Một số vấn đề về luỹ thừa của các iđêan đơn thức

R
a
i
{Ass(R/I
n
)}
n∈N
n(I) {Ass(R/I
n
)}
n∈N
n(I)
I
2002
I
n(I)
I
{Ass(R/I
n
)}
n∈N
{Ass(R/I
n
)}
n∈N
n
0
{Ass(R/I
n

)}
n∈N
n(I)
n(I) n(I)
I
n(I) I
1.3.4
1.4.1
2.1.2
I
M R
reg(I
n
M)
n n I
n
I
n
1
+ I
n
2
3.5.8 3.6.1
I, I
1
, . . . , I
p
R
reg(I +I
n

1
+···+I
n
p
) n
I
1
, . . . , I
p
I
dim R/I + I
n
1
+ ··· + I
n
p
 1
reg(I + I
n
1
+ ··· + I
n
p
)
I, I
1
, . . . , I
p
reg(I + I
n

1
+ ··· + I
n
p
)
n 3.5.8
n 3.3.4 2.2.1
a
i
3.3.2
2.1.4 ri(I
n
M)
reg(I
n
M) + 1
I R M
R
ri(I
n
) n n
reg(I
n
)
ri(I)
2.3.1
ri(I
n
M)
I

1 n(I)
1.1
{Ass(R/I
n
)}
n∈N
{Ass(R/I
n
)}
n∈N
1.2
n(I)
1.3 1.3.4
1.4
1.4.1
2
a
i
2.1 2.2
2.2.2 2.3
2.3.1
3
a
i
3.1
3.2
a
i
3.3 3.3.2
3.3.4

3.4
3.5
reg(I
n
1
+ ···+ I
n
p
) dim R/I
1
+ ···+ I
p
= 0
3.5.4
1
reg(I
n
1
+ ···+ I
n
p
) 3.6.1
3.6
R I
R
I I
x ∈ R
x
n
+ a

1
x
n−1
+ ···+ a
n−1
x + a
n
= 0,
a
i
∈ I
i
i = 1, . . . , n
6.5 ht(I)  1 {Ass(R/I
n
)}
n∈N
n
{Ass(R/I
n
)}
n∈N
n(I) = min{t| Ass(R/I
n
) = Ass(R/I
t
) n  t}.
{Ass(R/I
n
)}

n∈N
16.3 {Ass(R/I
n
)}
n∈N
R = K[X
1
, . . . , X
r
]
K R m = (X
1
, . . . , X
r
)
R
+
I R
n(I) NP (I)
I
I R
A ⊆ R A E(A) := {α| X
α
∈ A} ⊆ Z
r
.
I NP (I) := conv{E(I)}
I R
r
I

I R I
I E(I) = NP (I) ∩ Z
r
E(I) = {α| n  1 nα ∈ E(I
n
)}.
NP (I)
{x ∈ R
r
| a
i
, x  b
i
, i = 1, . . . , q},
a
i
, x = b
i
NP (I) t
i
E(G(I))
r −t
i
0 = a
i
∈ R
r
+
b
i

∈ R
+
i = 1, . . . , q t
i
a
i
R/I
n
R/I
16 r  2 Ass(R/I
n
) = Ass(R/I)
n  1
I R i = 1, . . . , r
I[i] = (X
α
[i]| X
α
∈ I).
I n  1
Ass(R/I
n
) \{m} =
r

i=1
Ass(R/I[i]
n
).
m ∈ Ass(R/I

n
)
I R NP (I)
1.2.7
I m ∈ Ass(R/I
n
)
n  1 a
i
> 0 1  i  q
I = (X
1
X
2
, X
1
X
3
, X
2
X
3
) ⊂ R = K[X
1
, X
2
, X
3
]
1.3.1 m /∈ Ass(R/I)

I R d(I)
G(I) δ = r2
r−1
[d(I)]
r−2
.
i a
i
> 0 m ∈ Ass(R/I
n
) n  δ.
1.2.8 1.2.12 1.3.1 1.3.2
r
I R
n
0
:=



1 r  2,
r2
r−1
[d(I)]
r−2
r > 2.
n  n
0
Ass(R/I
n

) = Ass(R/I
n
0
).
I R 1.3.4
n(I)  r2
r−1
d(I)
r−2
r
d, r r  4 d > r − 3
u = X
(
r−3
0
)
1
X
(
r−3
1
)
2
···X
(
r−3
r−4
)
r−3
v = X

