Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V\_K = K \otimes\_k V$ be the $KG$-module given by changing the base field.


Is it true that $H^n(G,V\_K) \cong K \otimes\_k H^n(G,V)$?


If no, what about with some extra assumptions (for example $G$ is finite and $k$ is finite)?


My main motivation is in computing some cohomology/ext groups for $KG$-modules, where $K$ is algebraically closed and $G$ is finite. When $k$ is a finite field, there are ways to compute these with GAP for example.



Extension of base field for modules of groups and cohomology
There is a commutative diagram
$\require{AMScd}$
\begin{CD}
k[G]\text{-Mod} @>{-\otimes\_{k[G]}K[G]}>> K[G]\text{-Mod}\\
@V{-^G}VV @V{-^G}VV\\
k\text{-Vect} @>{-\otimes\_kK}>> K\text{-Vect},
\end{CD}
i.e., $(K[G]\otimes\_{k[G]}V)^G\simeq K\otimes\_kV^G$, since $K$ is a free $k$-vector space (and hence $K[G]\simeq K\otimes\_kk[G]$ is a free $k[G]$-module). Now the horizontal arrows are exact, so the derived functor of the composition $k[G]\text{-Mod}\to K\text{-Vect}$ is isomorphic to both $H^i(G,V\otimes\_{k[G]}K[G])$ and $H^i(G,V)\otimes\_kK$.


