A Transfer morphism for Hermitian K-theory of schemes with involution

Abstract

In this paper, we consider the Hermitian $K$-theory of schemes with involution, for which we construct a transfer morphism and prove a version of the dévissage theorem. This theorem is then used to compute the Hermitian $K$-theory of ${\mathbb{P}}^1$ with involution given by $[X:Y]\mapsto [Y:X]$. We also prove the $C_2$-equivariant $\A^1$-invariance of Hermitian $K$-theory, which confirms the representability of Hermitian $K$-theory in the $C_2$-equivariant motivic homotopy category of Heller, Krishna and \{O}stvær \cite{HKO14}.

Final version. The title has been changed