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

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 a new version 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 Briançon-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.