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.
Categories: math.KT math.AG math.AT