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> k
M− > k
> k
> k
d − 1
H
i
I
(M) R− M I
i
H
i
I
(M)
Ass
R
(R/I

n
)
Ass
R
(M/I
n
M) Ass
R
(I
n
M/I
n+1
M)
n n
Att
R
(0 :
A
I
n
) Att
R
((0 :
A
I
n+1
)/(0 :
A
I
n
)
n A R−
Ext Tor
Ass
R
(J
n
M/J
n+1
M) Ass
R
(M/J
n
M)
depth(I, J
n
M/J
n+1
M) depth(I, M/J
n
M)
n
M− > k
dim M/IM > k M−
> k I
M− > k I
> k M I
depth
k
(I, M)
M = ⊕
n≥0
J
n
M/J
n+1
M r
k
(n) = depth
k
(I, J
n
M/J
n+1
M)
> k J
n
M/J
n+1
M I
i ≤ r
1
(n)
H
i
I
(J
n
M/J
n+1
M)
> k
J
n
M/J
n+1
M I
> k
r
k
(n) k = −1
k = 0 k = 1
J
n
M/J
n+1
M M/J
n
M
(R, m)
I, J R M
R− A R−
R = ⊕
n≥0
R
n
R
0
= R M = ⊕
n≥0
M
n
R−
> k
> k
> k J−
N
n
M
n
M/J
n
M
k ≥ −1 depth
k
(I, N
n
)
n
> k I−
N
n
depth(m, J
n
) depth(m, J
n
/J
n+1
) depth(m, R/J
n
)
lim
n→∞
depth J
n
lim
n→∞
depth J
n
/J
n+1
lim
n→∞
depth R/J
n
lim
n→∞
depth R/J
n
≤ lim
n→∞
depth J
n
/J
n+1
= lim
n→∞
depth J
n
− 1.
> k I
J− (M
n
)
(R, m) I, J R
M R− (M
n
) J− M
k ≥ −1
lim
n→∞
depth
k
(I, M
n
) lim
n→∞
depth
k
(I, M
n
/M
n+1
)
lim
n→∞
depth
k
(I, M/M
n
)
lim
n→∞
depth
k
(I, M/M
n
) ≤ lim
n→∞
depth
k
(I, M
n
/M
n+1
)
lim
n→∞
depth
k
(I, M
n
) − 1 ≤ lim
n→∞
depth
k
(I, M
n
/M
n+1
).
J ⊆ I lim
n→∞
depth
k
(I, M
n
/M
n+1
) = lim
n→∞
depth
k
(I, M
n
) − 1.
H
j
I
(M) j < i Ass
R
(H
i
I
(M))
depth
1
(I, M)
i H
i
I
(M) Ass
R
(H
i
I
(M))
i ≤ depth
1
(I, M)
i ≤ r
1
(n) = depth
1
(I, J
n
M/J
n+1
M) j ≤ s
1
(n) = depth
1
(I, M/J
n
M)
Ass
R
(H
i
I
(J
n
M/J
n+1
M)) Ass
R
(H
j
I
(M/J
n
M))
Ass
R
(H
i
I
(J
n
M/J
n+1
M)) Ass
R
(H
j
I
(M/J
n
M))
i ≤ r
1
(n) j ≤ s
1
(n) n
r
k
s
k
depth
k
(I, J
n
M/J
n+1
M) depth
k
(I, M/J
n
M)
Ass
R
(H
r
−1
I
(J
n
M/J
n+1
M)) Ass
R
(H
s
−1
I
(M/J
n
M))
n

i≤r
0
Ass
R
(H
i
I
(J
n
M/J
n+1
M))

i≤s
0
Ass
R
(H
i
I
(M/J
n
M))
n

t≤i
Ass
R
(H
t
I
(J
n
M/J
n+1
M)) ∪ {m}

t≤j
Ass
R
(H
t
I
(M/J
n
M)) ∪ {m}
n i ≤ r
1
j ≤ s
1
r
−1
≤ r
0
≤ r
1
s
−1
≤ s
0
≤ s
1
H
i
I
(J
n
M/J
n+1
M) H
j
I
(M/J
n
M) i < r
0
j < s
0
i < r
0
j < s
0
r
−1
= r
0
s
−1
= s
0
i = r
−1
j = s
−1
Ass
R
(H
i
I
(J
n
M/J
n+1
M)) Ass
R
(H
j
I
(M/J
n
M))
i ≤ r
0
j ≤ s
0
Ass
R
(H
i
I
(J
n
M/J
n+1
M)) Ass
R
(H
j
I
(M/J
n
M)) i ≤ r
1
j ≤ s
1
depth
−1
(I, M) depth
0
(I, M)
depth
1
(I, M)
depth(I, M) f-depth(I, M) gdepth(I, M)
R = ⊕
n≥0
R
n
R
0
= R M = ⊕
n≥0
M
n
R−
r
depth(I, M
n
) Ass
R
(H
r
I
(M
n
)) n
H
i
I
(M) = 0 i < depth(I, M)
R = ⊕
n≥0
R
n
R
0
= R M = ⊕
n≥0
M
n
R− I
R r
0
f-depth(I, M
n
)

