Cyclic homology for schemes

Abstract

Using hypercohomology, we can extend cyclic homology from al-gebras to all schemes over a ring k. By 'extend' we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme. The purpose of this paper is to show that there is a cyclic homology theory HC * of schemes over a commutative ring k, extending the usual cyclic homology HC * of k-algebras. By a cyclic homology theory for schemes over k we mean a family of graded k-modules HC n (X) associated to every scheme X over k which satisfy: (0.1) they are natural and contravariant in X; (0.2) for each affine scheme X = Spec A, there are natural isomorphisms HC n (X) ∼ = HC n (A) for all n;