Abstract
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 Kähler differentials for the cdh topology.
We use this to address Bass’ question, whether K n (R)=K n (R[t]) implies K n (R)=K n (R[t 1,t 2]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general.
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Cortiñas’ research was partially supported by Conicet and partially supported by grants PICT 2006-00836, UBACyT X051, PIP 112-200801-00900, and MTM2007-64704.
Haesemeyer’s research was partially supported by NSF grant DMS-0652860.
Walker’s research was partially supported by NSF grant DMS-0601666.
Weibel’s research was supported by NSA grant MSPF-04G-184 and the Oswald Veblen Fund.
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Cortiñas, G., Haesemeyer, C., Walker, M.E. et al. Bass’ NK groups and cdh-fibrant Hochschild homology. Invent. math. 181, 421–448 (2010). https://doi.org/10.1007/s00222-010-0253-z
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DOI: https://doi.org/10.1007/s00222-010-0253-z