Về tập iđêan nguyên tố gắn kết của môđun đối đồng điều địa phương Trần Đỗ Minh Châu.

M R.
H
i
I
(M)
i > dim Supp M (R, m)
M H
d
m
(M) = 0, d = dim M.
M i H
i
m
(M) = 0.
I
(R, m) dim R = n H
n
I
(R) = 0
dim

R/(I

R + P) ≥ 1 P
m

R.
M H
i

I
(M)
H
i
I
(M)
i H
i
I
(M)
(R, m)
M R dim M = d.
M
E(R/m) H
i
m
(M)
i ≥ 0.
H
d
I
(M).
H
i
m
(M) H
d
I
(M)
¨o

R
A Att
R
A
H
i
m
(M)
H
d
I
(M) M
R.
H
i
m
(M) H
d
I
(M)
R A
Ann
R
(0 :
A
p) = p p Ann
R
A.
H
i

m
(M)

R H
i
m
(M)

R
Att
R
H
i
m
(M)
Att

R
H
i
m
(M)
P

R Att

R
H
i
m

(M)
P ∩ R Att
R
H
i
m
(M).
H
i
m
(M) R.
Att
R
H
i
m
(M)
Att

R
H
i
m
(M)
R
H
i
m
(M) H
i−dim R/p

pR
p
(M
p
).
H
i
m
(M)
R
H
i
m
(M).
H
d
m
(M) R

R
H
d
I
(R)

R.
H
d
I
(M) R

H
d
I
(M)
R
R
H
i
m
(M)
H
d
I
(M)
H
d
I
(M)
(R, m)
M R dim M = d A
R
H
i
m
(M) m
Att
R
H
i
m

(M) Att

R
H
i
m
(M)
R
H
i
m
(M).
R
M

M
Ass

R

M =

p∈Ass M
Ass

R
(

R/p


R).
A
Att

R
A =

p∈Att
R
A
Ass

R
(

R/p

R) (1)
A = H
i
m
(M)
A
f
a
: Spec(

R) → Spec(R) R
(R


, m

) n
Att
R
H
i
m
(M)) Att

R
H
i
m
(M)
Att

R
(H
i
m
(M)) =

p∈Att
R
(H
i
m
(M))
Ass


R
(

R/p

R). (2)
R
R M i ≥ 0
R
Att

R
(H
i
m
(M)) Att
R
(H
i
m
(M))
i ≥ 0
M Psupp
i
R
(M)
Psupp
i
R

(M) = {p ∈ Spec R : H
i−dim R/p
pR
p
(M
p
) = 0}.
Psupp
i
R
(M) H
i
m
(M)
Psupp
i
R
(M)
Psupp
i

R
(

M)
p R p
R R
p
M
p

R
p
M
p
= 0 p ⊇ Ann
R
M.
p ⊇ Ann
R
A p = m A
p
= 0.
p ∈ Spec(R)
F
p
: M
R
→ M
R
p
R
R
p
F
p
R A
F
p
F
p

R
F
p
R R
p
F
p
(A) = 0 p ⊇ Ann
R
A R A
R →

R
R
F
p
p ∈ Spec(R)
R
R
p
H
i−dim(R/p)
pR
p
(M
p
)
H
i
m

(M)
H
i
m
(M).
H
i
m
(M)?
p ∈ Spec(R)
F
p
: M
R
→ M
R
p
R R
p
F
p
H
i
m
(M) R
R A
Ann
R
(0 :
A

p) = p p ⊇ Ann
R
A.
H
d
m
(M) R/ Ann
R
(H
d
m
(M))
R H
dim R
m
(R)
i
H
i
m
(M). H
i
m
(M)
Psupp
i
R
(M) = Var

Ann

R
H
i
m
(M)

H
d
I
(M)
R
H
d
I
(M)
M
H
dim R
I
(R)

R.
H
d
I
(M)
H
d
I
(M).

F
p
(−) = Hom
R

Hom
R
(−, E(R/m)), E(R/p)

R R
p
E(−)
H
d
I
(M) Cos
R
(H
d
I
(M)),
H
d
I
(M)
R A N-dim
R
A A
q R
(0 :

A
q) 
R
(0 :
A
q
n+1
) N-dim
R
A
n Θ
q
A
(n). N-dim
R
A = s.
Θ
q
A
(n) = (0 :
A
q
n+1
) =
e

(q, A)
s!
n
s

+ s
n  0 e

(q, A) e

(q, A)
A q
Psupp
i
R
(M)
H
i
m
(M).
Cos
R
(H
d
I
(M)) H
d
I
(M)
(R, m)
A R M R dim M = d.
I R Var(I) R I

