On the homology of algebras of Whitney functions over subanalytic sets

• 成果类型: Article
• 署名作者: Brasselet, Jean-Paul; Pflaum, Markus J.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.167.1
• 发表日期: 2008
• 页码: 1-52
• 关键词: HOCHSCHILD HOMOLOGY; COHOMOLOGY; FORMULA

摘要

In this article we study several homology theories of the algebra epsilon(infinity)(X) of Whitney functions over a subanalytic set X subset of 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 epsilon(infinity)(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 epsilon(infinity)(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 epsilon(infinity)(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.