j≤r
0
Ass
R
(H
j
I
(M
n
)) n
S =

n≥a
Ass
R
(H
r
0
I
(M
n
))
a p ∈ S, dim R/p > 0
p
R = ⊕
n≥0
R
n
R
0
= R M = ⊕
n≥0
M
n
R−
I R r
1
gdepth(I, M
n
)
l ≤ r
1

j≤l
Ass
R
(H
j
I
(M
n
)) ∪ {m} n
S =

n≥b

j≤l
Ass
R
(H
j
I
(M
n
)) b
p ∈ S, dim R/p > 0 p
Supp
R
(H
j
I
(M))
j < gdepth(I, M)
H
dim R−1
I
(M)
R R
R/ ann
R
(M) Supp
R
(H
dim M −1
I
(M))
Ass
R
(H
dim M −1
I
(M))
Ass
R
(H
d−1
I
(J
n
M/J
n+1
M)) Ass
R
(H
d

−1
I
(M/J
n
M))
d = dim(J
n
M/J
n+1
M) d

= dim(M/J
n
M)
N
n
R− J
n
M/J
n+1
M
M/J
n
M d dim N
n
Ass
R
(H
d−1
I
(N
n
)) n
H
i
I
(M)
i H
i
I
(M)
M I
f
I
(M) H
0
I
(M)
f
I
(M) ≥ 1 Ass
R
(H
i
I
(M))
i ≤ f
I
(M) Ass
R
(H
i
I
(J
n
M/J
n+1
M))
Ass
R
(H
j
I
(M/J
n
M)) i ≤ r(n) = f
I
(J
n
M/J
n+1
M)
j ≤ s(n) = f
I
(M/J
n
M)
Ass
R
(H
i
I
(J
n
M/J
n+1
M)) Ass
R
(H
j
I
(M/J
n
M))
i ≤ r(n) j ≤ s(n) n
M/J
n
M
R− M I, J
R Ass
R
(H
1
I
(M/J
n
M)) n
Ass
R
(H
1
I
(J
n
M/J
n+1
M)) n
J
n
M/J
n+1
M M/J
n
M
H
i
m
(M)
R− M
Att
R
(H
i
m
(M))
N
n
J
n
M/J
n+1
M M/J
n
M Att
R
(H
0
m
(N
n
))
n
Att
R
(H
d
m
(N
n
)) n d
dim N
n
Att
R
(H
i
m
(N
n
)) i n
n
(T, m) I T
Att
T
(H
3
m
(T/I
n
)) Att
T
(H
4
m
(I
n
)) n
M (R, m)
J R Att
R
(H
i
I
(J
n
M/J
n+1
M))
n i
Ext Tor
i ≥ 0 Ass
R
(Ext
i
R
(R/I
n
, M)) n
(T, m) I T

n≥0
Ass
T
(Ext
2
T
(T/I
n
, T ))
Ass
T
(Ext
2
T
(T/I
n
, T )) n
R I R
M R− p
M x ∈ M p = ann
R
(x).
M Ass
R
(M).
{p ∈ Spec(R)|M
p
= 0} M
Supp
R
(M)
R− M Ass
R
(M) = ∅
0 → M → N → K → 0 R−
Ass
R
(M) ⊆ Ass
R
(N) ⊆ Ass
R
(M) ∪ Ass
R
(K)
Supp
R
(M) ⊆ Supp
R
(N) = Supp
R
(M) ∪ Supp
R
(K).
M R−
Ass
R
(M)
Ass
R
(M) ⊆ Supp
R
(M)
p Ass
R
(M) p
Supp
R
(M)
M
p
M R− p ∈ Spec(R)
Ass
R
p
(M
p
) = {qR
p
|q ∈ Ass
R
(M), q ⊆ p}.
R dim R
dim R = sup{n|∃p
0
 p
1
 . . .  p
n
, p
i
∈ Spec(R)}.
M dim M = dim R/ ann
R
(M)
M R− Supp
R
M = V(ann
R
(M))
dim M = sup{dim R/p|p ∈ Supp
R
(M)}
= sup{dim R/p|p ∈ Ass
R
(M)}.
N R− M
Ass
R
(M/N) Ass
R
(M/N) = {p}
N p− M
N N =
N
1
∩ . . . ∩ N
r
N
i
p
i
− M
i = 1, . . . , r. N = N
1
∩ . . . ∩ N
r
N
i
p
i
= p
j
i = j.
p− p−
N
M R−
N M
N = N
1
∩ . . . ∩ N
r
N Ass
R
(M/N
i
) = {p
i
} i = 1, . . . , r
Ass
R
(M/N) = {p
1
, . . . , p
r
}
R− S S = 0 r ∈ R
r S S

ann
R
(S) = p S p−
R− M M =
M
1
+ . . . + M
r
p
i
− M
i
M M = 0 M
M = M
1
+ . . . + M
r
M
i
p
i
= p
j
i = j
p− p−
M
p R− M
M p−
M Att
R
(M)
M Att
R
(M)
M = M
1
+ . . . + M
r
M M
i
p
i

