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

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Let f:A→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)→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

$$ch_{*}:K_{*}(A,B:I)\otimes\mathbb{Q}\overset{\sim}{\to}HN_{*}(A\otimes\mathbb{Q},B\otimes\mathbb{Q}:I\otimes\mathbb{Q}).$$

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  1. Departamento de Matemática, Ciudad Universitaria Pab 1, 1428, Buenos Aires, Argentina

    Guillermo Cortiñas

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  1. Guillermo Cortiñas
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Cortiñas, G. The obstruction to excision in K-theory and in cyclic homology. Invent. math. 164, 143–173 (2006). https://doi.org/10.1007/s00222-005-0473-9

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