Group action on sheaves

Question

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)?

Answer 1

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 might be easier to instead work with the stack quotient $[X/G]$. Then the statement you are trying to show fits into the general framework of descent along torsors. See for instance Vistoli's notes from Fundamental algebraic geometry for a nice treatment. All of this amounts to a very general set-up for showing that G-equivariant sheaves on X are equivalent to sheaves on the quotient. This will for instance work for locally constant sheaves in the étale topology. Maybe you can also do it directly without invoking any general theory by imitating Mumford's proof: he passes to a completion to assume that the covering is trivial, but it seems that you could take an étale cover instead.

Depending on what you want to do with your sheaf, having a sheaf on the quotient stack might be just as good as having a sheaf on the scheme quotient. For instance, if you want to compute its cohomology you can apply Leray to the projection to the coarse moduli space. In any nice situation all higher derived functors of the pushforward will vanish so you get the cohomology on the scheme quotient as well.

Hope this helps.

Answer 2

One can prove this correspondence along these lines : let $a: G\times X\rightarrow X$ be the action. A $G$-linearisation on a sheaf $\mathcal F$ consists of an isomorphism $a^*\mathcal F\simeq pr_2^*\mathcal F$ on $G\times X$, satisfying a compatibility condition on $G\times G\times X$. When the action is free, $(a,pr_2): G\times X \rightarrow X\times _Y X$ is an isomorphism, so the $G$-linearisation translates into $pr_1^*\mathcal F\simeq pr_2^*\mathcal F$ on $X\times_Y X$ satisfying a compatibility condition on $X\times_Y X\times_Y X$, that is, a descent data relative to $X\rightarrow Y$. A final remark : the descent of quasi-coherent sheaves (Zarisky objects) along étale maps is not a trivial fact, because of the several topologies involved, but the descent of (locally constant) sheaves for the étale topology along étale maps is completely straightforward, you just need to write the definition of a sheaf. The slogan is "sheaves for a given topology form a stack for this topology", see Angelo's FAG notes.