TAILIEUCHUNG - Đề tài " On the homology of algebras of Whitney functions over subanalytic sets "

In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ Rn 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 Rn 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. . | Annals of Mathematics On the homology of algebras of Whitney functions over subanalytic sets By Jean-Paul Brasselet and Markus J. Pflaum Annals of Mathematics 167 2008 1 52 On the homology of algebras of Whitney functions over subanalytic sets By Jean-Paul Brasselet and Markus J. Pflaum Abstract In this article we study several homology theories of the algebra E X X of Whitney functions over a subanalytic set X c Rra 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 Rra 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 Introduction 1. Preliminaries on Whitney functions 2. Localization techniques 3. Peetre-like theorems 4. Hochschild homology of Whitney functions 5. Hochschild cohomology of Whitney functions 6. Cyclic homology of Whitney functions 7. Whitney-de Rham cohomology of subanalytic spaces 8. Bimeromorphic triangulations References 2 JEAN-PAUL BRASSELET AND MARKUS J. PFLAUM Introduction Methods originating from noncommutative differential geometry have proved to be very successful not only for the study of noncommutative algebras but also have given new insight to the geometric analysis of smooth manifolds which are the typical objects of commutative differential geometry. As

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