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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Splitting theorem for homology of $ {\rm GL}(R)$


Author: Stanisław Betley
Journal: Proc. Amer. Math. Soc. 108 (1990), 297-302
MSC: Primary 20J05
DOI: https://doi.org/10.1090/S0002-9939-1990-0984782-X
MathSciNet review: 984782
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Abstract: It is proved that if $ \left\{ {{M_n}} \right\}$ is a stable system of coefficients for $ {\text{G}}{{\text{l}}_n}\left( R \right)$ and $ {H_0}\left( {{\text{Gl}}\left( R \right),{\text{lim}}\left( {{M_n}} \right)} \right)$ contains $ {\mathbf{Z}}$, then for any $ j$, the group $ {H_j}\left( {{\text{Gl}}\left( R \right),{\text{lim}}\left( {{M_n}} \right)} \right)$ contains $ {H_j}\left( {{\text{Gl}}\left( R \right),Z} \right)$ as a direct summand. Now let $ {\text{Gl}}\left( {\mathbf{Z}} \right)$ act on $ M\left( {\mathbf{Z}} \right)$ (matrices over $ {\mathbf{Z}}$ ) by conjugation. Then our theorem implies that the trace map $ {\text{tr:}}M\left( {\mathbf{Z}} \right) \to {\mathbf{Z}}$ is a split epimorphism on homology.


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DOI: https://doi.org/10.1090/S0002-9939-1990-0984782-X
Keywords: Homology of a group, stable system of coefficients
Article copyright: © Copyright 1990 American Mathematical Society

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