The Chow ring for the classifying space of $GO(2n)$

TL;DR

This paper computes the Chow ring of the classifying space of the general orthogonal group scheme $GO(2n)$ over fields of characteristic not 2, extending known cohomology results to algebraic intersection theory.

Contribution

It provides an explicit description of the Chow ring $A^*_{GO(2n)}$ in terms of generators and relations, generalizing previous topological cohomology results.

Findings

Chow ring $A^*_{GO(2n)}$ computed explicitly

Generators and relations identified for the Chow ring

Extension of cohomology results to algebraic intersection theory

Abstract

Let $GO(2n)$ be the general orthogonal group scheme (the group of orthogonal similitudes). In the topological category, Y. Holla and N. Nitsure determined the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$ of the classifying space $BGO(2n,\mathbb C)$ of the corresponding complex Lie group $GO(2n,\mathbb C)$ in terms of explicit generators and relations. The author of the present note showed that over any algebraically closed field of characteristic not equal to $2$, the smooth-\'etale cohomology ring $H_{\rm sm-\'et}^*(BGO(2n),\mathbb F_2)$ of the classifying algebraic stack $BGO(2n)$ has the same description in terms of generators and relations as the singular cohomology ring $H^*_{\rm sing}(BGO(2n,\mathbb C),\mathbb F_2)$. Totaro defined for any reductive group $G$ over a field, the Chow ring $A^*_G$, which is canonically identified with the ring of characteristic classes in the sense of intersection theory, for principal $G$-bundles, locally trivial in \'etale topology. In this paper, we calculate the Chow group $A^*_{GO(2n)}$ over any field of characteristic different from $2$ in terms of generators and relations.