The Index Map in Algebraic K-Theory

Abstract

For a ring $R$, we construct a universal $K_R$-torsor ${\mathcal{T}}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group theory has features of $K$-theory (e.g. determinant bundles, tame symbol cocycle for Kac-Moody extension), the $K$-theory torsor relates higher loop groups with higher $K$-theory. We study the classifying "index" map of this torsor in detail. We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the $K$-theory torsor to previously studied dimension and determinant torsors.

43 pages. Final version. Removed Section 4 and turned this into separate article

References in corpus (1)

• Principal infinity-bundles -- General theory

Cited by in corpus (1)

• Determinant map for the prestack of Tate objects