The Briançon--Skoda property for singular rings via closure operations

Abstract: 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.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca