Algèbres simplicialesS1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs

Abstract

AbstractThis work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi,Chern character, loop spaces and derived algebraic geometry, inAlgebraic 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. IfAis a smooth commutativek-algebra andkhas characteristic 0, we show that two objects,S1⊗Aand ϵ(A), determine one another, functorially inA. The objectS1⊗Ais theS1-equivariant simplicialk-algebra obtained by tensoringAby the simplicial groupS1:=Bℤ, while the object ϵ(A) is the de Rham algebra ofA, endowed with the de Rham differential, and viewed as aϵ-dg-algebra(see the main text). We define an equivalence φ between the homotopy theory of simplicial commutativeS1-equivariantk-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying theS1-equivariant functions on the derived loop spaceLXof a smoothk-schemeXwith the algebraic de Rham cohomology of X/k. As corollaries, we obtainfunctorialandmultiplicativeversions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separatedk-schemes. By construction, these decompositions aremoreovercompatible with theS1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.