Geodesic
On the homology of algebras of Whitney functions over subanalytic sets
Jean-Paul Brasselet1 ; Markus J. Pflaum2
1 Institut de Mathématiques de Luminy<br/>13288 Marseille<br/>France
2 Department of Mathematics<br/>University of Colorado<br/>Boulder, CO 80309<br/>United States
Annals of mathematics, Tome 167 (2008) no. 1, pp. 1-52.

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In this article we study several homology theories of the algebra $\mathcal{E}^\infty (X)$ of Whitney functions over a subanalytic set $X\subset\mathbb{R}^n$ with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for $\mathcal{E}^\infty (X)$, when $X$ is a regular subset of $\mathbb{R}^n$ having regularly situated diagonals. This includes the case of subanalytic $X$. We also compute the Hochschild cohomology of $\mathcal{E}^\infty (X)$ 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 $\mathcal{E}^\infty (X)$ 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.
DOI : 10.4007/annals.2008.167.1

Jean-Paul Brasselet 1 ; Markus J. Pflaum 2

1 Institut de Mathématiques de Luminy<br/>13288 Marseille<br/>France
2 Department of Mathematics<br/>University of Colorado<br/>Boulder, CO 80309<br/>United States 
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Jean-Paul Brasselet; Markus J. Pflaum. On the homology of algebras of Whitney functions over subanalytic sets. Annals of mathematics, Tome 167 (2008) no. 1, pp. 1-52. doi : 10.4007/annals.2008.167.1. http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.1/

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