Simplicial S1-equivariant algebras, Rham theory and HKR multiplicative theorems

This work establishes a comparison between functions on derived loop spaces (Toen and Vezzosi, Chem, 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. Ostvaer (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 circle times A and epsilon(A), determine one another, functorially in A. The object S-1 circle times A is the S-1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S-1 := BZ, while the object epsilon(A) is the de R,ham algebra of A, endowed with the de Rham differential, and viewed as a epsilon-dg-algebra (see the main text). We define an equivalence phi between the homotopy theory of simplicial commutative S-1-equivariant k-algebras and the homotopy theory of epsilon-dg-algebras, and we show the existence of a functorial equivalence phi(S-1 circle times A) epsilon(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.