modular representation, field of definition

Question

Let $G$ be a finite group. Let $K \subset L$ be an extension of two algebraically closed fields of characteristic $p>0$. Let $V$ be a finite-dimensional $L[G]$-module. Does there always exist a $K[G]$-module $W$ such that $V=L \otimes_K W$? If it does, a reference or a sketch of a proof is appreciated. If not, a simple counterexample would be nice. Also, what if $V$ is assumed to be irreducible? Thanks in advance.

Answer 1

If $V$ is not semisimple, there may not be any such $W$.

Let $p=2$ and $G$ be the Klein $4$-group $C_2\times C_2$, generated by two elements $g$ and $h$.

For any $\lambda\in L$ there is a $2$-dimensional $L[G]$-module where $g$ and $h$ act as $V(g)=\pmatrix{1&1\\0&1}$ and $V(h)=\pmatrix{1&\lambda\\0&1}$ respectively (with respect to some basis).

Note that $V(h)-I=\lambda\left(V(g)-I\right)$.

If this module were of the form $L\otimes_KW$, then we could choose a basis with respect to which the actions of $g$ and $h$ were given by matrices $W(g)$ and $W(h)$ with entries in $K$ such that $W(h)-I=\lambda\left(W(g)-I\right)$. But this is clearly impossible if $\lambda\not\in K$.