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

TL;DR

This paper develops a new triangulated category framework for bivariant Hermitian K-theory over rings with involution, establishing a universal property and proving a version of Karoubi's fundamental theorem and exact sequence.

Contribution

It constructs a universal triangulated category for Hermitian K-theory and proves fundamental theorems extending Karoubi's work to this setting.

Findings

Established a triangulated category $kk^h$ with universal properties.

Proved a version of Karoubi's fundamental theorem in $kk^h$.

Derived a bivariant 12-term exact sequence for Hermitian K-theory.

Abstract

Let $\ell$ be a commutative ring with involution $*$ containing an element $\lambda$ such that $\lambda+\lambda^*=1$ and let $\operatorname{Alg}^*_\ell$ be the category of $\ell$-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category $kk^h$ and a functor $j^h:\operatorname{Alg}^*_\ell\to kk^h$ 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 $kk^h$. By the universal property of the latter, this implies that any functor $H:\operatorname{Alg}^*_\ell\to\mathfrak{T}$ 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.