Characteristic Classes for GO(2n) in \'Etale Cohomology

TL;DR

This paper computes the etale cohomology ring with mod 2 coefficients of the classifying stack BGO(2n) over algebraically closed fields, extending topological results to the algebraic setting using advanced cohomological techniques.

Contribution

It extends the known topological cohomology results of GO(2n) to the algebraic category by developing new methods in etale cohomology and algebraic stacks.

Findings

Determined the etale cohomology ring of BGO(2n) with mod 2 coefficients.

Extended Totaro's ideas to etale cohomology for algebraic stacks.

Constructed a Gysin sequence for the G_m fibration in algebraic stacks.

Abstract

Let GO(2n) be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic not equal to 2. We determine the etale cohomology ring with mod 2 coefficients of the algebraic stack BGO(2n). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring of the classifying space BGO(2n) of the complex Lie group GO(2n) in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are (i) an extension to etale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the G_m fibration BO(2n) to BGO(2n) of algebraic stacks.