On the homology of algebras of Whitney functions over subanalytic sets

Abstract

In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ R n with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for E ∞ (X), when X is a regular subset of R n having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (X) for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. It is shown that the periodic cyclic homology coincides with the de Rham cohomology, thus generalizing a result of Feigin-Tsygan. Motivated by the algebraic de Rham theory of Grothendieck we finally prove that for subanalytic sets the de Rham cohomology of E ∞ (X) coincides with the singular cohomology. For the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation. Contents