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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On $(K_\ast (\textbf {Z}/p^{2}\textbf {Z})$ 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
DOI: https://doi.org/10.1090/S0002-9947-1982-0642328-X
MathSciNet review: 642328
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Abstract: It is shown that, for $p \geqslant 5$, \[ R = {\mathbf {Z}} / {p^2}{\mathbf {Z}}, {K_3}(R) = {\mathbf {Z}} / {p^2}{\mathbf {Z}} + {\mathbf {Z}} / ({p^2} - 1){\mathbf {Z}}\] 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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Keywords: <IMG WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$K$">-theory, homology of groups, Lyndon-Hochschild-Serre spectral sequence, Charlap-Vasquez theory, linear groups, dual numbers, <!โ€“ MATH ${\mathbf {Z}} / {p^2}{\mathbf {Z}}$ โ€“> <IMG WIDTH="63" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${\mathbf {Z}} / {p^2}{\mathbf {Z}}$">
Article copyright: © Copyright 1982 American Mathematical Society