The tangent complex of K-theory

TL;DR

This paper establishes that the tangent complex of algebraic K-theory over a characteristic zero field is equivalent to cyclic homology, revealing deep connections between deformation theory, cyclic homology, and K-theory.

Contribution

It proves the equivalence of the tangent complex of K-theory with cyclic homology and relates the Loday-Quillen-Tsygan trace to the tangent morphism of a canonical map.

Findings

Tangent complex of K-theory is cyclic homology over characteristic zero fields.

The relative algebraic K-theory determines absolute cyclic homology.

Loday-Quillen-Tsygan trace arises as a tangent morphism.

Abstract

We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $\lambda$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\infty \to K$. The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory \`a la Lurie-Pridham.