On the cyclic homology of ringed spaces and schemes, by Bernhard Keller

In their recent proof of Schapira-Schneider's conjecture, Bressler-Nest-Tsygan construct a (generalized) Chern character from K-theory to the negative cyclic homology of a sheaf of algebras A on a topological space.

The first aim of this paper is to show how to construct a (classical) Chern character defined on the Grothendieck group of A with values in the mixed negative cyclic homology of A using the methods of previous work on the cyclic homology of exact categories.

The second aim of the paper is to prove that the cyclic homology of a quasi-compact separated scheme as defined by Loday and Weibel coincides with the cyclic homology of the "localization pair" of perfect complexes on the scheme. In particular, if the scheme admits an ample line bundle, its cyclic homology coincides with the cyclic homology of the exact category of algebraic vector bundles.

In an appendix, we prove the useful technical result that hypercohomology of (unbounded) complexes with quasi-coherent homology on a scheme may be computed using Cartan-Eilenberg resolutions. For module categories, this was independently observed by C. Weibel (unpublished). Among other things, it yields a (partially) new proof of Boekstedt-Neeman's theorem which states that for a quasi-compact separated scheme X, the (unbounded) derived category of quasi-coherent sheaves on X is equivalent to the full subcategory of the unbounded derived category of all modules on X whose objects are the complexes with quasi-coherent homology. A different proof of this is due to Alonso-Jeremías-Lipman.

Bernhard Keller <keller@math.jussieu.fr>