Excision in Cyclic Homology and in Rational Algebraic K-theory

Abstract

(for a precise definition, see ?1 below). By replacing everywhere K*( ) by K*( ) ? Q, one obtains the corresponding notion in rational algebraic K-theory. The above definition has an obvious counterpart for cyclic homology and algebras over a fixed field. In the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable (due to the well-known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring). An extension of k-algebras will be called pure if it is pure as an extension of k-modules (in the sense of P. M. Cohn [6]; cf. also Appendix A.3 below). The class of pure extensions contains, e.g., (i) extensions which admit a k-module splitting, (ii) extensions A >-e R -* S with S flat over k. In fact, one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k-modules is an inductive limit of split extensions (see Theorem A.4 of Appendix A below). Everywhere in this paper the word used in the context of cyclic homology will mean with respect to the class of pure extensions. The second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology. Before stating the corresponding result we need the following definition. Let us