DMS Algebra Seminar

NOTE: NEW TIME, NEW PLACE

Speaker: Ben Briggs (MSRI)

Title: Cohomological jump loci for commutative rings

Abstract: Inspired by Quillen’s use of cohomology to study the representation theory of finite groups geometrically, Avramov and Buchweitz established some remarkable facts about modules over complete intersection rings using "cohomological support varieties." The utility and scope these varieties have continued to grow since - for example, Pollitz used (a generalization of) them to characterise complete intersection rings purely in terms the structure of their derived categories. More recently, in joint work with McCormick and Pollitz, we enrich the support varieties to a nested sequence of varieties called "cohomological jump loci," which can detect much finer information than the support varieties alone. We use this to enhance known symmetries in the homological algebra of complete intersection rings. To be precise: let M be a finitely generated Maximal Cohen-Macaulay module over a local complete intersection ring. The sequence of Betti numbers of M grows like a polynomial, and the same goes for the dual module M\(^*\). Avramov and Buchweitz showed that these polynomials have the same degree for M and M\(^*\). We use jump loci to prove that they even have the same leading term. I will (try to) introduce all of this, survey the history a bit, and explain why it's very cool.