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

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 $H^{*}_{\textrm{sing}} (BGO(2n, \mathbb{C}), \mathbb{F}_2)$ 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-étale cohomology ring $H^{*}_{\textrm{sm-ét}} (BGO(2n), \mathbb{F}_2)$ of the classifying algebraic stack $BGO(2n)$ has the same description in terms of generators and relations as the singular cohomology ring $ H^{*}_{\textrm{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 étale 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.

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Accepted 1 September 2014

Published 24 July 2015