Bass' $NK$ groups and $cdh$-fibrant Hochschild homology

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Cortiñas, G.
Haesemeyer, C.
Walker, Mark E.
Weibel, C.
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Abstract
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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.
Comment: The article was split into two parts on referee's suggestion in 4/2010. This is the first part; the second can be found at arXiv:1004.3829
Keywords
Mathematics - K-Theory and Homology, Mathematics - Algebraic Geometry, 19D35 (Primary), 14F10, 14J17, 19D55 (Secondary)
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