(a
1
, . . . , a
i−1
)M i = 1, . . . , n.
a
1
, . . . , a
n
∈ R M− a
1
, . . . , a
n
M− M/(a
1

, . . . , a
n
)M = 0 I
R M I
M
I M
I depth(I, M). M
m M depth M.
depth M  dim M. dim M = Sup{i : H
i
m
(M) = 0},
depth M = inf{i :
H
i
m
(M) = 0}
M
H
i
I
(M)
M R
R H
i
m
(M) i
E(k)
R k k = R/m D
R

(−) Hom(−, E(k))
R M D
R
(M) M
Ann
R
(M) = Ann
R
(D
R
(M)).
(R, m)
M A R
M D
R
(M)
M
∼
=
D
R
(D
R
(M))
A D
R
(A) A
∼
=
D

R
(D
R
(A))
R
R R
Ext
M R M
inj dim M inj dim
R
M n
E
•
M E
m
= 0 m > n
n M
R
inj dim R < ∞.
(R, m)
(R

, m

) n

f : R

−→ R M R
Ext

j
R

(M, R

) R
H
i
m
(M)
∼
=
D
R
(Ext
n

−i
R

(M, R

)).
R A A = 0
x A x ∈ R.
x ∈ R x A
p A p−
p
p
0 p p

p Ann A p
A
1
, . . . , A
n
p x ∈ p.
x A
1
, . . . , A
n
∃t ∈ N x
t
A
i
=
0, ∀i. x
t
(

n
i=1
A
i
) = 0, x

n
i=1
A
i
x /∈ p. xA

i
= A
i
, ∀i. x(

n
i=1
A
i
) =

n
i=1
A
i
,
x

n
i=1
A
i

n
i=1
A
i
p
A p B A A/B = 0.
x ∈ p. ∃t ∈ N x

t
A = 0,
x
t
(A/B) = (x
t
A + B)/B = 0,
x A/B x /∈ p. xA = A,
x(A/B) = (xA + B)/B = (A + B)/B = A/B,
x A/B A/B p
1.A = A, 1 /∈ Ann A, Ann A = R.
xy ∈ Ann A x /∈ Ann A y
n
A = 0, ∀n A
y A yA = A 0 = xyA = xA
x ∈ Ann A y
n
∈ Ann A n Ann A
p
p S R
A
ϕ
−→ S
−1
A m −→ m/1
S ∩ p = ∅ S
−1
A = 0.
S ∩ p = ∅ S
−1

A S
−1
p
m/s ∈ S
−1
A, m ∈ A, s ∈ S. s ∈ p
∃n ∈ N s
n
A = 0.
m/s = s
n
m/s
n
s = 0/s
n
s = 0/1 = ϕ(0).
s /∈ p sA = A m ∈ A m ∈ sA. ∃y ∈ A
m = sy, m/s = sy/s = y/1 = ϕ(y). ϕ
S ∩ p = ∅ ∃s ∈ S ∩ p. s ∈ S ∃n ∈ N
s
n
A = 0. ∀m/t ∈ S
−1
A, m ∈ A, t ∈ S
m/t = s
n
m/s
n
t = 0/s
n

t = 0/1.
S
−1
A = 0
S ∩ p = ∅ S
−1
p S
−1
R. a/s ∈
S
−1
p a ∈ p, s ∈ S ∃n ∈ N a
n
A = 0.
(a/s)
n
(b/t) = a
n
b/s
n
t = 0/s
n
t = 0/1
b/t ∈ S
−1
A. a/s S
−1
A
a/s /∈ S
−1

p a ∈ R s ∈ S. a /∈ p. A p
aA = A. x/t ∈ S
−1
A, t ∈ S, x ∈ A
y ∈ A x = ay.
x/t = ay/t = (a/s)(sy/t) ∈ a/sS
−1
A.
a/sS
−1
A = S
−1
A, a/s S
−1
A
S
−1
A S
−1
p
A R p R A
1
, . . . , A
r
p A. B = A
1
+ . . . + A
r
p
A.

B = 0
ϕ :
r

i=1
A
i
−→ A
1
+ . . . + A
r
ϕ(x
1
, . . . , x
r
) −→ x
1
+ . . . + x
r
. ϕ
B

r
i=1
A
i
.

r
i=1

A
i
p B
p
A A
A = A
1
+ . . . + A
n
A
i
p
i
−
∀i = 1, n. A
i
p
i
A A = A
1
+ . . . + A
n
A. i = j A
i
A
j
p− A
i
+ A
j

p−
A
A = A
1
+ . . . + A
n
A
i
p
i
∀i = 1, n
Ann A p
i
−
Ass(R/ Ann A) ⊆ {p
1
, . . . , p
n
}
Ann A
i
= q
i
i = 1, . . . , n.
q
i
p
i
−
Ann A = Ann(

n

i=1
A
i
) =
n

i=1
Ann A
i
=
n

i=1
q
i
.
q
i
p
i
Ann A =

n
i=1
q
i
Ann A
Ass(R/ Ann A) = {p

1
, . . . , p
r
} ⊆ {p
1
, . . . , p
n
}
n ≥ r).
Q A
Q = Q
1
+ . . . Q
r
Q
i
q
i
i = 1, . . . , n
{q
1
, . . . , q
r
} ⊆ {p
1
, . . . , p
n
}.
Q = A/P P A
Q = (

n

i=1
A
i
)/P =
n

i=1
(A
i
+ P )/P.
(A
i
+ P)/P
∼
=
A
i
/(A
i
∩ P).
(A
i
+ P)/P = 0 (A
i
+ P)/P p
i
−
Q.

Q = A/P = (A
1
+ P )/P + . . . + (A
r
+ P )/P
Q (A
i
+ P )/P p
i
Q
ℵ(A) = {x ∈ R : ∃n ∈ N x
n
A = 0}
I R V (I)
R I.
p
p ∈ {p
1
, . . . , p
n
}.
A p
A Q ℵ(Q) = p.
A Q p
V (Ann Q).
⇒ i = 1, . . . , n, P
i
=

j=i

A
j
A
i
A =
n

i=1
A
i
, A/P
i
= 0.
A/P
i
= (A
i
+ P
i
)/P
i
∼
=
A
i
/(A
i
∩ P
i
).

A/P
i
A
i
.
A/P
i
p
i
A
⇒ Q p A
ℵ(Q) = p.
⇒ Q A ℵ(Q) = p.
Rad(Ann Q) = {x ∈ R : ∃n ∈ N x
n
∈ Ann Q}
= {x ∈ R : ∃n ∈ N x
n
Q = 0}
= ℵ(Q) = p
Rad(Ann Q)
Ann Q p V (Ann Q).
⇒ Q A p
V (Ann Q).
Q
Q =
m

i=1
Q

i
, Q
i
p
i
− , i = 1, . . . , m
m  n Ann(Q
i
) = q
i
,
q
i
p
i
∀i = 1, m. p
V (Ann Q) p ∈ Ass(R/ Ann Q).
p ∈ Ass(R/ Ann Q) ⊆ {p
1
, . . . , p
m
} ⊆ {p
1
, . . . , p
n
}.
{p
1
, . . . , p
n

}
A A
A, Att A.
A
A =
m

i=1
A
i
=
n

j=1
A

j
.
m = n ℵ(A
i
) = ℵ(A

i
)
i = 1, . . . , n.
p ∈ {p
1
, , p
n
} A Q

p = Ann(Q).
A Q p = Ann(Q).
p ∈ {p
1
, , p
n
} p = p
i
P = A/P
i
P p
i
R p
i
n p
n
i
P = p
i
P = P.
Q := P/p
i
P p
i
A
p
i
⊆ Ann(Q) ⊆ Rad(Ann(Q)) = p
i
.

Ann(Q) = p
i
= p.
p Att A p
A.
p A. p
i
A A
i
A.
R/ Ann A A
A
Ann A.
p Ann A. A
A ⇒
p ∈ {p
1
, . . . , p
n
} = Att A. p Att A
q ⊂ p q = p q ∈ Att A.
⇒ Q A q
Ann Q. Ann A ⊆ Ann Q. q ∈ V (Ann A).
p. p
Att A.
p Att A.
⇒ Q A p
Ann Q. Ann A ⊆ Ann Q.
p ∈ V (Ann A). p V (Ann A)
q ∈ V (Ann A) q ⊂ p q = p.

q ∈ Att A. p.
p V (Ann A).
x ∈ R
x A x /∈
n

i=1
p
i
.
x A x ∈
n

i=1
p
i
,
ℵ(A) =
n

i=1
p
i
.
x /∈
n

i=1
p
i

x /∈ p
i
, ∀i = 1, n
x A
i
xA
i
= A
i
, ∀i = 1, n.
xA = x
n

i=1
A
i
=
n

i=1
xA
i
=
n

i=1
A
i
= A.
x A x ∈

n

i=1
p
i
x ∈ p
i
i ∃ t ∈ N x
t
A
i
= 0.
x
t
A =
n

i=1
x
t
A
i
=

i=j
x
t
A
j
⊆


i=j
A
j
= A.
x
t
x
x ∈
n

i=1
p
i
x ∈ p
i
, ∀i = 1, n. x
A
i
i ∃t ∈ N x
t
A
i
= 0, ∀i = 1, n.
x
t
A =
n

i=1

x
t
A
i
= 0. x A
x /∈
n

i=1
p
i
, x /∈ p
i
i xA
i
= A
i
.
x
t
A
i
= A
i
t ∈ N. ∀t ∈ N
x
t
A =
n


i=1
x
t
A
i
⊇ A
i
= 0.
x A
I R
A = IA.
x ∈ I A = xA.
⇒ I ⊆ p
i
, ∀i = 1, n
I ⊆ p
i
, i I
I = (x
1
, . . . , x
k
). k, x
k
∈ I ⊆ p
i
∃r
k
∈ N
x

r
k
N
i
= 0. r r
k
t = rk.
I
t
N
i
= 0. A = IA
A = I
t
A.
A = I
t
A = I
t
n

i=1
A
i
= I
t

j=i
A
j

⊆

j=i
A
j
= A.
I ⊆ p
i
, ∀i = 1, n.
I /∈ ∪
n
i=1
p
i
. x ∈ I x /∈ p
i
i.
x A
xA = A.
⇒
S R p
i
S ∩ p
i
= ∅ i = 1, . . . , r S ∩ p
j
= ∅ i = r + 1, . . . , n.
L
1
=


s∈S
sA, L
2
=
r

i=1
A
i
L
3
p− A p∩S = ∅
L
1
= L
2
= L
3
.
j = r+1, . . . , n, S ∩p
j
= ∅
x
j
∈ S ∩ p
j
. x
j
∈ p

j
k
j
∈ N x
k
j
j
A
j
= 0
j = r + 1, . . . , n. x =
n

j=r+1
x
k
j
j
. S
x ∈ S
L
1
=

s∈S
sA ⊆ xA = x
r

i=1
A

i
⊆
r

i=1
A
i
= L
2
.