Differential forms on singular varieties and cyclic homology. (English) Zbl 0949.55003

Brasselet, J. P.; Legrand, A.

Bruce, Bill (ed.) et al., Singularity theory. Proceedings of the European singularities conference, Liverpool, UK, August 18-24, 1996. Dedicated to C.T.C. Wall on the occasion of his 60th birthday. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 263, 175-187 (1999).

Summary: A classic result of A. Connes [Publ. Math., Inst. Hautes Étud. Sci. 62, 41-144 (1985; Zbl 0592.46056)] asserts that the Fréchet algebra of smooth functions on a smooth compact manifold \(X\) provides, by a purely algebraic procedure, the de Rham cohomology of \(X\). The procedure uses the Hochschild and cyclic homology of this algebra.
For a Thom-Mather stratified variety, we construct a Fréchet algebra of functions on the regular part and a module of poles along the singular part. We associate to these objects a complex of differential forms and a Hochschild mixed complex, on the regular part, both with poles along the singular part. The de Rham cohomology of the first complex and the periodic cyclic homology of the second one are the intersection homology of the variety, the corresponding perversity being determined by the orders of poles.
Detailed proofs will appear in a forthcoming publication.
For the entire collection see [Zbl 0919.00048].

MSC:

55N33	Intersection homology and cohomology in algebraic topology
14F43	Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
57N80	Stratifications in topological manifolds
14B05	Singularities in algebraic geometry
18G60	Other (co)homology theories (MSC2010)
14F40	de Rham cohomology and algebraic geometry

Keywords:

de Rham cohomology; Thom-Mather stratified variety; Fréchet algebra of functions; complex of differential forms; Hochschild mixed complex; cyclic homology; intersection homology; perversity