The tangent complex of Ktheory
Abstract
We prove that the tangent complex of Ktheory, 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 Ktheory functor fully determines the absolute cyclic homology over any field k of characteristic 0. We also show that the LodayQuillenTsygan 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 à la LuriePridham.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.08268
 Bibcode:
 2019arXiv190408268H
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Algebraic Topology
 EPrint:
 36 pages. Final version. To appear in Journal de l'\'Ecole Polytechnique