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Algèbres simpliciales S1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs

Published online by Cambridge University Press:  29 July 2011

Bertrand Toën
Affiliation:
I3M UMR 5149, Université de Montpellier 2, France
Gabriele Vezzosi
Affiliation:
Dipartimento di Sistemi ed Informatica, Sezione di Matematica, Università di Firenze, Italy (email: gabriele.vezzosi@unifi.it)
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Abstract

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This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1A and ϵ(A), determine one another, functorially in A. The object S1A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 :=Bℤ, while the object ϵ(A) is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a ϵ-dg-algebra (see the main text). We define an equivalence φ between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

Résumé

Ce travail a pour objectif d’etablir une comparaison entre fonctions sur les espaces des lacets dérivés (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) et théorie de de Rham. Pour une k-algèbre commutative A, lisse sur k de caractéristique nulle, nous montrons que deux objets, S1A et ϵ(A), se déterminent mutuellement, et ce fonctoriellement en A. L’objet S1A est la k-algèbre simpliciale S1-équivariante obtenue en tensorisant A par le groupe simplicial S1 :=Bℤ. L’objet ϵ(A) est l’algèbre de de Rham de A, munie de la différentielle de de Rham et considérée comme une ϵ-dg-algèbre (i.e. une algèbre dans une certaine catégorie monoïdale de k[ϵ] -dg-modules, où k[ϵ]:=H* (S1,k) ). Nous construisons une équivalence ϕ, entre la théorie homotopique des k-algèbres simpliciales S1-équivariantes et celle des ϵ-dg-algèbres, et nous montrons l’existence d’une équivalence fonctorielle ϕ(S1A)∼ϵ(A) . Nous déduisons de cela la comparaison annoncée, identifiant les fonctions S1-équivariantes sur LX, l’espace des lacets dérivé d’un k-schéma X lisse, et la cohomologie de de Rham algébrique de X/k. Cela nous permet aussi de prouver des versions fonctorielles et multiplicatives des théorèmes de décomposition de l’homologie de Hochschild (du type Hochschild–Kostant–Rosenberg), pour des k-schémas semi-séparés quelconques. Par construction, ces décompositions sont de plus compatibles avec d’une part l’action naturelle de S1 sur le complexe de Hochschild, et d’autre part la différentielle de de Rham.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

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