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 M. Schlichting and G. S. Tripathi [Math. Ann. 362, No. 3–4, 1143–1167 (2015; Zbl 1331.14028)]. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_{\infty }$ motivic ring spectrum ${\mathbf{KR}}^{\text{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that ${\mathbf{KR}}^{\text{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence.