April 15 : Julia Pevtsova

Title: Support theory for finite dimensional Hopf (super)algebras or how to sniff out projective modules in the wild.

Abstract:

Singularity categories for non-semisimple finite dimensional Hopf algebras are usually “wild”, that is, classifying indecomposable modules is a fairly hopeless task. Supports, the geometric invariants associated to representations, allow us to bring at least some structure to this wild territory. In particular, they lead to the computation of the spectrum of the singularity category, in the sense of P. Balmer. To construct a useful support theory one needs to effectively detect vanishing of objects as well as tensor nilpotence of maps. In the representation theoretic context these problems translate into detection of projectivity of modules and nilpotence of cohomology classes.

I’ll review the classical detection results in the subject due to D. Quillen, A. Suslin and others and discuss some recent progress for the singularity category of a finite supergroup scheme. Based on joint work with D. Benson, S. Iyengar and H. Krause.