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

We prove that, given a sufficiently functorial assignment from rings to big Cohen-Macaulay algebras R ↦ B, that the associated big Cohen-Macaulay closure operation on ideals I ↦ I B ∩ R necessarily satisfies the Brian{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.