The tangent complex of K-theory
[Le complexe tangent de la K-théorie]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 895-932.

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.

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.

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Accepté le :
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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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