class="bi x0 y0 w1 h1"
class="bi x0 y1 w1 h2"
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


∞.
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
class="bi x1 y3a w3 he"
M R
M
M
0 = 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

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
∼
=

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


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

/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

(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}.

) > 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 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

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) →
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

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

−→ 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


(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

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
(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

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

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

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

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