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Speaker: Julia Pevtsova, (University of Oregan)

Title: Geometry of finite group schemes

Abstract: For the purposes of this talk, and to simplify the title, "finite group scheme" will be an abbreviation for the cocommutative finite dimensional Hopf algebra over a field of positive characteristic $p$. The most familiar example of such is a finite group. I shall give examples of other classes of finite group schemes as well, such as restricted Lie algebras and Frobenius kernels.

I shall explain an attempt to study representation theory of such Hopf algebras via the "local approach". Namely, we determine a family of very simple subalgebras inside our Hopf algebra and study restrictions of representations to these small subalgebras. We construct a geometric invariant of a module which contains some vital information about the original module, the most crucial of which is detection of projectivity of the module. Furthermore, we identify this geometric invariant with the cohomological construction known as support variety.

Even though the results are valid for arbitrary finite group schemes, interesting geometric properties occur already for products of cyclic groups of order $p$. This is what all my examples will be about.

This is joint work with Eric Friedlander.

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