The Brian\c{c}on-Skoda Theorem via weak functoriality of big Cohen-Macaulay algebras

TL;DR

This paper demonstrates that a functorial approach to big Cohen-Macaulay algebras ensures the Briançon-Skoda property for ideal closures, connecting algebraic functoriality with ideal-theoretic properties.

Contribution

It establishes that functorial big Cohen-Macaulay algebra assignments imply the Briançon-Skoda property for ideal closures, extending previous results in mixed characteristic.

Findings

Functorial big Cohen-Macaulay algebras imply Briançon-Skoda property

Connects functoriality with ideal closure properties

Recovers a strengthened result of Heitmann in mixed characteristic

Abstract

We prove that, given a sufficiently functorial assignment from rings to big Cohen-Macaulay algebras $R \mapsto B$, that the associated big Cohen-Macaulay closure operation on ideals $I \mapsto I B \cap R$ necessarily satisfies the Brian\c{c}on-Skoda type property. The proof combines arguments of Lipman-Teissier, Hochster, Ma, and Hochster-Huneke. Specializing to mixed characteristic, and utilizing a result of Bhatt on absolute integral closures, this recovers a slight strengthening of a result of Heitmann.