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I hope this question is not too elementary, but I'm a bit lost :

Let $S$ be a finite set of closed points of $\mathbb{P^1_{C}}$ and $j : U \longrightarrow \mathbb{P^1_{C}}$ be the inclusion of the complement. I have an etale $\mathbb{Q}_{\ell}$-sheaf $\mathcal{F}$ on $\mathbb{P^1_{C}}$ such that $j^{*} \mathcal{F}$ is locally constant (hence corresponds to some $\ell$-adic representation $V$ of $\pi_{1,et}(U)$), and I would like to understand the fiber of $j_{*}j^{*} \mathcal{F}$ over a point $s \in S$. Is there a description of this fiber in term of the monodromy representation $V$ ?

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Yes, it is just the invariants under the inertia group corresponding to the point. The reason is that the stalk of $j_*\mathcal{G}$ can be computed as global sections of $\mathcal{G}$ over $\text{Spec}(K^{sh})$ where $K^{sh}$ is the fraction field of the strictly henselian local ring $O_{\mathbf{P}^1, s}^{sh}$ at $s$, see Theorem Tag 03Q9. To be sure, I am using that $s$ is not contained in $U$. Finally, you observe that the absolute Galois group of $K^{sh}$ is the inertia group and use Lemma Tag 04JM.

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  • $\begingroup$ Thanks ! That's exactly what I was looking for. $\endgroup$
    – js21
    Apr 22 '14 at 17:42

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