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

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

2001-11-08

arxiv.org/abs/math/0111096v5

Invent. Math. 454 (2006) 143-173

Mathematics

K-Theory and Homology

Final version to appear in Inventiones

Scientific paper

10.1007/s00222-005-0473-9

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.

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