I am trying to get familiar with group (Tate) cohomology. I am for instance reading Brown's Cohomology of Groups. Now something seems unclear to me
What information do we hope to attain from studying $\hat{H}^*(G,\mathbb{Z})$ vs. $\hat{H}^*(G,k)$ ?
Here $k$ is some field, say $\mathbb{F}_p$.
In chapter VI.8 (cohomologically trivial modules) Brown starts considering $k[G]$-modules, without (seemingly) much introduction. What causes the need to change from $\mathbb{Z}[G]$-modules to $k[G]$-modules? What is really happening in that change?
Also, when are $\mathbb{Z}[G]$-modules interesting to study versus $k[G]$-modules?
It is claimed that the results for $k[G]$-modules have applications to class field theory, algebraic K-theory and homotopy theory. I am interested in number theory, so any illustrations there would be appreciated.
You posted a similar question not long ago ; if the answer I gave did not satisfy you, I don't really know what more I can say about the relevance of $\hat{H}^*(G,\mathbb{Z})$ or the relation between $\hat{H}^*(G,\mathbb{Z})$ $\hat{H}^*(G,\mathbb{F}_p)$. The thing is, why would you feel that one is studied rather than the other ? They are both basic subjects of study in group cohomology. It feels like for you $\hat{H}^*(G,\mathbb{F}_p)$ is the really important object and people study $\hat{H}^*(G,\mathbb{Z})$ instead, but that is not the case.
As for $k[G]$-modules, it's simple : $\mathbb{Z}[G]$-modules are abelian groups with a $G$-action, and $k[G]$-modules are $k$-vector spaces with a $G$-action (or $k$-modules if $k$ is just a ring). So if you believe that $k$-modules are important, and that group actions on $k$-modules are important, then there you go for $k[G]$-modules. The case $k=\mathbb{Z}$ is just more general and universal (since it covers all abelian groups).