, Volume 164, Issue 1, pp 143-173
Date: 25 Oct 2005

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

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Abstract

Let f:AB 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)→K *(B:f(I)) to be an isomorphism; it is measured by the birelative groups K *(A,B:I). Similarly the groups HN *(A,B:I) measure the obstruction to excision in negative cyclic homology. We show that the rational Jones-Goodwillie Chern character induces an isomorphism
$$ch_{*}:K_{*}(A,B:I)\otimes\mathbb{Q}\overset{\sim}{\to}HN_{*}(A\otimes\mathbb{Q},B\otimes\mathbb{Q}:I\otimes\mathbb{Q}).$$