Att
R
(M) = {p
1
, . . . , p
r
}.
R−
M R− x ∈ R
M x.m = 0 0 = m ∈ M
x
1
, . . . , x
r
R M
x
i
M/(x
1
, . . . , x
i−1
)M i = 1, . . . , r
M/(x
1
, . . . , x
r
)M = 0.
x ∈ R
M x ∈ p p ∈ Ass
R
(M). (R, m)
x
1
, . . . , x
r
∈ m M R−
M R−
M I
M R−
I R IM = M
M I n
n = inf{i| Ext
i
R
(R/I, M) = 0}.
M R− I
R IM = M
M I M I
depth(I, M) IM = M depth(I, M) = +∞
M I
depth(I, M) = inf{i| Ext
i
R
(R/I, M) = 0}.
(R, m)
depth(m, M) M depth(M).
M R−
x
1
, . . . , x
r
M I
depth(I, M/(x
1
, . . . , x
r
)M) = depth(I, M) − r.
x = x
1
, . . . , x
t
R H
i
(x; M)
i K

(x; M)
M R− x =
x
1
, . . . , x
n
R
H
0
(x; M)

=
M/xM
H
n
(x; M)

=
(0 :
M
x)
(x).H
i
(x; M) = 0, i.
(R, m)
x
1
, . . . , x
n
∈ m M R−
x = x
1
, . . . , x
n
M
H
i
(x; M) = 0 i > 0
I x = (x
1
, . . . , x
n
)
M R− IM = M
depth(I, M) = n − sup{i|H
i
(x; M) = 0}.
I−
I R
I− Γ
I
(−) Γ
I
(M) =

n≥0
(0 :
M
I
n
)
R− M Γ
I
(M) I−
M
i ≥ 0
i Γ
I
(−) H
i
I
(−)
i I
M R− H
i
I
(−) M
H
i
I
(M) i M
I
M I− Γ
I
(M) = M
I− Γ
I
(M) = 0
M R−
M I− I
M
M R−
M/Γ
I
(M) I−
H
i
I
(M)

=
H
i
I
(M/Γ
I
(M)) i > 0.
M R− I, J R
Γ
I

J
(M)) = Γ
I+J
(M). Γ
I
(M) I−
I−
I− H
i
I
(M) I−
i
R

R−
M

R

− I ⊂ R M

R−
i ≥ 0 R− H
i
IR

(M

)

=
H
i
I
(M

)
M J JM = 0
M R/J− R−
H
i
I
(M)

=
H
i
I(R/J)
(M) = H
i
(I+J)/J
(M)
R− H
i
I+J
(M)

=
H
i
(I+J)/J
(M)
R− H
i
I
(M)

=
H
i
I+J
(M)
R

R−
M R− i ≥ 0 R


H
i
IR

(M ⊗
R
R

)

=
H
i
I
(M) ⊗
R
R

.
M R− p R
p
R
p
R− R
p

H
i
I
(M) ⊗
R
R
p

=
H
i
IR
p
(M ⊗
R
R
p
).
M ⊗
R
R
p

=
M
p
R− M
(H
i
I
(M))
p

=
H
i
IR
p
(M
p
).
M R−
H
i
I
(M) = 0 i > dim M
(R, m)
M R− H
i
m
(M) R−
i
(R, m) I
R M R− d
H
d
I
(M) R−
(R, m) M
R− d H
d
m
(M) = 0
Att
R
(H
d
m
(M)) = {p ∈ Ass
R
(M)| dim R/p = d}.
R− M
I R Ass
R
(H
i
I
(M)) i
M R−
r H
i
I
(M) i < r
Ass
R
(H
r
I
(M))
x = x
1
, . . . , x
n
R u ∈ N
x
u
= x
u
1
, . . . , x
u
n
M (x)
i ≥ 0 R− M
H
i
(x)
(M)

=
lim
−→
u
H
n−i
(x
u
, M).
H
n
(x)
(M) = lim
−→
u
M/(x
u
)M.
R = ⊕
n≥0
R
n
R
0
R R
0

R = ⊕
n≥0
R
n
M = ⊕
n≥0
M
n

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