Closure-theoretic proofs of uniform bounds on symbolic powers in regular rings

TL;DR

This paper presents concise, closure-theoretic proofs for uniform bounds on the growth of symbolic powers of ideals in regular rings across all characteristics, simplifying previous complex methods.

Contribution

It introduces a unified closure-theoretic approach using Heitmann's epf closure and related techniques to establish bounds on symbolic powers in regular rings, extending previous results.

Findings

Provides new proofs of uniform bounds in mixed and equal characteristic cases.

Applies to any Dietz closure satisfying specific axioms and the Briançon-Skoda theorem.

Simplifies existing proofs using closure-theoretic methods.

Abstract

We give short, closure-theoretic proofs for uniform bounds on the growth of symbolic powers of ideals in regular rings. The author recently proved these bounds in mixed characteristic using various versions of perfectoid/big Cohen-Macaulay test ideals, with special cases obtained earlier by Ma and Schwede. In mixed characteristic, we instead use Heitmann's full extended plus (epf) closure, Jiang's weak epf (wepf) closure, and R.G.'s results on closure operations that induce big Cohen-Macaulay algebras. Our strategy also applies to any Dietz closure satisfying R.G.'s algebra axiom and a Brian\c{c}on-Skoda-type theorem, and hence yields new proofs of these results on uniform bounds on the growth of symbolic powers of ideals in regular rings of all characteristics. In equal characteristic, these results on symbolic powers are due to Ein-Lazarsfeld-Smith, Hochster-Huneke, Takagi-Yoshida, and Johnson.