β
0
1
X
β
1
2
···X
β
r−4
r−3
X
d−r+2
r−2
,
β
i
=



0 r − 3 −i ,
2

r−3
i

r − 3 −i .
I = (uX
d

1
, uX
d−1
2
X
r
, . . . , uX
d−r+3
r−2
X
r−3
r
, uX
r−1
X
d−1
r
, vX
r−3
r
)
r d + 2
r−3
− 1
δ :=
d(d −1) ···(d − r + 3)
(r − 3)!
.
n(I)  δ
r  4 d  r 1.4.1

n(I) = O([d(I)]
r−2
) n(I) 1.3.4
n(I)
A =

m0
A
m
A
0
A = A
0
[A
1
] M =

m∈Z
M
m
A dim(M) = d
A
+
=

m>0
A
m
H
i

A
+
(M)
i M A
+
H
i
A
+
(M) A
i  0 A
M
a
i
(M) :=



max{m| H
i
A
+
(M)
m
= 0} H
i
A
+
(M) = 0,
−∞ H

i
A
+
(M) = 0.
M
reg(M) = max{a
i
(M) + i| i  0}.
2005
reg(I
n
M) n n I
A
M H
M
: Z −→ Z
H
M
(m) := 
A
0
(M
m
).
P
M
(x) d −1
M H
M
(m) = P

M
(m) m
A
M
ri(M) := min{m
0
| H
M
(m) = P
M
(m) ∀m  m
0
}.
M
ri(M)  reg(M) + 1,
M
(2.2) ri(I
n
M)  reg(I
n
M) + 1
ri(I
n
M)
n
ri(I
n
M) n n
I
R(I) I

2.3.1
2.2.2
3
−∞
f : N −→ Q ∪ {−∞}
N
a
0
, a
1
, . . . , a
N−1
, b
0
, b
1
, . . . , b
N−1
0  i  N −1
n n ≡ i mod N, 0  i  N − 1
f(n) = a
i
n + b
i
f(n) = −∞ f
−∞ f
i
(n) = a
i
n+b

i
, i =
0, . . . , N − 1 f
P
0
(x), P
1
(x), . . . ∈ Q[x]
α
1
, . . . , α
s
, β
1
, . . . , β
s
, n
1
, . . . , n
s
α
i
+ β
i
= 0 i =
1, . . . , s P (x, y) ∈ Q[x, y]

n0
P
n

(x)y
n
=
P (x, y)
(1 −x
α
1
y
β
1
)
n
1
···(1 − x
α
s
y
β
s
)
n
s
,
deg P
n
(x) n n
β
1
= ··· = β
s

= 1
deg P
n
(x) n n
α
i
ri(I
n
M)
I
A A
0
M A
I
n
M = 0 n  1 ri(I
n
M)
n n
I
2.3.1
N
r
2.2.2
A = K[X
1
, . . . , X
r
] K
I A f

1
, . . . , f
s
I d
i
= deg f
i
i = 1, . . . , s
ri(I
n
) n n  0
ri(I
n
) n n  0
d
1
, . . . , d
s
3.5.8
[ ] a
i
3.3.2 3.3.4
reg(I
n
1
+ . . . + I
n
p
)
R = K[X

1
, . . . , X
r
] r K
m = (X
1
, . . . , X
r
) R
3.3.1
p i = 1, . . . , p
I
i
, I
i1
, . . . , I
iq
i
, J
i1
, . . . , J
iq
i
R
n  0
M
n
=
p


i=1


I
i
∩
q
i

j=1
J
ij
I
n
ij


.
α
1
, . . . , α
s
, β
1
, . . . , β
s
α
i
+ β
i

=
0, ∀i = 1, . . . , s P (x, y) ∈ Z[x, y]

a,n0
dim
K
[M
n
]
a
x
a
y
n
=
P (x, y)
(1 −x
α
1
y
β
1
) ···(1 − x
α
s
y
β
s
)
α = (α

1
, . . . , α
r
) ∈ Z
r
F ⊆ {1, . . . , r}
X
α
[F ] =

i/∈F
X
α
i
i
.
1.2.9 I
R
I[F ] = (X
β
[F ]| X
β
∈ I).
∆(I)
√
I
i  0 I
B
i
(I) ={(F, Γ)| F ∈ ∆(I) Γ {1, . . . , r} \ F


H
i−|F |−1
(Γ; K) = 0}.
B
i
(I)
(F, Γ) ∈ B
i
(I) {j
1
, . . . , j
s
} = {1, . . . , r} \ F
S = K[X
j
1
, . . . , X
j
s
].
F
1
, . . . , F
p
Γ H
1
, . . . , H
q
Γ I

