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For me, a local system on a topological space $X$ is a locally constant sheaf on $X$, i.e., a sheaf of sets $\mathscr{F}$ for which there is an open cover $X=\bigcup_i U_i$ so that $\mathscr{F}_i\cong\underline{\mathscr{F}(U_i)}_{U_i}$, the constant sheaf on $U_i$ associated to the set $\mathscr{F}(U_i)$.

The local systems I'm interested in arise as follows (and perhaps they all arise this way, maybe under some additional connectedness assumptions on $X$). Let $G$ be a group acting (on the left, let's say) freely and properly discontinuously on $X$ (I'm not sure if the definition of the latter term is universally agreed upon, but the condition I want is that each $x\in X$ admits an open $x\in U\subseteq X$ such that $gU\cap U=\emptyset$ for $g\neq 1$) and $V$ an $R$-module (for some commutative ring $R$) with an $R$-linear action of $G$. Then the natural projection $q:G\setminus(X\times V)\rightarrow G\setminus X$ (here I regard $V$ as having the discrete topology and let $G$ act on $X\times V$ diagonally) is then a covering space (because of the assumption on the action of $G$ on $X$), and all the fibers are in bijection with $V$. So the sheaf of continuous sections $\mathscr{F}_V$ on $G\setminus X$ is going to be locally on $X$ isomorphic to the constant sheaf $\underline{V}_X$. Now, my understanding is that something like this is called a local system of $R$-modules, and my question is:

What is the $R$-module structure on the set of sections $\mathscr{F}_V(U)$ for $U\subseteq X$ open?

The only thing that I can think of (and I guess I'm motivated by the vector space structure on the sections of a vector bundle) is that for each $x\in X$, the restriction of $X\times V\rightarrow G\setminus(X\times V)$ to $\{x\}\times V$ is a bijection onto the fiber over $Gx$ of $q$. So one can make the fiber into an $R$-module by declaring that this bijection should be an isomorphism of $R$-modules. Then the $R$-module structure on $\mathscr{F}_V(U)$ would be defined pointwise (like for vector bundles), i.e., if $s,t\in\mathscr{F}_V(U)$, so that $s(u),t(u)\in q^{-1}(u)$, then $(s+t)(u)=s(u)+t(u)$, the addition taking place in the fiber $q^{-1}(u)$, and likewise, for $r\in R$, $(rs)(u)=rs(u)$, the scalar multiplication taking place in $q^{-1}(u)$.

I think that with this definition, if $U$ is an open set of $X$ over which the covering $G\setminus(X\times V)\rightarrow G\setminus X$ can be trivialized, then the induced isomorphism $\mathscr{F}_V\vert_U\cong\underline{V}_U$ will be one of sheaves of $R$-modules, which suggests I have the right definition...but I haven't checked the details. Do I have the right definition of the $R$-module structure?

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$X \times V$ is a constant local system of $R$-modules, just using the $R$-module structure on $V$. (A section on $U$ is just a locally constant $V$-valued function on $U$, with the $R$-module structure defined pointwise.)

The action of $G$ on $V$ is $R$-linear, and so $G\backslash (X \times V)$ is a (typically no longer constant) local system of $R$-modules too. A section of this local system over an open set $U$ of $G\backslash X$ is a $G$-equivariant locally constant $V$-valued function on the preimage of $U$ in $X$, again with the $R$-module structure defined pointwise.


In short, you have the correct $R$-module structure.

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  • $\begingroup$ Ah, I see. I like this description better than mine. Less fiber-by-fiber and sort of more global. Thank you. $\endgroup$ – Keenan Kidwell Aug 18 '13 at 2:07

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