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

TL;DR

This paper develops a geometric model for classifying spaces of automorphisms of Hermitian vector bundles over rings with involution, proves a periodicity theorem, and constructs a motivic ring spectrum representing homotopy Hermitian K-theory, establishing cdh descent.

Contribution

It introduces a geometric model for automorphism classifying spaces, proves periodicity, and constructs a motivic spectrum for homotopy Hermitian K-theory, extending previous results to rings with involution.

Findings

Constructed a geometric model for classifying spaces over rings with involution.

Proved a periodicity theorem for Hermitian K-theory.

Established cdh descent for homotopy Hermitian K-theory.

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.