Chiều hữu hạn và tập các Iđêan nguyên tố liên kết của môđun hữu hạn sinh

I

R
I R M R
H
i
I
(M)
H
i
I
(M)
H
i
I
(M) i ≥ 0
H
t
I
(M)
H
i
I
(M) i < t H
i
I
(M)
i < t
H

t
I
(M)
H
t
I
(M)
H
t
I
(M)
t H
t
I
(M)
H
t
I
(M)
M I f
I
(M)
f
I
(M) =

i ∈ N|H
i
I
(M)


∞.
I R M R
t ≤ f
I
(M) x
1
, , x
t
I
M

n
1
, ,n
t
AssM/(x
n
1
1
, . . . , x
n
d
d
)M
M R
M
M
0 = M

0
⊂ M
1
⊂ M
2
⊂ . . .
M ∃k ∈ N M
n
= M
k
n ≥ k.
R R R
R R
m R/m
R. (R, m)
(R, m, k) (k = R/m)
R R
R
m = R R R
m x ∈ R \m
R
m R
1 + m = {1 + a|a ∈ m} R R
m.
R S
f : R → S f(m
R
) ⊆ m
S
m

R
R m
S
S
R
R
R
R
R
M R S
M M S
M S S
S
S < S >
< S > M S
< S > R S
< S >=

n

i=0
r
i
s
i
|r
i
∈ R, s
i
∈ S, n ∈ N


.
R M M
R
R
0
//
N
//
M
//
P
//
0.
M
N P M
N M
M M/N
N M/N M
M N
M R M
n ∈N
M R p
R M x ∈ M, x = 0
p = Ann
R
(x) = {a ∈ R| ax = 0}.
M Ass
R
M

AssM R
p ∈ Ass
R
M ⇔

 p ∈ SpecR,
 ∃x ∈ M, x = 0 : p = Ann
R
(x).
R R.
p R
R/p R p
R/p ¯x R/p
¯x = x + p x ∈ R, x /∈ p
Ann
R
(¯x) = {a ∈ R| a¯x =
¯
0} = {a ∈ R| ax ∈ p} = {a ∈ R| a ∈ p} = p.
p
R/p Ass
R
(R/p) = {p}
M R p
R p ∈ Ass
R
M M N
N
∼
=

R/p

R
Ann(x) x ∈ M, x = 0 P

P ∈ AssM
M = 0 Ass
R
M = ∅
M = 0 Ass
R
M = ∅ D
M D =

p∈Ass
R
M
p.
0 → M

→ M → M

→ 0
R
Ass
R
M

⊆ Ass
R

M ⊆ Ass
R
M

∪ Ass
R
M

.
M R Ass
R
M
R
(M/I
n
M)
R
(I
n−1
M/I
n
M)
n n
S R
R ×S = {(r, s) |r ∈ R, s ∈ S}
∼
(r, s) ∼ (r

, s


) ⇔ ∃t ∈ S : t(rs

− r

s) = 0.
∼ R × S R × S
(r, s) = {(r

, s

) ∈ R ×S |(r

, s

) ∼ (r, s)}.
r/s (r, s)
S
−1
R = R ×S/ ∼= {r/s |r ∈ R, s ∈ S}
R ×S ∼ .
S
−1
R S
−1
R
R
S
−1
R S
−1

I I
R S
−1
I = S
−1
R ⇔ I ∩ S = φ S
−1
I
S
−1
R I ∩ S = φ.
p ∈ SpecR S = R\p R
S
−1
R R
p
pRp = {a/s |a ∈ p, s ∈ R\p}
R p.
M R M × S
(m, s) ∼ (m

, s

) ⇔ ∃t ∈ S : t(s

m −sm

) = 0.
∼ M × S M × S
(m, s) ∈ M × S

m/s (m, s)
(m/s) = {(m

, s

) ∈ M ×S |(m

, s

) ∼ (m, s)}.
= {(m

, s

) ∈ M ×S |∃t ∈ S : t(s

m −sm

) = 0}.
S
−1
M = M × S/ ∼ M × S
∼
S
−1
M = M × S/ ∼= {m/s |m ∈ M, s ∈ S}.
S
−1
M : m/s = m


/s

⇔ ∃t ∈ S : t(s

m −sm

) = 0.
S
−1
M S
−1
M
S
−1
R M
S 0/1 = 0
M
/s, ∀s ∈ S
S
−1
M R
rm/s = r/1.m/s = rm/s,
r ∈ R m/s ∈ S
−1
M.
p ∈ SpecR S = R\p R
R
p
S
−1

