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

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 $H_{\mathrm{sing}}^{\ast }(BGO(2n,\mathbb{C}),{\mathbb{F}}_2)$ 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-étale cohomology ring $H_{\mathrm{sm}-\acute{\mathrm{e}}\mathrm{t}}^{\ast }(BGO(2n),{\mathbb{F}}_2)$ of the classifying algebraic stack $BGO(2n)$ has the same description in terms of generators and relations as the singular cohomology ring $H_{\mathrm{sing}}^{\ast }(BGO(2n,\mathbb{C}),{\mathbb{F}}_2)$. Totaro defined for any reductive group $G$ over a field, the Chow ring $A_G^{\ast }$, which is canonically identified with the ring of characteristic classes in the sense of intersection theory, for principal $G$-bundles, locally trivial in étale topology. In this paper, we calculate the Chow group $A_{GO(2n)}^{\ast }$ over any field of characteristic different from $2$ in terms of generators and relations.

arXiv comment11 pages