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On $ K\sb 3$ of truncated polynomial rings


Author: Janet Aisbett
Journal: Trans. Amer. Math. Soc. 294 (1986), 517-536
MSC: Primary 18F25; Secondary 13D15, 19D55, 20G10, 20J06
MathSciNet review: 825719
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Abstract: Group homology spectral sequences are used to investigate $ {K_3}$ of truncated polynomial rings. If $ F$ is a finite field of odd characteristic, we show that relative $ {K_2}$ of the pair $ (F\left[ t \right]/({t^q}),\,({t^k}))$, which has been identified by van der Kallen and Stienstra, is isomorphic to $ {K_3}(F\left[ t \right]/({t^k}),\,(t))$ when $ q$ is sufficiently large. We also show that $ {H_3}({\text{SL}}\,{\mathbf{Z}}\left[ t \right]/({t^k});{\mathbf{Z}}) = {{\mathbf{Z}}^{k - 1}} \oplus {\mathbf{Z}}/24$ and is isomorphic to the associated $ {K_3}$ group modulo an elementary abelian $ 2$-group.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0825719-7
Article copyright: © Copyright 1986 American Mathematical Society