Integral closures of powers of sums of ideals

TL;DR

This paper develops a binomial expansion formula for rational powers of sums of monomial ideals in polynomial rings and provides explicit formulas for their integral closures, depth, and regularity.

Contribution

It introduces a binomial expansion for rational powers of sums of ideals and establishes conditions for similar formulas for their integral closures, with explicit depth and regularity calculations.

Findings

Established a binomial expansion for rational powers of sums of ideals.

Derived conditions under which the expansion applies to integral closures.

Provided explicit formulas for depth and regularity of integral closures.

Abstract

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S= A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J \subseteq S$ in terms of those of $I$ and $J$. Particularly, for a positive rational number $u$, we prove that $(I+J)_u = \sum_{0 \le \omega \le u, \ \omega \in \mathbb{Q}} I_\omega J_{u-\omega},$ 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 $\overline{(I+J)^k}$ in terms of those of powers of $I$ and $J$.