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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 $\tilde{X} \times_G V$?

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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.

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    $\begingroup$ so does all the local system arise in this way? Does this way more natural than the original way(the usual definition of sheaf)? $\endgroup$
    – abc
    Commented Dec 7, 2010 at 7:35
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    $\begingroup$ @abc: Yes, all local systems arise in this way, and this a very natural way to think about them. Very often people will speak of the local system corresponding to a $G$-module $V$, and this is what they mean. $\endgroup$
    – Matt E
    Commented Dec 7, 2010 at 8:58
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    $\begingroup$ Can I ask how one defines $\tilde{X}\times_G V $ ? The notation makes it seem like a pullback but I don't see any natural maps $\tilde{X}\to G$ or $V\to G$ ? Is it just the product modulo $(x,gv) = (xg,v)$ for $x\in\tilde{X},v\in V,g\in G$ ? $\endgroup$ Commented Jan 25, 2019 at 17:14
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    $\begingroup$ For the future reader, the answer to Maxime's comment is yes. $\endgroup$
    – shubhankar
    Commented Jul 24, 2021 at 20:31

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