Arithmetic Geometry Seminar - Roy Joshua

Ohio State Garden of Constants
January 14, 2025
3:00PM - 4:00PM
Smith Lab 2180

Date Range
2025-01-14 15:00:00 2025-01-14 16:00:00 Arithmetic Geometry Seminar - Roy Joshua Roy JoshuaThe Ohio State UniversityTitleMotivic Euler characteristic and the transferAbstractThis concerns the following conjecture of Fabien Morel. Given a linear algebraic group G defined over a field k, with a maximal torus T, the conjecture was that a suitable form of the Euler Characteristic of G/N(T), where N(T) is the normalizer of T in G, is 1, when viewed as a class in the Grothendieck-Witt group of the field k. In a paper in 2023, together with P. Pelaez, we settled this conjecture in the affirmative for split groups G, provided the field k has a square root of -1. We also showed that this implies the same Euler characteristic is a unit in general.This has numerous applications to splitting in various forms of Borel-style equivariant cohomology theories, for example, equivariant K-theory, equivariant (higher) Chow groups, equivariant Brauer groups etc., and greatly facilitates the computation of all such equivariant cohomology theories.The talk will sketch a proof of the conjecture and outline some of the applications. Smith Lab 2180 America/New_York public

Roy Joshua
The Ohio State University

Title
Motivic Euler characteristic and the transfer

Abstract
This concerns the following conjecture of Fabien Morel. Given a linear algebraic group G defined over a field k, with a maximal torus T, the conjecture was that a suitable form of the Euler Characteristic of G/N(T), where N(T) is the normalizer of T in G, is 1, when viewed as a class in the Grothendieck-Witt group of the field k. In a paper in 2023, together with P. Pelaez, we settled this conjecture in the affirmative for split groups G, provided the field k has a square root of -1. We also showed that this implies the same Euler characteristic is a unit in general.

This has numerous applications to splitting in various forms of Borel-style equivariant cohomology theories, for example, equivariant K-theory, equivariant (higher) Chow groups, equivariant Brauer groups etc., and greatly facilitates the computation of all such equivariant cohomology theories.

The talk will sketch a proof of the conjecture and outline some of the applications.

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