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Let $\mathcal F$ be a locally constant sheaf with values in $\mathbb C$ on a nice enough space, say a compact manifold. The etale space of $\mathcal F$ defines a covering $p: \tilde X \to X$.

Is there any relation between the (cech) cohomology $H^*(X, \mathcal F)$ and the (singular) cohomology $H^*(\tilde X, \mathbb C)$?

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    $\begingroup$ Welcome to MathOverflow! I suggest that you try to work out a simple example, where $X=S^1$ and $\mathcal F$ has rank $1$. Then $\tilde X$ has $\mathbb R$ many connected components. If the monodromy of $\mathcal F$ is a root of unity, they are all circles. Otherwise, only one is a circle. This allows you to understand $H^*(\tilde X,\mathbb C)$. On the other hand, $H^*(X,\mathcal F)$ is nontrivial only if the monodromy of $\mathcal F$ is trivial. $\endgroup$
    – Sebastian Goette
    Commented Aug 27, 2019 at 19:00

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