class="bi x0 y0 w1 h1"
class="bi x1 y1 w2 h2"
S = K[x
1
, . . . , x
n
] M
S M =

i∈Z
M
i
M
reg(M) = inf{p | H
i
m
(M)
j
= 0 ∀i, j : i + j > p},
H
i
m
(M) i
m = (x
1
, . . . , x
n
)
a
k
k

reg(K
i
(M)) reg(M)
reg(K
i
(M))
l(H
i
m
(M)
j
)
l(H
i
m
(M)
j
)
reg(K
i
(M))
M
k reg(K
M
)
reg(K
R
) reg(R)
a
K

(M)
K
i
(M)
M
class="bi x2 y54 w3 hc"
M S
M
0 −→
β
q

i=1
S(−a
qi
)
ϕ
q
−→ · · · −→
β
1

i=1
S(−a
1i
)
ϕ
1
−→
β

, . . . , a
0β
0
}
indeg(0) = ∞
R = S/I I M R
d m = R
+
=

i>0
R
i
R H
i
m
(M) i
M m
N
a(N) = sup{t ∈ Z | [N]
t
= 0},
a(0) = −∞
a
i
(M) := a(H
i
m
(M)).
M

reg(M) ≤ max{reg(N), reg(P ) + 1}
reg(N) ≤ max{reg(M), reg(P )}
reg(P ) ≤ max{reg(N), reg(M) − 1}.
dim(M) > 0 y ∈ S
1
M p ≥ 1
reg
p
(M/yM) ≤ reg
p
(M) ≤ reg
p−1
(M/yM).
P
M
(t) M P
M
(t)
P
M
(t) = e
0

t + d − 1
d − 1

− e
1

t + d − 2

dim
K
(H
i
m
(M)
t
).
k
k
0 ≤ i ≤ d D
i
M
dim D
i
≤ i D
−1
= 0.
0 = D
−1
⊆ D
0
⊆ · · · ⊆ D
d
= M
M
M
i
= D
i

adeg(M) =

p∈Ass(M)
mult
M
(p)e(S/p),
mult
M
(p) = l(H
0
m
p
(M
p
)) p M
adeg(M) deg(M)
D = {D
i
}
−1≤i≤d
M
adeg(M) = deg(M
d
) + adeg(D
d−1
) =
d

i=0
deg(M

,
t ≥ 1. R (2t−1) adeg(R) =
2, reg(R) = t, 2t
adeg(R)
R =
K[x, y, u, v]
(x
s
, y) ∩ (u, v) ∩ (x, y
t
, u)
, s, t ≥ 2.
s 2, adeg(R) = s + t, reg(R) =
max{s, t} 2s + t − 1.
K
i
(M) = Ext
n−i
S
(M, S)(−n).
K
M
:= K
d
(M)
reg(K
M
) reg(M) M
D = {D
i

m

N/(y
1
, . . . , y
j
)N

= 0 ∀i, j : i + j < d.
k > 0 m
k
M M
k M k m
2k
M
k ≥ 1 d > 0 m
k
M
d
reg(K
M
) ≤ − indeg(M) + (d − 1)k + 2.
M
reg(K
M
) ≤ − indeg(M) + d + 1.
M
d
reg(K
M

(M/xM) −→ 0 :
K
i
(M)
x −→ 0
d
R d ≥ 2
a
d
(K
R
) ≤ [(deg R)
c
− 1] reg(R).
R
reg(K
R
) ≤ [(deg R)
c
− 1] reg(R) + d.
K
i
(M). reg(K
i
(M))
reg(M)
M = R = S/I
M = R
h
i


r − 1 − t
i − 1

H
M/(y
1
, ,y
i−1
)M
(r).
M
n S = K[x
1
, . . . , x
n
], n ≥ 2. r = reg(M),
i ≥ 1
h
i
M
(t) ≤ µ(M)

r − 1 − t
i − 1

r + n − i
n − i

.

M
(t),
d
i
M
(t) = h
i+1
M
(t), i ≥ 1.
K
i
(M) S q
i
M
(t)
d
i
M
(t) = q
i
M
(t) = P
K
i
(M)
(−t) t  0.
i ≥ 0,
∆
i
=

i
:= K
i
(M).
i ≥ 1
ri(K
i
) ≤ [2(1 + ∆
i−1
)]
2
i−1
− 2.
i
i = 0
reg(K
0
(M)) ≤ − indeg(M).
i = 1
reg(K
1
) < 4µ(M)(r + 2)
n
− 4µ(M)(r + 2)
n−1
.
i = 2
d ≥ 3.
∆
1

<
1
2

2µ(M)(r + 2)

n···(n+i)2
i(i−1)
2
−[µ(M)(r + 2)]
n
− n,
reg(K
i+1
) < [2µ(M)(r + 2)]
n···(n+i)2
i(i+1)
2
− 2[µ(M)(r + 2)]
n
− 2n.
i = d
d ≥ 2.
reg(K
d
) < [2µ(M)(r+2)]
n···(n+d−1)2
(d−1)(d−2)
2
−2[µ(M)(r+2)]

+
+ indeg(M) i ≥ 2.
k
k
M M
M
k M
d
a
· · ·
k