Bivariant Hermitian $K$-theory and Karoubi's fundamental theorem

Abstract

Let $\ell$ be a commutative ring with involution $\ast$ containing an element $\lambda$ such that $\lambda +{\lambda }^{\ast }=1$ and let ${\mathrm{Alg}}_{\ell }^{\ast }$ be the category of $\ell$-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category $k{k}^h$ and a functor $j^h:{\mathrm{Alg}}_{\ell }^{\ast }\to k{k}^h$ that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in $k{k}^h$. By the universal property of the latter, this implies that any functor $H:{\mathrm{Alg}}_{\ell }^{\ast }\to \mathfrak{T}$ with values in a triangulated category which is homotopy invariant, matricially and hermitian stable and excisive satisfies the fundamental theorem. We also prove a bivariant version of Karoubi's $12$-term exact sequence.

27 pages. Various minor, mostly expository changes in second version