Algèbres simpliciales S 1 -équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs

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, S 1 ⊗ A and ϵ( A ), determine one another, functorially in A . The object S 1 ⊗ A is the S 1 -equivariant simplicial k -algebra obtained by tensoring A by the simplicial group S 1 := B ℤ, 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 S 1 -equivariant k -algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ( S 1 ⊗ A )∼ϵ( A ) . We deduce from this the comparison mentioned above, identifying the S 1 -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 S 1 -action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.