The Briançon-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 $R\mapsto B$, that the associated big Cohen-Macaulay closure operation on ideals $I\mapsto IB\cap R$ necessarily satisfies the Briançon-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.