Periodic cyclic homology as sheaf cohomology
Abstract
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 JonesGoodwillie character which goes from $K$theory to negative cyclic homology.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1998
 DOI:
 10.48550/arXiv.math/9810144
 arXiv:
 arXiv:math/9810144
 Bibcode:
 1998math.....10144C
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 Mathematics  Quantum Algebra;
 19D55 (Primary) 18F10 (Secondary)
 EPrint:
 23 pages, AmsTeX