Cyclic homology and algebraic K-theory of spaces—II

Detecting K-theory by Cyclic Homology

We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.

On the K · Theory and Cyclic Homology of a Square

The purpose of this paper is to establish an integral connection between the relative K-theory and cyclic homology of a square-zero ideal IeR, R a ring with unit. We restrict attention to the case when R is a split extension of R/I by I. The outline of the paper is as follows: in Section 2, we compute a direct summand of HC*(R, I), where HC*(R, 1) is graded in analogy to relative K-theory and f...

Pro cdh-descent for cyclic homology and K-theory

In this paper we prove that cyclic homology, topological cyclic homology, and algebraic K-theory satisfy a pro Mayer–Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of K-theory with compact support.

Cyclic Homology Theory, Part II

Bivariant algebraic K-theory

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞-stable, homotopy-invariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) 7→ kk∗(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel’s homotopy algebraic K-theory, KH....