Chỉ số chính quy Castelnuovo-Mumford bậc lũy linh

m
m I
I
R =

n≥0
R
n
R
0
R
+
R M R
i
(M) =

{n | H
i
R
+
(M)
n
= 0} H
i
R
+
(M) = 0
−∞ H
i
R

+
(M) = 0
H
i
R
+
(M) i M R
+
M
(M) := {a
i
(M) + i|i ≥ 0}
(R, m) I m
R M R
G
I
(M) =

n≥0
I
n
M/I
n+1
M
M I
(G
I
(M))
M I
(G

I
(M))
D(M)
M m
m I
G
I
(M)
I M
I M
R Z R =

i∈Z
R
i
R
i
R
j
⊆ R
i+j
∀i, j ∈ Z R
i
= 0
i < 0 R N
M Z R Z M =

i∈Z
M
i

R
i
M
j
⊆ M
i+j
i, j ∈ Z M
R x R
i
M
i
i deg(x) = i
0 a ∈ R x ∈ M
deg(ax) = deg(a) + deg(x), ax = 0.
R
0
R
M
i
R
0
x ∈ M
x = x
i
+ x
i+1
+ + x
j
x
k

∈ M
k
, i ≤ k ≤ j, i, j ∈ Z
x
k
x
k
= 0
k x
S R
R a
1
, , a
n
∈ R S[a
1
, , a
n
]
S a
p
1
1
, , a
p
n
n
(p
1
, , p

n
) ∈ N
n
R
a
1
, , a
n
a
1
, , a
n
∈ R R = S[a
1
, , a
n
] R
R =

i≥0
R
i
R
0
R = R
0
[R
1
].
n

R = K[x
1
, x
2
, , x
n
]
R
t
t
R =

t≥0
R
t
t s
t + s K[x
1
, x
2
, , x
n
]
K[x
1
, x
2
, , x
n
] R = R

0
[R
1
]
R
0
= K, R
1
N ⊆ M
N
x ∈ N N
N =

i∈Z
(N ∩ M
i
)
m
I R I
R
I  R I R
∀a, b ∈ R ab ∈ I a ∈ I b ∈
√
I
∀a, b ∈ R ab ∈ I a ∈ I ∃n ∈ N b
n
∈ I
R
I R P :=
√

I
R I P
P I R
R I P
I R
√
I = m
R I R
I m R
I ⊆
√
I = m ⊂ R I
R a, b ∈ R ab ∈ I b ∈
√
I
√
I = m b ∈ m m + Rb = R
√
I +
√
Rb = R
√
I = m,
√
Rb ⊇ Rb
√
I +
√
Rb ⊇ m + Rb
I + Rb = R

√
I +
√
J = R I + J = R
∃d ∈ I, c ∈ r d+cb = 1 a = a.1 = a(d+cb) = ad+c(ab) ∈
I d, ab ∈ I) I m
m
n
(n ∈ N) m
m
R
p
0
⊃ p
1
⊃ ⊃ p
n
n
p ∈ R
p
0
= p p ht(p)
ht(p) = { p
0
= p}.
I R I
ht(I) = {ht(p) |p ∈ R, p ⊇ I}.
R
R R
M R R/

R
M
M M M ≤ R
K = 0 K
Z = 1 Z
pZ p 0 pZ
p Z = 1
0 ⊂ (x
1
) ⊂ (x
1
, x
2
) ⊂ ⊂ (x
1
, , x
n
)
K[x
1
, x
2
, , x
n
] ≥ n.
K[x
1
, x
2
, , x

n
] = n.
R = K[x
1
, x
2
, , x
n
, ]. R =
∞
0 ⊂ (x
1
)  (x
1
, x
2
)   (x
1
, , x
n
) 
R
M = {p ∈ R|M
p
= 0} R
M
x ∈ M
R
(x) = {a ∈ R|ax = 0};
R

M = {a ∈ R|aM = 0} = {a ∈ R|ax = 0, ∀x ∈ M}.
R
(x)
R
(M) (x) (M)
M (M) M M
R
M = V (
R
(M)) = {p ∈ R|p ⊇
R
(M)}.
R M R
p R M
0 = x ∈ M
p =
R
(x) = {r ∈ R|rx = 0}
M
R
(M)
(R, m) M
R M = d {x
1
, , x
d
} m
M (M/(x
1
, , x

d
)M) < ∞
q = (x
1
, , x
d
)R
M
(R, m) x
1
, , x
d
M
(M/(x
1
, , x
i
)) = d − i, ∀1 ≤ i ≤ d.
R R[[x]]
∞

i=0
a
i
x
i
= a
0
+ a
1

x + + a
n
x
n
+
a
0
, a
1
, , a
n
, ∈ R R[[x]]
∞

i=0
a
i
x
i
+
∞

i=0
b
i
x
i
=
∞


i=0
(a
i
+ b
i
)x
i

∞

i=0
a
i
x
i

∞

i=0
b
i
x
i

=
∞

i=0
c
k

x
k
,
k ≥ 0
c
k
= a
0
b
k
+ a
1
b
k−1
+ + a
k
b
0
.
R[[x]] 0

∞
i=0
0x
i
0
1 + 0x + 0x
2
+ + 0x
n

+
R[[x
1
, x
2
, , x
n
]]
K[[x
1
, x
2
, , x
n
]] K
(x
1
, x
2
, , x
n
)
K[[x
1
, x
2
, , x
n
]] K[[x
1

, x
2
, , x
n
]] =
n {x
1
, x
2
, , x
n
} K[[x
1
, x
2
, , x
n
]]
q M
d a
1
, a
2
, , a
d
∈ m (M/(a
1
, a
2
, , a

d
)M) < ∞
H
q,M
(n) = (M/q
n
M), n ∈ Z
P
q,M
(n) n  0
P
q,M
(n) = M = d
P
q,M
(n) = e
0
(q, M

d + n
d

+e
1
(q, M

d + n − 1
d − 1

+ +(−1)

d
e
d
(q, M) (∗)
e
0
(q, M), e
1
(q, M), , e
d
(q, M) e
0
(q, M) >
0
a
0
P
q,M
(n) e
0
(q, M) = a
0
d!
e
0
(q, M) P
q,M
(n)
M q
q = m e(q, M) = e

0
(q, M) = e(M)
M
e
i
(q, M) M I
R = k[x
1
, x
2
, , x
n
] 1
gr
m
(R) = R/ m ⊕m/m
2
⊕ m
2
/m
3
⊕
R = k ⊕ R
1
⊕ ⊕ R
n
⊕
H
R
(t) = (k) + (R

1
) + + (R
t
)
= 1 + + + .
H
R
(n) = ≤ n
=

d + n
n

=
(d + n)!
n!d!
=
(n + 1) (n + d)
d!
=
n
d
d!
+ .
H
R
(n) =
e(R)
d!
n

d
+ .
e(R) = 1.
M R
x ∈ R, x = 0 M
m ∈ M, m = 0 xm = 0.
x ∈ R M M = xM x
0 M
{x
1
, , x
t
} R
M M M/(x
1
, , x
t
)M = 0 x
i
0
M/(x
1
, , x
i−1
)M, ∀ i = 1, 2, , t
I ⊆ R x
1
, , x
t
∈ I

{x
1
, , x
t
} M
y ∈ I {x
1
, , x
t
, y} M t
R I ⊆ R
M I
(R, m)
m (m, M) (M)
M
M R
(M) ≤ R/p, ∀p ∈
R
M.
M R
M ≤ M.
M = max{ R/p| p ∈ M}.
M ≤ M.
M R a ∈ R
M (0 :
M
a) < ∞
x ∈ M M x ∈ p, ∀p ∈ M
M m ∈
R

M
a ∈ R M a ∈ p, ∀p ∈
M \ m
R M
M = dim M R
R
M R
dim R/p = d, ∀p ∈
R
(M)
(x
1
, , x
i
) M M/(x
1
, , x
i
)M
M
p
p ∈ M
R = K[x
1
, , x
n
] R R
{x
1
, , x

n
} K[x
1
, , x
n
]
K[x
1
, , x
n
] = n M ≤ M {x
1
, , x
n
}
R = n = R
K[x
1
, , x
n
]
M =
k[x, y, z]
(x) ∩ (x
2
, y
2
, z)
k[x, y, z]
M = {(x), (x, y, z)} m = (x, y, z) ∈ M

M M = 0
M = { R/p | p ∈ M} = 2 M
(R, m) M R
M = d
x = (x
1
, , x
d
) M
q = xR = (x
1
, , x
d
)R M
e
0
(x, M) = e(q, M) ≤ (M/qM).
I
M
(q) = (M/qM) − e(q, M) ≥ 0.
I(M) = I
M
(q) q
M
x = (x
1
, , x
d
) M I
M

