Bivariant algebraic K-theory

Abstract

These notes aim to serve as an introduction to algebraic kk-theory, defined by Cortiñas–Thom (2007) as an algebraic analogue of Kasparov’s KK-theory. After discussing its universal properties, we introduce $kk$ as a localisation of the category of algebras and describe its triangulated structure. We then introduce the $G$-equivariant version of kk-theory and discuss the adjointness theorems and the relation between kk-theory and the isomorphism conjecture for Weibel’s homotopy K-theory. Finally, we look at kk-theory from a homotopical point of view and prove that $kk$ is the homotopy category of a stable $\infty$-category.