Về tập iđêan nguyên tố gắn kết của môđun đối đồng điều địa phương

M R.
H
i
I
(M) i > dim Supp
R
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
R
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
K
i
(M) R Ext
n−i
R

(M, R

) i ≥ 0.
K
i
(M) i M K(M) := K
d
(M)
M.
H
i
m
(M)
∼
=

Hom
R
(K
i
(M), E(R/m)) i
E(R/m) R/m.
K
i
(M)
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
R A
N-dim
R
A A
R 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
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)
M
Psupp
i
R
(M) H
i
m
(M)
Psupp
i
R

(M)
Psupp
i

R
(

M).
R R
Psupp
i
R
M = {P ∩R | P ∈ Psupp
i

R
(

M)} R
M i ≥ 0
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
p, q ∈ Spec(R), Q ∈ Spec(

R)
q ⊆ p Q ∩ R = q P ∈ Spec(

R) Q ⊆ P
P ∩ R = p.
p ∈ Spec(R)
F
p
: M
R
→ M
R
p
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 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
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.
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 N
0
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).
H
dim R
I
(R)

R
Att

R
H
dim R
I
(R) = {P ∈ Ass(

R) | dim(

R/P) = dim R,

I

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

F
p
(−) = Hom
R

Hom
R
(−, E(R/m)), E(R/p)

R R
p
E(−)
R F
p
(H
d
I
(M))
∼
=
H
d−dim R/p
pR
p
(M/N)
p
.
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

,
N
H
d
I
(M)
H
d
I

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

Cos
R
(H
d
I
(M)) H
d
I
(M)
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) = 
R
(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) N
Ass
R
(I, M)
q m H
d
I
(M)
e

(q, H
d
I
(M)) =

p∈Cos
R
(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).
(R, m)

R m R M R

d A R L R
I R Var(I)
R I.
R L L = 0
x ∈ R x L
x ∈ R x L
p L p
L R L = L
1
+ . . . + L
n
L
i
p
i
L,
L = 0 L L
p
i
L
i
i = 1, . . . , n.
L
1
, L
2
p L L
1
+L
2

p L. L
{p
1
, . . . , p
n
}
L
L Att
R
L. L
i
,
i = 1, . . . , n L. p
i
Att
R
L p
i
L
L
i
L.
L R
Att
R
L = ∅ L = 0.
min Att
R
L = min Var(Ann
R

L)
dim(R/ Ann
R
L) = max{dim(R/p) | p ∈ Att
R
L}.
0 → L

→ L → L

→ 0 R
Att
R
L

⊆ Att
R
L ⊆ Att
R
L

∪ Att
R
L

.
A R r ∈

R u ∈ A. (r
n

)
n∈N
R r. Ru k
m
k
u = 0. n
0
r
n
− r
m
∈ m
k
m, n ≥ n
0
. r
n
u = r
n
0
u n ≥ n
0
.
ru = r
n
0
u. A

R
A R

A

R A

R

R A
R R →

R
R A. A
R

R
Att
R
A = {P ∩ R |P ∈ Att

R
A}.
A A.
A = 0 
R
(A) < ∞
Att
R
A = {m}.
I R
R L Γ
I

(L) =

n≥0
(0 :
L
I
n
) f : L → L

R f(Γ
I
(L)) ⊆ Γ
I
(L

))
Γ
I
(f) : Γ
I
(L) → Γ
I
(L

) Γ
I
(f)(x) = f(x)
x ∈ Γ
I
(L) Γ

I
(−)
R I
i ≥ 0
i Γ
I
(−)
i I H
i
I
(−).
H
i
I
(−) R L H
i
I
(L)
i L I
H
i
I
(−) H
0
I
(L)
∼
=
Γ
I

(L).
I, J R
√
I =
√
J H
i
I
(L) = H
i
J
(L)
f : R → R

L

R

L

R f
r ∈ R m

∈ L

f(r)m

.
R H
i

IR

(L

) H
i
I
(L

), IR

R

f(I).
i L

R R

f : R → R

L

R

I R. i ≥ 0
H
i
IR

(L


)
∼
=
H
i
I
(L

) R
f : R → R

f : R → R

R

H
i
I
(L) ⊗
R
R

∼
=
H
i
IR

(L ⊗

R
R

)
i ≥ 0
H
i
I
(L) = 0 i > dim Supp
R
L
M = 0 d = max{i|H
i
m
(M) = 0}
M = 0 depth(I, M) = min{i|H
i
I
(M) = 0}.
I
dim R = n I R
H
n
I
(R) = 0
P

R dim

R/P = n

dim

R/(I

R + P) > 0.
H
i
m
(M) R i ≥ 0
H
d
I
(M) R I R
M = 0 dim M = d.
H
d
m
(M) = 0
Att
R
(H
d
m
(M)) = {p ∈ Ass
R
M | dim(R/p) = d}.
p ∈ Supp
R
(M) dim R/p = t
i ≥ 0 q q ⊆ p

qR
p
∈ Att
R
p
(H
i
pR
p
(M
p
)) q ∈ Att
R
(H
i+t
m
(M))
p ∈ Ass
R
M dim R/p = t.
H
t
m
(M) = 0 p ∈ Att
R
H
t
m
(M).
H

dim R
I
(R)

R.
I R
Att

R
(H
d
I
(M)) = {P ∈ Ass

R

M | dim

R/P = d,

I

R + P = m

R}.
R
p ⊂ q R
p = p
0
⊂ p

1
⊂ . . . ⊂ p
n
= q p q
i = 0, . . . , n −1 p
i
p
i+1
.
n
R
p ⊂ q R p q
K K
K
R R
p
p ∈ Spec(R).
R R
R dim R/q = dim R/p + ht p/q
p, q q ⊆ p
R
R
R n
R[x
1
, . . . , x
n
]
R
R

R
dim

R/P = dim

R P ∈ min Ass

R R
depth R = dim R.
R
R
R