R


M m R M
H
i
m
(M) m
H
i
m
(M)
R M
M

M
Ass

R

M =

p∈Ass M
Ass

R
(

R/p

R);
Ass
R

M =

P ∩ R | P ∈ Ass

R

M

.
A R A

R
A

R
Att
R
A Att

R
A
Att
R
A = {P ∩ R | P ∈ Att

R
A}.
Att

R

A =

p∈Att
R
A
Ass

R
(

R/p

R) (1)
A
R A.
Att

R
A =

p∈Att
R
A
Ass

R
(

R/p


R) R A
f
a
: Spec(

R) → Spec(R)
(R, m)
Att

R
A =

p∈Att
R
A
Ass

R
(

R/p

R)
R A R
R
Att

R
(H
i

m
(M)) =

p∈Att
R
(H
i
m
(M))
Ass

R
(

R/p

R)
Att
R
H
i
m
(M) Att

R
H
i
m
(M).
R

R M i ≥ 0
Att

R
(H
i
m
(M)) =

p∈Att
R
(H
i
m
(M))
Ass

R
(

R/p

R). (2)
(R, m)
R H
i
m
(M) R
R
R

R
R R
p ∈ Spec(R) P ∈ Spec(

R) P ∩R = p
R →

R f : R
p
→

R
P

R
P
⊗(R
p
/pR
p
)
∼
=

R
P
/p

R
P

f
pR
p
R
p
R p P.
H
i
m
(M) R

R.
i M
i ≥ 0 i M
Psupp
i
R
(M)
Psupp
i
R
M = {p ∈ Spec(R) | H
i−dim(R/p)
pR
p
(M
p
) = 0}.
R
min Att


R
(H
i
m
(M)) = min

p∈Att
R
(H
i
m
(M))
Ass

R
(

R/p

R) R
M i ≥ 0
dim(R/ Ann
R
H
i
m
(M)) = N-dim
R
H

i
m
(M) R
M i ≥ 0
R
i M

M
Psupp
i
R
(M) Psupp
i

R
(

M).
i ≥ 0
Psupp
i
R
M ⊆ {P ∩ R | P ∈ Psupp
i

R
(

M)}.
i M.

R
Psupp
i
R
M = {P ∩R | P ∈ Psupp
i

R
(

M)} R
M i ≥ 0
R
H
i
m
(M) H
i−dim R/p
pR
p
(M
p
)
H
i−dim R/p
pR
p
(M
p
) H

i
m
(M)
p
H
i
m
(M)
R
H
i
m
(M).
p R p
R R
p
M
p
R
p
M
p
= 0 p ⊇ Ann
R
M.
R A p = m A
p
= 0 Supp(A) ⊆ {m}.
p ∈ Spec(R)
F

p
: M
R
→ M
R
p
F
p
R
A F
p
F
p
R
F
p
R R
p
F
p
(A) = 0 p ⊇ Ann
R
A R A
F
p
p ∈ Spec(R)
p ∈ Att
R
A Ann
R

(0 :
A
p) = p
A = A
1
+ . . . + A
n
A A
i
p
i
r 0 < r < n.
B = A
1
+ . . . + A
r
.
Att
R
(A/B) = {p
r+1
, . . . , p
n
}.
R M p ∈ Spec(R)
p ⊇ Ann
R
M M
p
= 0 F

p
p ∈ Spec(R). p ⊇ Ann
R
A F
p
(A) = 0
F
p
: M
R
→ M
R
p
p ∈ Spec(R). F
p
: M
R
→ M
R
p
R A R
F
p
(A) R
p
0 Ann
R
(0 :
A
p) = p

R A
N-dim
R
A ≤ dim(R/ Ann
R
A). R A
N-dim
R
A < dim(R/ Ann
R
A).
¨o
dim(R/ Ann
R
A) = N-dim
R
A R A
dim(R/ Ann
R
A) = N-dim
R
A
A = H
k
m

R
(

R/P) P ∈ Spec(


R) k = dim(

R/P)
dim(

R/P) = dim(R/(P ∩ R)) P ∈ Spec(

R)
p ∈ Spec(R)
F
p
: M
R
→ M
R
p
R →

R R
H
i
m
(M)
N-dim
R
(H
i
m
(M)) ≤ dim(R/ Ann

R
H
i
m
(M)).
(R, m) N-dim
R
(H
1
m
(R)) < dim(R/ Ann
R
H
1
m
(R)).
i H
i
m
(M)
N-dim
R
(H
i
m
(M)) = dim(R/ Ann
R
H
i
m

