Algebraic Kasparov K-theory. I

TL;DR

This paper constructs various bivariant algebraic Kasparov K-theory spectra for k-algebras, demonstrating their homotopy invariance, excisiveness, and universality in representing different types of bivariant homology theories.

Contribution

It introduces unstable, Morita stable, and stable bivariant algebraic Kasparov K-theory spectra and proves their key properties and universality.

Findings

Spectra are homotopy invariant.

Spectra are excisive in each variable.

Spectra represent universal bivariant homology theories.

Abstract

This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.