The tangent complex of K-theory
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 895-932.

We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is the cyclic homology (over k). This equivalence is compatible with λ-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 K. The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology and formal deformation theory à la Lurie-Pridham.

Dans cet article, nous prouvons que le complexe tangent de la K-théorie, en termes de problèmes de déformations formels et sur un corps k de caractéristique nulle, n’est autre que l’homologie cyclique sur k. Cette équivalence est de plus compatible aux λ-opérations. Nous démontrons également que le morphisme tangent du morphisme canonique BGL K est homotope au morphisme de trace généralisée de Loday-Quillen et Tsygan. La démonstration s’appuie sur des résultats de Goodwillie, à l’aide du théorème d’excision pour l’homologie cyclique de Wodzicki et de la théorie des déformations formelles à la Lurie-Pridham.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.161
Classification: 19E20
Keywords: K-theory, cyclic homology, formal moduli problem
Benjamin Hennion 1

1 IMO - Université Paris-Saclay F-91405 Orsay Cedex, France
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Benjamin Hennion. The tangent complex of K-theory. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 895-932. doi : 10.5802/jep.161. https://jep.centre-mersenne.org/articles/10.5802/jep.161/

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