What local system really is

Question

I know a local system is a locally constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, $\tilde{X}$ is the universal covering space of $X$, and $V$ is a $G$-module? How do you recover the locally free sheaf from $\tilde{X} \times_G V$?

Answer 1

The group $G$ acts properly discontinuously on $\tilde{X}$, and so if $x$ is any point of $\tilde{X}$, it admits a neighbourhood $U$ s.t. that $U g$ is disjoint from $U$ if $g \in G$ is non-trivial. Thus the natural map from $U$ to $\tilde{X}/G = X$ is an embedding.

Thus the natural map from $U \times V$ to $\tilde{X}\times_G V$ is also an embedding, and so $\tilde{X}\times_G V$ is locally constant (i.e. locally a product).

More detailed remarks:

• We should equip $V$ with its discrete topology

• The object $\tilde{X}\times_G V$ is not itself actually a sheaf, but is rather the espace etale of a sheaf. To get the actual sheaf we consider the natural projection $\tilde{X}\times_G V \to \tilde{X}/G = X$, and form the associated sheaf of sections. Over the open set $U \hookrightarrow X,$ this restricts to the sheaf of sections to the projection $U\times V \to U$, whose sections are precisely the constant sheaf on $U$ attached to the vector space $V$. (Here is where we see that it is important to equip $V$ with the discrete topology.) Thus our original sheaf of sections is locally constant, as claimed.