UBC Algebra and Algebraic Geometry Seminar: Peter McDonald

The Briançon-Skoda theorem is a comparison relating the integral closure of powers of a finitely generated ideal with its ordinary powers. Originally proved using analytic methods for coordinate rings of smooth varieties over the complex numbers in 1974, it took until 1981 for Lipman and Sathaye to provide an algebraic proof for all regular local rings, regardless of characteristic. Since then, there have been other proofs and generalizations to mild singularities, most notably using tight closure theory in positive characteristic and reduction mod p. In this talk, we prove a general Briançon-Skoda containment for pseudo-rational singularities in all characteristics. Our method is quite simple, and it recovers and unifies many previously known results while also extending them to mixed characteristic. It also yields some new results on F-pure and Du Bois singularities (as well as a characteristic-free analog) and settles a conjecture of Huneke on uniform bounds regardless of the singularities of the ring. This is based on joint work with Linquan Ma, Rebecca R.G., and Karl Schwede.