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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On $ (K\sb\ast ({\bf Z}/p\sp{2}{\bf Z})$ and related homology groups


Authors: Leonard Evens and Eric M. Friedlander
Journal: Trans. Amer. Math. Soc. 270 (1982), 1-46
MSC: Primary 18F25; Secondary 20G10, 20J06, 20J10
MathSciNet review: 642328
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Abstract: It is shown that, for $ p \geqslant 5$,

$\displaystyle R = {\mathbf{Z}} / {p^2}{\mathbf{Z}},\,{K_3}(R) = {\mathbf{Z}} / {p^2}{\mathbf{Z}} + {\mathbf{Z}} / ({p^2} - 1){\mathbf{Z}}$

and $ {K_4}(R) = 0$. Similar calculations are made for $ R$ the ring of dual numbers over $ {\mathbf{Z}} / p{\mathbf{Z}}$. The calculation reduces to finding homology groups of $ \operatorname{Sl} (R)$. A key tool is the spectral sequence of the group extension of $ \operatorname{Sl} (n,\,{p^2})$ over $ \operatorname{Sl} (n,\,p)$. The terms of this spectral sequence depend in turn on the homology of $ \operatorname{Gl} (n,\,p)$ with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of $ {d^2}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0642328-X
PII: S 0002-9947(1982)0642328-X
Keywords: $ K$-theory, homology of groups, Lyndon-Hochschild-Serre spectral sequence, Charlap-Vasquez theory, linear groups, dual numbers, $ {\mathbf{Z}} / {p^2}{\mathbf{Z}}$
Article copyright: © Copyright 1982 American Mathematical Society