Bivariant Hermitian K-theory and Karoubi's fundamental theorem

Abstract

Let ℓ be a commutative ring with involution ⁎ containing an element λ such that λ+λ⁎=1 and let Algℓ⁎ be the category of ℓ-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category kkh and a functor jh:Algℓ⁎→kkh 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 kkh. By the universal property of the latter, this implies that any functor H:Algℓ⁎→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.