Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution

Abstract

We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution $R$ such that $\frac{1}{2}\in R$; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_{\infty }$ motivic ring spectrum ${\mathbf{KR}}^{\mathrm{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that ${\mathbf{KR}}^{\mathrm{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence.

34 pages: Final version - removed unnecessary hypothesis in theorem 5.1