The Index Map in Algebraic K-Theory

TL;DR

This paper constructs a universal torsor in algebraic K-theory for Tate modules over a ring, linking higher loop groups with K-theory and generalizing classical index maps.

Contribution

It introduces a universal K_R-torsor on the K-theory space of Tate R-modules, connecting higher loop groups with algebraic K-theory in a novel way.

Findings

Constructed a universal K_R-torsor on K-theory space of Tate R-modules.

Related the torsor to classical central extensions of loop groups.

Analyzed the classifying index map and its analogy with Fredholm operators.

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.