## EVENT DETAILS AND ABSTRACT

**Geometry/Physics Seminar**
**Title:** Stacky abelianization of a connected algebraic group and Serre duality

**Speaker:** Masoud Kamgarpour

**Speaker Info:** U of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

Let G be a connected algebraic group over an algebraically closed field k. Let A be a discrete group scheme over k. In view of Grothendieck's function-sheaf correspondence, it is clear that the sheaf-theoretic analogue of a homomorphism G--->A is a "multplicative A-torsor" on G. We will provide examples of multplicative torsors L, such the restriction of L to the commutator subgroup [G,G] is non-trivial. This leads us to define a new commutator of G, called the etale commutator. The etale commutator will be a central extension of [G,G] by a finite group scheme. The quotient stack of G by the action of the etale commutator will be called the etale abelianization of G. We will see that the etale abelinaization is the "universal" Deligne-Mumford Picard stack to which G maps. (Compare this to the statement that G^ab is the universal abelian group to which G maps). As an independent application of these ideas, we will generalize Serre duality for commutative perfect unipotent groups to the non-commutative setting.

**Date:** Thursday, May 3, 2007

**Time:** 3:00pm

**Where:** Lunt 107

**Contact Person:** Prof. Kevin Costello

**Contact email:** costello@math.northwestern.edu

**Contact Phone:** 773-910-8788

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