(x) = 0
x = (x
1
, , x
d
) M I
M
(x) = 0
I(M) = 0
M I(M) = ∞
M
I(M) < ∞
M
(H
i
m
(M)) < ∞, ∀i = d d = M
M = 1
(H
0
m
(M)) < ∞
R I
R M R
0 :
M
I ⊆ 0 :
M
I
2

⊆ ⊆ 0 :
M
I
n
⊆
M

n∈N
(0 :
M
I
n
)
M Γ
I
(M)
Γ
I
(M)
I M
R f : M −→ N M
R
.
f(Γ
I
(M)) ⊆ Γ
I
(N).
Γ
I

(f) f
∗
f Γ
I
(M)
Γ
I
(M)
Γ
I
(f)
−→ Γ
I
(N).
Γ
I
= Γ
I
(•) : (M
f
−→ N) −→ (Γ
I
(M)
Γ
I
(f)
−→ Γ
I
(N))
Γ

I
(•) R
Γ
I
= Γ
I
(•)
I
M
E
•
: 0 −→ M −→ E
0
d
0
−→ E
1
d
1
−→ −→ E
i
d
i
−→ E
i+1
−→
Γ
I
(E
•

) : 0 −→ Γ
I
(M) −→ Γ
I
(E
0
)
Γ
I
(d
0
)
−→ Γ
I
(E
1
)
Γ
I
(d
1
)
−→
Γ
I
(d
i−1
)
−→ Γ
I

(E
i
)
Γ
I
(d
i
)
−→
Γ
I
(E
i+1
) −→
H
i
(Γ
I
(E
◦
)) = Γ
I
(d
i
)/ Γ
I
(d
i−1
)
i M I

i I Γ
I
I H
i
I
(−)
x
1
, , x
r
∈ I M H
i
I
(M) =
0, ∀i < r
H
i
I
(M) = 0, ∀i < (M)
I R M R
d
H
i
I
(M) = 0, ∀i > d.
R
0 −→ N
f
−→ M
g

−→ P −→ 0
0 −→ H
0
I
(N)
H
0
I
(f)
−→ H
0
I
(M)
H
0
I
(g)
−→ H
0
I
(P )
δ
0
−→ H
1
I
(N)
H
1
I

(f)
−→ H
1
I
(M)
H
1
I
(g)
−→
H
1
I
(P )
δ
1
−→ H
2
I
(N) −→ −→ H
n−1
I
(P )
δ
n−1
−→ H
n
I
(N)
H

n
I
(f)
−→ H
n
I
(M)
H
n
I
(g)
−→
H
n
I
(P ) −→
δ
0
, δ
1
,
M R M
R
M (M)
M
0 −→ M −→ E
0
−→ E
1
−→ −→ E

n
−→ 0.
R
M = ∞
R R
R
M < ∞
p R = Zp
Zp R
0 −→ Zp
i
−→ Zp −→ 0.
i
Zp
Zp = 0 < ∞
R = Zp
M(R) R M(R)
I D(I, .) M(R)
M ∈ M(R)
D(I, M) = D(I, M/L) + (L) L M
D(I, M) ≥ D(I, M/xM) x M I
D(I, M) = e(I, M) M R
e(I, M) M I
D(I, M) D(I, M) ≥ e(I, M)
M
D(I, M)
D(M) := D(m, M)
I R M R
I
R

∗
= G
r
I
(R) := R/I ⊕ I/I
2
⊕ =

∞
n=0
I
n
/I
n+1
(R
∗
)
0
= R/I) R
∗
a ∈ I
n
/I
n+1
, b ∈ I
m
/I
m+1
a.b = ab I
n+m+1

R
∗
= G
r
I
(R) =

∞
n=0
(G
r
I
(R))
n
R I
G
r
I
(M) =

∞
n=0
I
n
M/I
n+1
M R
∗
a ∈ I
m

/I
m+1
, x ∈ I
n
M/I
n+1
M
a.x = ax ∈ I
m+n
M/I
m+n+1
M
M I
R =

n≥0
R
n
R
0
R
+
R M R
i
(M) =

{n | H
i
R
+

(M)
n
= 0} H
i
R
+
(M) = 0
−∞ H
i
R
+
(M) = 0
H
i
R
+
(M) i M R
+
M
(M) := {a
i
(M) + i|i ≥ 0}
(G
I
(M)) M
I
M R
M = d ≥ 1 D(I, M) M I
(G
I

