Symbolic powers via extension

TL;DR

This paper explores conditions under which symbolic powers commute with extension of ideals, generalizing known cases, and applies these results to compute resurgence for sums of homogeneous ideals in algebraic domains.

Contribution

It generalizes existing results on symbolic powers and provides formulas for resurgence of sums of homogeneous ideals in finitely generated algebra domains.

Findings

Established conditions for symbolic powers to commute with extension

Generalized known results to broader algebraic settings

Derived formulas for resurgence of sums of homogeneous ideals

Abstract

This article investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the resurgence of sum of two homogeneous ideals in finitely generated k-algebra domains, where k is algebraically closed. Initially, these were known for ideals in polynomial rings.