Let $K$ be a field, $L/K$ be a finite Galois extension with Galois group $G$ such that $(char(K),|G|)=1$ and $K$ contains all $|G|$th roots of unity. Let $B$ be a $L$-algebra of finite type endowed with a $K[G]$-module structure such that the $G$-action also preserves multiplication, and the restriction of the $K[G]$-module structure on $L$ coincides with the natural Galois representation. Must $B$ be a free $K[G]$-module?
I was considering the quotient of a quasi-projective variety by a finite group. Let $V$ be such a variety over $L$, $G$ be a finite subgroup of $Aut(L)$, and $K$ be the fixed subfield. Consider a $G$-action on $V$ extending that one on $L$. I would like to ask whether we have $$ V/G\times_K L\cong V $$ under some assumptions. An exercise on Hartshorne's book states the case when $L=\mathbb{C}$, $K=\mathbb{R}$. I can do this when $G$ is abelian and $K$ contains sufficiently many roots of unity. However, I have difficulty applying my idea to the nonabelian case.