The tangent complex of K-theory

Hennion, Benjamin

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 $\lambda$-opérations. Nous démontrons également que le morphisme tangent du morphisme canonique ${\mathrm{BGL}}_{\infty }\to \mathrm{K}$ 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 $\lambda$-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 ${\mathrm{BGL}}_{\infty }\to \mathrm{K}$. The proof builds on results of Goodwillie, using Wodzicki’s excision for cyclic homology and formal deformation theory à la Lurie-Pridham.

Classification : 19E20
Keywords: K-theory, cyclic homology, formal moduli problem
Mot clés : K-théorie, homologie cyclique, espace de module formel

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