On the Cylic Homology of Ringed Spaces and Schemes

In their recent proof [2] of Schapira-Schneider’s conjecture [19], Bressler-Nest-Tsygan construct a (generalized) Chern character from the Ktheory of perfect complexes to the negative cyclic homology HC− ∗ (A) 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 perfect complexes over A with values in the mixed negative cyclic homology HC − mix,∗ (A) using the methods of previous work [10] on the cyclic homology of exact categories. The mixed negative cyclic homology groups HC− mix,∗ (A) are slightly different from the groups HC− ∗ (A). They are deduced from the mixed complex associated with A. The main virtue of the groups HC− mix,∗ is that they are preserved under morphisms of ringed spaces inducing isomorphisms in Hochschild homology. The analogous statement for HC− ∗ does not seem obvious. The second aim of the paper is to prove that the cyclic homology of a quasicompact separated scheme as defined by Loday [12] and Weibel [23] 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, a result announced in [10]. 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 the case of module categories, this was independently observed by C. Weibel (cf. the footnote on page 2 of [21]). Among other things, it yields a (partially) new proof of Boekstedt-Neeman’s theorem [1] 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 OX-modules whose objects are the complexes with quasi-coherent homology. A different proof of this was given by Alonso-Jeremı́as-Lipman in [21, Prop. 1.3]. 1. Notations In sections 1 to 4, k will denote a field, X a topological space and A a sheaf of k-algebras on X . 2. Homology theories for ringed spaces 2.1. Hochschild and cyclic homologies. The Hochschild complex C(A), and the bicomplexes CC(A), CC−(A) and CCper(A) are defined in [2, 4.1] by composing Date: February 9, 1998. 1991 Mathematics Subject Classification. 16E40 (Primary), 18E30, 14F05 (Secondary).