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?

  • $\begingroup$ I assume you mean $L$ is a direct factor of $\pi_*\mbb{C}$ restricted to $U$? $\endgroup$ – Daniel Litt Jul 3 '13 at 21:43
  • $\begingroup$ In "Faisceaux pervers" BBD write direct factor of $\pi_\ast \mathbb{C}$ but I guess they really mean the restriction to U. $\endgroup$ – user36461 Jul 3 '13 at 22:07

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).

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  • $\begingroup$ Thanks! I don't understand what's $n$ in your answer, the dimension of $X$ or the order of the monodromy? $\endgroup$ – user36461 Jul 3 '13 at 19:33
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    $\begingroup$ Two other question: what's the ramification of the cover you describe? And are you really considering $L$ when you write $Hom(L, \pi_\ast \underline{\mathbb{C}}^n)$. This objets live in different places: L only on U, $\pi_\ast \underline{\mathbb{C}}^n$, so I guess some extension of $L$ to $X$ is needed. $\endgroup$ – user36461 Jul 3 '13 at 19:38
  • $\begingroup$ $n$ is the rank of $L$, and hopefully this edit fixes the error you mention. Sorry, read the question too hastily. $\endgroup$ – Daniel Litt Jul 3 '13 at 21:43
  • $\begingroup$ And the cover might be ramified outside of $U$; it is \'etale over $U$. $\endgroup$ – Daniel Litt Jul 3 '13 at 21:44

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