A kind of $KK$-theory for rings

TL;DR

This paper introduces a new $KK$-theory for rings, extending classical $KK$-theory for $C^*$-algebras by incorporating homotopy invariance, matrix invertibility, and split exact sequences, with applications like Morita equivalence.

Contribution

It develops a novel $KK$-theory framework for rings, paralleling Kasparov's theory, and demonstrates its utility through key isomorphisms and equivalences.

Findings

Proves Morita equivalence induces isomorphisms in the new theory

Establishes a Green-Julg type isomorphism within this framework

Shows the theory linearizes the category of rings with added structure

Abstract

A group equivariant $KK$-theory for rings will be defined and studied in analogy to Kasparov's $KK$-theory for $C^*$-algebras. It is a kind of linearization of the category of rings by allowing addition of homomorphisms, imposing also homotopy invariance, invertibility of matrix corner embeddings, and allowing morphisms which are the opposite split of split exact sequences. We demonstrate the potential of this theory by proving for example equivalence induced by Morita equivalence and a Green-Julg isomorphism in this framework.