R M
p
S
−1
M
M
p
M
p.
R
R.
I R.
V (I) = {p ∈ SpecR |p ⊇ I}.
(R) R.
T R (T )
T
Supp(M) R
M
p
= 0 Supp(M) M
M = ∅ M = 0
Ann
R
M = {a ∈ R|aM = 0} = {a ∈ R|ax = 0, ∀x ∈ M}
R M
M R
Supp(M) = {p ∈ SpecR|p ⊇ Ann
R
(M)} = V (Ann
R

(M)).
0 −→ M

−→ M −→ M

−→ 0
Supp(M) = Supp(M

) ∪Supp(M

).
R
p
0
⊃ p
1
⊃ p
2
⊃ ⊃ p
n
n.
p ∈ SpecR
p
0
= p p ht(p)
I R
ht(I) = inf{ht(p)|p ∈ SpecR, p ⊇ I}.
R
R dim R
dim R = sup {ht(p) | p ∈ SpecR}.

M R dim(R/Ann
R
(M))
M dim
R
(M) dim M M
SuppM = V (Ann
R
(M)
dim M = dim(R/Ann
R
(M)) = sup
p∈ M
dim(R/p).
R K
dim(SuppK) = max{dim(R/p ) | p ∈ SuppK}.
p ⊆ q p ∈ SuppK q ∈ SuppK
dim(SuppK)
SuppK
T SpecR
(T )
i
:= {p ∈ T | dim(R/p) = i} (T)
≥i
:= {p ∈ T | dim(R/p) ≥ i}
I R dim(M/IM) >
s m R dim(M
m
/IM
m

) > s.
M R
a M
ax = 0 x ∈ M, x = 0.
x
1
, x
2
, . . . , x
n
R
R M M
M/(x
1
, x
2
, . . . , x
n
)M = 0 x
i
M/(x
1
, x
2
, . . . , x
n
)M
i = 1, . . . , n.
a ∈ R M x /∈ p
p ∈ AssM x

1
, x
2
, . . . , x
n
M
M/(x
1
, x
2
, . . . , x
n
)M = 0 x
i
∈ p p ∈ AssM/(x
1
, x
2
, . . . , x
n
)M
i = 1, . . . , n.
x
1
, x
2
, . . . , x
n
M n
I R IM = M x

1
, x
2
, . . . , x
n
M I x
1
, x
2
, . . . , x
n
I y ∈ I x
1
, x
2
, . . . , x
n
, y
M
I M
I
I
M
• I = m
m
(M) M
M
• x
1
, x

2
, . . . , x
n
M
M M ≤ M
R m M
R
a ∈ m M
M a /∈ p p ∈
R
(M)\{m}
x
1
, x
2
, , x
n
R
M M x
i
M/(x
1
, , x
i−1
)M i = 1, , n
a ∈ m M 0:
M
a
a ∈ m M dim(0:
M

a) ≤ 0
M m
M m
I R dim(M/IM) > 0 M
I
M I
a ∈ R
M a /∈ p p ∈
R
(M)
dim(R/p) > 1
x
1
, , x
n
R M
a
i
M/(x
1
, , x
i−1
)M
i = 1, , n
a ∈ m M
dim(0:
M
a) ≤ 1.
I R dim(R/I) ≤ 1 M
I

M I
I R dim(M/IM) > 1 M
I M
M I
R m
M R dim M = d
I R R M
Γ
I
(M) :=

n∈N
(0:
M
I
n
) ={x ∈ M|∃n ∈ N, xI
n
= 0}.
Γ
I
(M) M R f : M →
N f(Γ
I
(M)) ⊆ Γ
I
(N) R Γ
I
(f) : Γ
I

(M) →
Γ
I
(N), x → Γ
I
(f)(x) = f(x), ∀x ∈ Γ
I
(M) Γ
I
(−)
R
R Γ
I
(−) i
i Γ
I
(−) H
i
I
(−)
i I
M H
i
I
(M) R H
i
I
(M)
i M I
dim M = d H

d
m
(M)
M
H
i
I
(M)
M :
0 −→ M −→ E
0
−→ E
1
−→ E
2
−→
Γ
I
(−)
0 −→ Γ(E
0
)
u
∗
0
−→ Γ(E
1
)
u
∗

