### Periodic cyclic homology as sheaf cohomology, by Guillermo Corti~nas

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.

Guillermo Corti~nas <gcorti@dm.uba.ar>