1
$\begingroup$

I am studying Group Cohomology, but I am a little confused about $\mathbb{Z} G$-module and $G$-module. Some text uses the $G$-module for group cohomology, but I thought group cohomology of $G$ is defined on $\mathbb{Z} G$-module. Are they isomorphic in some way? Why could we use only $G$-module instead of $\mathbb{Z} G$-module?

$\endgroup$
  • 1
    $\begingroup$ When people say $G$-module, they probably have already fixed a ground ring/field $R$ beforehand. Then $RG$-cohomology is just the tenosring of $\mathbb{Z}G$-cohomology with $R$ over $\mathbb{Z}$. $\endgroup$ – Aaron Nov 11 '15 at 13:35
2
$\begingroup$

$G$-modules, defined as abelian groups with an associative unital distributive action of $G$, are the same thing as $\mathbb{Z}G$-modules. For a $\mathbb{Z}G$-module restricts to a $G$-module via the inclusion of $G$ into the group ring, a $G$-module induces a $\mathbb{Z}G$-module by linearity, and these operations are mutually inverse.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.