Bass' \(NK\) groups and \(cdh\)-fibrant Hochschild homology

The \(K\)-theory of a polynomial ring \(R[t]\) contains the \(K\)-theory of \(R\) as a summand. For \(R\) commutative and containing \(\Q\), we describe \(K_*(R[t])/K_*(R)\) in terms of Hochschild homology and the cohomology of K\"ahler differentials for the \(cdh\) topology. We use this to add...

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Bibliographic Details
Published in:arXiv.org 2010-04
Main Authors: Cortiñas, G, Haesemeyer, C, Walker, Mark E, Weibel, C
Format: Article
Language:English
Subjects:
Homology
Number theory
Polynomials
Rings (mathematics)
Topology
Online Access:Get full text
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Summary:The \(K\)-theory of a polynomial ring \(R[t]\) contains the \(K\)-theory of \(R\) as a summand. For \(R\) commutative and containing \(\Q\), we describe \(K_*(R[t])/K_*(R)\) in terms of Hochschild homology and the cohomology of K\"ahler differentials for the \(cdh\) topology. We use this to address Bass' question, on whether \(K_n(R)=K_n(R[t])\) implies \(K_n(R)=K_n(R[t_1,t_2])\). The answer is positive over fields of infinite transcendence degree; the companion paper arXiv:1004.3829 provides a counterexample over a number field.
ISSN:2331-8422