The tangent complex of K-theory

Benjamin Hennion

doi:10.5802/jep.161

The tangent complex of K-theory
[Le complexe tangent de la K-théorie]

Benjamin Hennion

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 895-932.

Résumé
Abstract

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 ${BGL}_{\infty} \to K$ 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.

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 ${BGL}_{\infty} \to K$ . The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology and formal deformation theory à la Lurie-Pridham.

Reçu le : 2020-05-08
Accepté le : 2021-03-16
Publié le : 2021-03-30

DOI : https://doi.org/10.5802/jep.161
Classification : 19E20
Mots clés : K-théorie, homologie cyclique, espace de module formel

@article{JEP_2021__8__895_0,
     author = {Benjamin Hennion},
     title = {The tangent complex of K-theory},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     pages = {895--932},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.161},
     language = {en},
     url = {https://jep.centre-mersenne.org/item/JEP_2021__8__895_0/}
}

Benjamin Hennion. The tangent complex of K-theory. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 895-932. doi : 10.5802/jep.161. https://jep.centre-mersenne.org/item/JEP_2021__8__895_0/

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