(M))
N-dim
R
(H
i
m
(M)) =
dim(R/ Ann
R
H
i
m
(M))
H
i
m
(M)
M i
dim(R/ Ann
R
H
i
m
(M)) = N-dim
R
(H
i
m
(M)) i
R M;

H
i
m
(M) i
R M;
R
p ∈ Spec(R), F
p
R
F
p
(H
i
m
(M)) = H
i−dim(R/p)
pR
p
(M
p
). R
R
p
H
i−dim(R/p)
pR
p
(M
p
)

H
i
m
(M)
H
i
m
(M)
H
i
m
(M)
H
i
m
(M)
p ∈ Spec(R)
F
p
: M
R
→ M
R
p
R F
p
(H
i
m
(M))

F
p
(H
i
m
(M)) = 0 p ⊇ Ann
R
H
i
m
(M) i
R M R
R
H
i
m
(M)
Att

R
H
i
m
(M)
Att
R
H
i
m
(M) Psupp

i
(M) Psupp
i
(

M)
R
R
H
i
m
(M).
(R, m)
m M R dim M = d
A R I R Var(I)
R I

R

M m
R M
H
d
m
(M) R

R
H
dim R
I

(R)

R.
H
d
I
(M) R
H
d
I
(M)
H
d
I
(M)
M.
R A
Ann
R
(0 :
A
p) = p p ⊇ Ann
R
A.
R
H
d
I
(M)
M.

0 M.
0 =

p∈Ass
R
M
N(p)
0 M
Ass
R
(I, M) =

p ∈ Ass
R
M | dim(R/p) = d,

I + p = m

.
N =

p∈Ass
R
(I,M)
N(p). Ass
R
(I, M)
M R N
0
H

d
I
(M)
H
d
I
(M)
R N
H
d
I
(M)
R/ Ann
R
H
d
I
(M)
√
I + p = m
p H
d
I
(M);
R/ Ann
R
H
d
I
(M) H

d
I
(M)
∼
=
H
d
m
(M/N).
R H
d
I
(M)
Ass
R
(I, M) Att
R
H
d
I
(M)
Ass
R
(I, M)
Ass
R
(I, M) ⊆ Att
R
H
d

I
(M). Ass
R
(I, M) = ∅
H
d
I
(M) = 0.
Ass
R
(I, M) = ∅ H
d
I
(M)
H
d
I
(M) = 0
H
d
I
(M)
Att
R
H
d
I
(M) =

p ∈ Ass

R
M | dim(R/p) = d,

I + p = m

.
Ass
R
(I, M)
Att

R
H
d
I
(M) =

p∈Att
R
H
d
I
(M)
Ass

R
(

R/p


R);
H
d
I
(M) R/p
p ∈ Ass
R
(I, M).
p ∈ Spec(R).
F
p
(−) = Hom
R

Hom
R
(−, E(R/m)), E(R/p)

R R
p
E(−)
p ∈ Spec(R) F
p
(−)
N R
F
p
(H
d
I

(M))
∼
=
H
d−dim(R/p)
pR
p
(M/N)
p
.
H
d
I
(M)
N
H
d
I
(M) Cos
R
(H
d
I
(M)),
Cos
R
(H
d
I
(M)) =


p ∈ Spec(R) | H
d−dim(R/p)
pR
p
(M/N)
p
= 0

.
Cos
R
(H
d
I
(M))
Var(Ann
R
H
d
I
(M)).
Cos
R
(H
d
I
(M)) ⊆ Var(Ann
R
H

d
I
(M)).
H
d
I
(M) Cos
R
(H
d
I
(M))
H
d
I
(M)
Cos
R
(H
d
I
(M)) = Var(Ann
R
H
d
I
(M)).
H
d
I

(M)
Spec(R)
q R (0 :
A
q)
(0 :
A
q
n+1
) N-dim
R
A n
Θ
q
A
(n) N-dim A = s.
Θ
q
A
(n) = (0 :
A
q
n+1
) =
e

(q, A)
s!
n
s

+ s
n  0 e

(q, A) e

(q, A)
A q
Psupp
i
R
(M) i psd
i
(M)
H
i
m
(M).
H
i
m
(M)
H
d
I
(M)
q m Ass
R
(I, M) N
H
d

I
(M)
e

(q, H
d
I
(M)) =

p∈Cos H
d
I
(M)
dim(R/p)=d

R
p

H
0
pR
p
(M/N)
p

e(q, R/p).
e

(q, H
d

I
(M)) = e(q, M/N) =

p∈Ass
R
(I,M)

R
p
(M
p
)e(q, R/p).
H
d
I
(M)
H
d
I
(M)
H
d
I
(M)
H
d
I
(M)
R