Chỉ số chính quy Castelnuovo-Mumford của một số lớp môđun

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

k
C : reg(C) ≤ deg(C) − codim(C) + 1
deg(C) codim(C) V 1
reg(V ) ≤ deg(V ) − codim(V ) + 1
V V
deg(V ) ≤ codim(V ) + 2
¨u V
k
k
k
reg(M) k
k
S M
K
d
(M) = Ext
n−d
S
(M, S)(−n)
reg(K
d
(M)) K
d
(M) reg(M)
K
i
(M) = Ext
n−i
S
(M, S)(−n), i < d

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
R
reg(K
i
(R)) K
i
(R)
reg(K
i
(M)) reg(M) M
i + 1 M
i M M/xM
i M = R
k
k
adeg(R) deg(R)
k
k
adeg(R)
K
M
reg(M)
M M
K
M
M
a K
R
K
i

(M)
K
i
(M)
M
M S
M
0 −→
β
q

i=1
S(−a
qi
)
ϕ
q
−→ · · · −→
β
1

i=1
S(−a
1i
)
ϕ
1
−→
β

0

i=1
S(−a
0i
)
ϕ
0
−→ M −→ 0,
a
ki
, k = 1, . . . , q; 1 ≤ i ≤ β
k
β
i
i M β
0
M
µ(M)
gen(M) = max{a
01
, . . . , a
0β
0
}.
M
indeg(M) = inf{n | [M]
n
= 0} = inf{a
01

, . . . , 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{i + a
i
(M) | 0 ≤ i ≤ d}.
a
ij
M
S
reg(M) = max{a
ij
− i | i = 0, . . . , q j = 1, . . . , β
i
},
a
ij
reg(M) ≥ max{a
0j
| j = 1, . . . , β
0
} = gen(M).
reg(M)
M
r ≥ gen(M). H
i
m
(M)
r−i
=
0 i ≤ d reg(M) ≤ r
0 −→ M −→ N −→ P −→ 0
S

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

d − 2

+ · · · + (−1)
d−1
e
d−1
,
e
0
, e
1
, . . . , e
d−1
e
0
> 0 e
0
M deg(M) e(M) d = 0
e(M) = l(M)
t
H
M
(t) − P
M
(t) =
d

i=0
(−1)
i

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

/D
i−1
0 ≤ i ≤ d.
M
i
= 0 dim M
i
= i
k M
k M
i
, 0 ≤ i ≤ d
k
m
k
H
j
m
(M
i
) = 0 j < dim M
i
M M
i
, 0 ≤
i ≤ d R = S/I k
k
k
k
M

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

i
).
M
reg(M) ≤ gen(M) + deg(M).
dim(M) = 0 M
reg(M) ≤ gen(M) + deg(M) − 1.
M S
reg(M) ≤ gen(M) + adeg(M) − 1.
R
reg(R) ≤ adeg(R) − 1.
k
M k k ≥ 1.
M
k k ≥ 1
reg(M) ≤ gen(M) + deg(M) + (d − depth(M))k − 1.
M S k
reg(M) ≤ gen(M) + adeg(M) +
d(d − 1)
2
k − 1.
k
R =
K[x, y, u, v]
((x, y)
2
, xu
t
+ yv
t
)

,
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

}
−1≤i≤d
M
H
d
m
(M)
∼
=
H
d
m
(M
d
) K
M
∼
=
K
M
d
N S d
m q N
y
1
, . . . , y
d
N q
qH
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

) ≤ − indeg(M) + d.
M = R M
d
reg(K
R
) = d.
R
M
K
M
R
x ∈ S
1

x M
K
i
(M)

i ≥ 0
0 −→

K
i+1
(M)/xK
i+1
(M)

(1) −→ K
i

(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

M
(t) = l(H
i
m
(M)
t
).
M
indeg(M) ≥ 0 M =

i≥0
M
i
.
M
S = K[x
1
, . . . , x
n
] n ≥ 2. r = reg(M). y
1
, . . . , y
d
∈ S
1
M i ≥ 1
h
i
M
(t) ≤


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
S = K[x
1
, . . . , x
n
], n ≥ 2. r = reg(M),
reg(K
i
(M)) <

4µ(M)(r + 2)
n
− 4µ(M)(r + 2)
n−1
i = 1,
[2µ(M)(r + 2)]
n···(n+i−1)2
i(i−1)
2
i ≥ 2.
d
0
M
(t) = H
M
(t) − h
0
M
(t) + h
1

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

j=0

i
j


d
j
M
(−j)+ | q
j
M
(−j) |

.
M S
ri(M) = max{j ∈ Z | H
M
(j) = P
M
(j)}.
y ∈ S
1
M.
reg(M) = max{reg(M/yM), ri(M)}
M d reg(M) = ri(M)+
d.
K

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+1)
−

µ(M)(r + 2)

n
−n,
reg(K
2
) <

2µ(M)(r + 2)

2n(n+1)
−2

µ(M)(r + 2)

n
−2n.
i ≤ d − 1
1 ≤ i < d − 1.
∆
i

<
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)]

n
−2n+2.
M
S = K[x
1
, . . . , x
n
], n ≥ 2. r = reg(M),
reg(K
i
(M)) <









4µ(M)(r − indeg(M) + 2)
n
− 4µ(M)×
×(r − indeg(M) + 2)
n−1
+ indeg(M) i = 1,
[2µ(M)(r − indeg(M) + 2)]
n···(n+i−1)2
i(i−1)
2

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