SFU Number Theory and Algebraic Geometry Seminar: Peter McDonald

In 1974, Briançon and Skoda answered a question of Mather, showing that for $I=(f_1,\dots,f_n)$ an ideal of the coordinate ring at a smooth point on a complex algebraic variety, there is a containment $\overline{I^{n+k-1}}\subseteq I^k$ for all $k\geq1$. To the dismay of algebraists, this was achieved using analytic techniques, leading Lipman and Sathaye in 1981 to supply an algebraic proof to give a similar bound for ideals in regular rings in all characteristics. Generally, this containment fails for singular rings, though work of many people have given results for singular rings in various settings. In this talk, I'll discuss recent joint work with Neil Epstein, Rebecca RG, and Karl Schwede where we give a characteristic-free proof of the desired containment for a large class of singular rings, implying many of the previously-known Brian\c{c}on--Skoda type results.