Cdh Descent in Equivariant Homotopy $K$-Theory

Abstract

We construct geometric models for classifying spaces of linear algebraic groups in $G$-equivariant motivic homotopy theory, where $G$ is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy $K$-theory of $G$-schemes (which we construct as an $E_{\infty }$-ring) is stable under arbitrary base change, and we deduce that the homotopy $K$-theory of $G$-schemes satisfies cdh descent.