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

i
I
(M) = 0 i ≥ 1
M = Γ
I
(M) H
i
I
(M) = 0 i ≥ 1
H
i
I
(M) I H
i
I
(M) = Γ
I
(H
i
I
(M))
H
j
I
(H
i
I
(M)) = 0
0 −→ M


−→ M −→ M

−→ 0
R
δ
n
: H
n
I
(M

) −→ H
n+1
I
(M

)
0 −→ H
0
I
(M

) −→ H
0
I
(M) −→ H
0
I
(M


)
δ
0
−→ H
1
I
(M

)
−→ H
1
I
(M) −→ H
1
I
(M

)
δ
1
−→ H
2
I
(M

) −→
δ
n
M = M/Γ
1

(M). ≥ 1
H
n
I
(M)
∼
=
H
n
I
(M)
H
n
I
(M) M M
I
H
n
I
(M) M I M
x ∈ I.
M I R
depth
I
(M) = inf{i | H
i
I
(M) = 0}.
I R H
i

I
(M) = 0 i >
dim(SuppM) M (R, m)
m dim M = sup{i | H
i
I
(M) = 0}.
p R p ∈ AssH
n
I
(M)
pR
p
∈ AssH
n
IR
p
(M
p
)
H
i
I
(M)
H
i
I
(M)
i ≥ 0
H

t
I
(M) H
i
I
(M)
i < t H
i
I
(M) i < t
H
t
I
(M) H
t
I
(M)
H
t
I
(M)
H
i
I
(M)
i R
I = m M H
i
m
(M)

i = dim M
M I
f
I
(M) =

i ∈ N | H
i
I
(M)

∞
f
I
(M) IM = M
M I t = f
I
(M). H
t
I
(M)
IM = M M I
f
I
(M) = inf

i ∈ N | I
n
H
i

I
(M) = 0, ∀n ∈ N

n
0
I
n
0
H
i
I
(M) = 0 i < f
I
(M).
I
I R x
1
, . . . , x
t
I x
1
, . . . , x
t
I
M
((x
1
, . . . , x
i−1
)M : x

i
/(x
1
, . . . , x
i−1
)M) ⊆ V (I)
i = 1, . . . , t V (I)
I
n
I M n
I
x
1
, . . . , x
t
I M, i =
1, . . . , t. y ∈ I x
1
, . . . , x
t
, y
I M
H
0
I
(M) = M y ∈ I.
H
0
I
(M) = M H

0
I
(M/H
0
I
(M)) = 0 (M/H
0
I
(M)) > 0
y ∈ I M/H
0
I
(M) x
1
, . . . , x
t
, y
I M
x
1
, . . . , x
t
I M
∀p ∈ SpecR\V (I)
x
1
1
, . . . ,
x
t

1
M
p
i = 1, . . . , t x
i
M/(x
1
, . . . , x
t−1
)M
x
n
1
1
, . . . , x
n
t
t
I
n
1
, . . . , n
t
.
(M/(x
n
1
1
, , x
n

t
t
)M)\V (I) = (M/(x
1
, , x
t
)M)\V (I). (∗)
x
n
1
1
, . . . , x
n
t
t
I
n
1
, . . . , n
t
.
p ∈ (M/(x
1
, . . . , x
t
)M)\V (I). p
pR
p
∈ (M
p

/(
x
1
1
, . . . ,
x
t
1
)M
p
)
x
1
1
, . . . ,
x
t
1
M
p
(x
1
, . . . , x
t
)
(M/(x
n
1
1
, . . . , x

n
t
t
)M) = (M/(x
1
, . . . , x
t
)M)
n
1
, . . . , n
t
.
(R, m) m
m f
I
I
H
i
I
(M).
a
1
, . . . , a
t
√
I =

(a
1

, . . . , a
t
)R. I x
1
, . . . , x
t
M
(a
1
, . . . , a
t
)R = (x
1
, . . . , x
t
)R.
√
I =

(a
1
, . . . , a
t
)R a
1
R + (a
2
, . . . , a
t
)R  p

p ∈ AssM\V (I) x
1
= a
1
+ b
1
b
1
∈ (a
2
, . . . , a
t
)R x
1
/∈ p p ∈ AssM\V (I) x
1
I M (a
1
, . . . , a
t
)R = (x
1
, a
2
, . . . , a
t
)R.
I M
I R x
1

, . . . , x
t
I M j ≤ t
H
j
I
(M) =

H
j
(x
1
, ,x
t
)R
(M) j < t
H
j−t
I
(H
j
(x
1
, ,x
t
)R
(M)) j ≥ t
.