(M)) ≤ D(I, M) − 1 d = 1
(G
I
(M)) ≤ 2
(d−1)!
D(I, M)
3(d−1)!−1
d ≥ 2
R S
S = n M ∈ M(R) dim(M) = d M
I h (I, M)
d = 0 h (I, M) = (M)
d > 0
h (I, M) := e(I, M) +
d−1

i=0

d − 1
i

h (I, Ext
n−i
s
(M, S))
dim Ext
n−i
s
(M, S) ≤ i
A

h (I, M) := h (I, M ⊗
A

A),

A m A
h (I, M)
M I h (m, M) = h (M) h (M)
M
M
h (I, M) = e(I, M) +
d−1

i=0

d − i
i

(H
i
m
(M)).
R E R
D(E) = D(m, E) m := R
+
E
E m
h (I, M) h (M)
d = M ≥ 1
e(M) ≤ e(I, M) ≤ n(I)

d
e(M)
n(I) I n(I)
m
n(I)
⊆ I
M A I
m A
(M/m
m+1
M) ≤ (M/I
m+1
M) ≤ (M/m
n(I)(m+1)
M)
m
n(I)
⊆ I
e(M)
d!
m
d
+
≤
e(I, M)n(I)
d
d!
m
d
+

≤
e(M)n(I)
d
d!
m
d
+
n
e(M) ≤ e(I, M) ≤ n(I)
d
e(M).
h (M) h (I, M)
(R, m) I m
R M R dimM = d
h (M) ≤ h (I, M) ≤ n(I)
d
h (M).
R S
R
d = 0 h (M) = h (I, M) = (M)
d ≥ 1 M
i
:=
n−1
S
(M, S) n = S (M
i
) ≤ i
d
h (M

i
) ≤ h (I, M
i
) ≤ n(I)
d
h (M
i
), i = 0, , d − 1.
h (M) = e(M) +
d−1

i=0

d − 1
i

h (M
i
)
≤ e(I, M) +
d−1

i=0

d − 1
i

h (I, M)
= h (I, M) (1).
e(I, M) ≤ n(I)

d
e(M).
h (I, M) = e(I, M) +
d−1

i=0

d − 1
i

h (I, M
i
)
≤ n(I)
d
e(M) +
d−1

i=0

d − 1
i

n(I)
i
h (M
i
)
≤ n(I)
d

[e(M) +
d−1

i=0

d − 1
i

h (M
i
)]
= n(I)
d
h (M) (2).

G
I
(M) n(I)
(R, m) I m
R M R d = dimM ≥ 1
(G
I
(M)) ≤ n(I)h (M) − 1 d = 1
(G
I
(M)) ≤ 2
(d−1)!
h (M)
3(d−1)!−1
n(I)

3d!−d
− 1 d ≥ 2
h (I, M) = D(I, M)
(G
I
(M)) ≤ h (I, M) − 1 d = 1
(G
I
(M)) ≤ 2
(d−1)!
h (I, M)
3(d−1)!−1
n(I)
3d!−d
− 1 d ≥ 2
(G
I
(M)) ≤ n(I)h (M) − 1 d = 1
(G
I
(M)) ≤ 2
(d−1)!
h (M)
3(d−1)!−1
n(I)
3d!−d
− 1 d ≥ 2
d = 1
M = R I = m
(G

I
(M)) = 0 (n(m) = h (R) = 1).
ρ
M
(I) M I m
H
M
(n) = P
M
(n) n ≥ m
h
G
I
(M)
(n) P
G
I
(M)
(n)
G
I
(M) r := (G
I
(M))
h
G
I
(M)
(n) − P
G

I
(M)
(n) =
d

i=0
(H
G
I
(R)
+
(G
I
(M))).
h
G
I
(M)
(n) = P
G
I
(M)
(n), ∀ n ≥ r + 1.
H
M
(n) = (M/I
n+1
M) =
r


i=0
h
G
I
(M)
(i) +
n

i=r+1
h
G
I
(M)
(i)
n ≥ r + 1
H
M
(n) = P
M
(n), ∀ n ≥ r.
ρ
M
(I) ≤ (G
I
(M))
M R d = M ≥ 1
ρ
M
(I) ≤ n(I)h (m) −1 d = 1
ρ

M
(I) ≤ 2
(d−1)!
h (M)
3(d−1)!−1
n(I)
3d!−d
− 1 d ≥ 2
ρ
R
(I) n(I) R 0
R
I
(R) R I
(G
I
(R)) = (R
I
(R) R
I
(R) = R[T ]/J
R[T ] J R[T ]
(I) I
J (I) (G
I
(R))