J = I[F ] ∩S S
M(F, Γ, I) =


q

j=1
J[H
j
] +
p

i=1
J[F
i
]


/
p

i=1
J[F
i
].
a(M(F, Γ, I)) := max{|β|| M(F, Γ, I)
β
= 0}.
1.1
i  0 I ⊆ R

a
i
(R/I) = max{a(M(F, Γ, I)) − |F | | (F, Γ) ∈ B
i
(I)}.
a
i
(R/I + I
n
1
+ ···+ I
n
p
)
I, I
1
, . . . , I
p
R =
K[X
1
, . . . , X
r
] K n  0 L
n
=
I + I
n
1
+ ··· + I

n
p
i  0 (F, Γ) ∈ B
i
(L
1
)
S = K[X
i
| i ∈ {1, . . . , r}\F ] P (x, y) ∈ Z[x, y]
α
1
, . . . , α
t
, β
1
, . . . , β
t
 0 α
i
+ β
i
= 0

a,n0
dim
K
[M(F, Γ, L
n
)]

a
x
a
y
n
=
P (x, y)
(1 −x
α
1
y
β
1
) ···(1 − x
α
t
y
β
t
)
,
M(F, Γ, L
n
) S (3.2)
3.2.4 3.3.1 2.2.2
I, I
1
, . . . , I
p
R = K[X

1
, . . . , X
r
] K i  0 d =
dim(R/I + I
1
+ ···+ I
p
)
a
i
(R/I + I
n
1
+ ···+ I
n
p
) n n
lim
n→∞
a
d
(R/I + I
n
1
+ ···+ I
n
p
)
n

I, I
1
, . . . , I
p
R = K[X
1
, . . . , X
r
] K reg(I + I
n
1
+ ··· + I
n
p
)
n n
3.3.2 lim
n→∞
a
d
(R/I + I
n
1
+ ···+ I
n
p
)
n
d = 0
I

1
, . . . , I
p
R I
1
+ ···+ I
p
m
NP (I
1
, . . . , I
p
) =
p

i=1
NP (I
i
)
I
1
, . . . , I
p
R
r
+
d(x, y) =
|x − y| = |x
1
− y

1
| + ··· + |x
r
− y
r
| x = (x
1
, . . . , x
r
), y =
(y
1
, . . . , y
r
) ∈ R
r
∂
o
(NP (I
1
, . . . , I
p
)) NP (I
1
, . . . , I
p
)
R
r
+

Γ(I
1
, . . . , I
p
) = ∂
o
(NP (I
1
, . . . , I
p
))
I
1
, . . . , I
p
Γ(I
1
, . . . , I
p
) R
r
I
1
, . . . , I
p
R I
1
+ ··· + I
p
m n

J
n
= I
n
1
+ ···+ I
n
p
.
R/J
n
reg(J
n
) = reg(R/J
n
) + 1 = a
0
(R/J
n
) + 1.
3.3.2 lim
n→∞
reg(R/J
n
)
n
∈ Q
4 1
m(I
1

, . . . , I
p
) := max{|z| | z ∈ Γ(I
1
, . . . , I
p
)}.
m(I
1
, . . . , I
p
)
I
1
, . . . , I
p
R
I
1
+ ···+ I
p
m
lim
n→∞
reg(I
n
1
+ ···+ I
n
p

)
n
= m(I
1
, . . . , I
p
).
3.5.4 reg(J
n
)
n n
I
1
= (x, y
2
) I
2
= (x
2
, y)
R = K[x, y]
lim
n→∞
reg(I
n
1
+ I
n
2
)

n
= 4/3.
reg(I
n
1
+ I
n
2
) n n  0
reg(J
n
) =

4
3
n −
1
3

+ 1 n  1.
3.5.4 lim
n→∞
reg(J
n
)
n
= m(I
1
, . . . , I
p

) ∈ Q
m(I
1
, . . . , I
p
)  1 3.5.8
m(I
1
, . . . , I
p
)
1
s  1
R = K[t
1
, . . . , t
r
] I
1
, . . . , I
p
⊂ R
lim
n→∞
reg(J
n
)
n
= s.
Ass(R/I

n
) I
R
ri(I
n
M) n n
reg(I + I
n
1
+ ···+ I
n
p
)
I, I
1
, . . . , I
p
I
1
+ ···+ I
p
lim
n→∞
reg(I
n
1
+···+I
n
p
)

n
Ass(M/I
n
M)
4
a
i