Binomial expansion for saturated and symbolic powers of sums of ideals

TL;DR

This paper introduces a binomial expansion formula for saturated powers of sums of ideals, unifying various results and criteria related to symbolic powers in algebraic geometry and commutative algebra.

Contribution

It provides a novel binomial expansion formula for saturated powers of sums of ideals, offering a unified approach to existing and new results on symbolic powers.

Findings

Derived binomial expansion formulas for saturated powers

Computed depth and regularity of symbolic powers

Established criteria for equality of ordinary and symbolic powers

Abstract

There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers.