This is a question on a sentence in the paper "Faisceaux pervers", p. 163.
The say that if $j: U \hookrightarrow X$ is a Zariski open subset and $L$ is a local system on $U$ with finite monodromy, then $L$ is a direct factor of $\pi_\ast \mathbb{C}$ for $\pi: \tilde{X} \to X$ a finite covering.
I don't understand how to construct $\tilde{X}$ out of the data of $L$ on $U$.
Can anybody explain the procedure to me?
First, suppose $L$ is simple. $L$ is trivialied by some finite etale map $\tilde U\to U$ (this is what it means to have finite monodromy); then applying Grauert-Remmert (or SGA I, as mentioned in Georges Elencwajg's answer to this question), we may extend the map to a (possibly ramified) map $\pi: \tilde X\to X$. That is, $\pi^*L|_{\tilde U}=\underline{\mathbb{C}}^n$. Then $\operatorname{Hom}_U(L, \pi_*(\underline{\mathbb{C}}^n)|_{U})\simeq\operatorname{Hom}_{\tilde U}(\pi^*L|_{\tilde U}, \underline{\mathbb{C}}^n)$ is non-zero, so $L$ is a factor of $\pi_*\underline{\mathbb{C}}^n$ (using simplicity). Again, as $L$ is simple, it is in fact a factor of $\pi_*\underline{\mathbb{C}}$.
Now if $L$ is not simple, decompose it into simple factors $L=\oplus L_i$, and let $\pi: {\tilde X_i}\to X$ be covers trivializing the $L_i$. Setting $\tilde X=\bigsqcup {\tilde X_i}$ does the trick.
BTW, just a silly but perhaps cute remark; one may apply this same argument to the map $\pi: *\to BG$, for $G$ a finite group, to obtain that every irreducible representation of $G$ is a subrepresentation of the regular representation (this is basically the same as the usual argument using the adjointness of induction and restriction).
