Integral closures of powers of sums of ideals
Let k be a field, let A and B be polynomial rings over k , and let S = A ⊗ k B . Let I ⊆ A and J ⊆ B be monomial ideals. We establish a binomial expansion for rational powers of I + J ⊆ S in terms of those of I and J . Particularly, for u ∈ Q + , we prove that ( I + J ) u = ∑ 0 ≤ ω ≤ u , ω ∈ Q I ω J...
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| Published in | Journal of algebraic combinatorics Vol. 58; no. 1; pp. 307 - 323 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.08.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Summary: | Let
k
be a field, let
A
and
B
be polynomial rings over
k
, and let
S
=
A
⊗
k
B
. Let
I
⊆
A
and
J
⊆
B
be monomial ideals. We establish a binomial expansion for rational powers of
I
+
J
⊆
S
in terms of those of
I
and
J
. Particularly, for
u
∈
Q
+
, we prove that
(
I
+
J
)
u
=
∑
0
≤
ω
≤
u
,
ω
∈
Q
I
ω
J
u
-
ω
,
and that the sum on the right-hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of
I
and
J
. We further give sufficient conditions for this formula to hold for the integral closures of powers of
I
+
J
in terms of those of
I
and
J
. Under these conditions, we provide explicit formulas for the depth and regularity of
(
I
+
J
)
k
¯
in terms of those of powers of
I
and
J
. |
|---|---|
| ISSN: | 0925-9899 1572-9192 |
| DOI: | 10.1007/s10801-023-01252-4 |