Periodic cyclic homology as sheaf cohomology

Details Periodic cyclic homology as sheaf cohomology Periodic cyclic homology as sheaf cohomology

1998-10-23

arxiv.org/abs/math/9810144v1

K-theory 20 (2000) 175-200

Mathematics

K-Theory and Homology

23 pages, AmsTeX

Scientific paper

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example we show that hypercohomology with coefficients in $K$-theory gives the fiber of the Jones-Goodwillie character which goes from $K$-theory to negative cyclic homology.

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