Take the 2-minute tour ×  I am currently reading D. Mumford´s
Abelian Varieties and it came up the following question: let $X$
be an algebraic variety over an algebraically closed field $k$
and $G$ a finite group acting on $X$. Assume we are in a
situation that there exists a quotient $(Y, \pi:X \rightarrow
Y)$, he then proves a proposition that there is a one-to-one
correspondence between coherent $\mathcal{O}_{Y}$-modules and
$G$-equivariant coherent $\mathcal{O}_{X}$-modules. In the
course of the proof he shows that for such a $G$-sheaf
$\mathfrak{F}$ on $X$ the natural morphism
$\pi^{-1}((\pi_{*}\mathfrak{F})^{G}) \rightarrow \mathfrak{F}$
is an isomorphism.  What about other kind of sheaves, especially
locally constant sheaves (regarding whether this natural
morphism is an isomorphism)?  share|improve this question add
comment  2 Answers  up vote 5 down vote accepted  Hi Peter!  I
just checked in Mumford's book and there is an assumption in the
theorem that G acts freely. If you don't assume this, it mig