The obstruction to excision in K-theory and in cyclic homology

Abstract

Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and $I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups $K_*(A:I) \to K_*(B:f(I))$ to be an isomorphism; it is measured by the birelative groups $K_*(A,B:I)$. We show that these are rationally isomorphic to the corresponding birelative groups for cyclic homology up to a dimension shift. In the particular case when A and B are $\Q$-algebras we obtain an integral isomorphism.Comment: Final version to appear in Inventione

Algebraic Topology: Applications and New Directions

“…We finally mention that for k a field of characteristic zero, the cyclic homology groups of A were calculated by Geller, Reid, and Weibel [12,Theorem 9.2] and that, in view of the affirmation by Cortiñas [4] of the KABI-conjecture made in [12], this gives a complete calculation of the groups K q (A, a).…”

Infinitesimal cohomology and the Chern character to negative cyclic homology

“…10 The construction of the basis B p (σ ) is also enough for the following proof to work; it is the original proof of [42, 1.4]. For each q, the simplicial homotopy…”

The 𝐾-theory of toric varieties

“…We will see in the next section that their Zariski cohomology groups turn out to give the cdh-cohomology groups of Ω p . Given a fan ∆, let ∆(1) denote the collection of rays in ∆; the 1-skeleton of ∆ is the fan ∆(1) ∪ {0} and its toric variety X (1) lies in the smooth locus of X(∆).…”

“…License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use HC of HC (1) → H cdh (X, HC (1) ) is also the mapping cone of L X →Ω 1 X . This proves the first assertion; the second assertion follows from this, Proposition 5.6 and [3, 1.6].…”