Let $S$ be the spectrum of a Henselian discrete valuation ring (called a Henselian trait). Let $f:X\to S$ be a finite type, separated morphism of schemes. Let $\eta\in S$ be the generic point. Let $s\in S$ be the closed point. Let $\ell$ be a prime invertible on $S$. Consider $\mathbb{Q}_{\ell}$-étale sheaves. Then there is the nearby cycles functor $R\Psi_f:D_c^b(X_{\eta})\to D_c^b(X_{\bar{s}})$ between the triangulated categories of constructible sheaves. (Section 1.2, Grothendieck and vanishing cycles, by Illusie). Does this functor factor through $D_c^b(X_s)$?
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$\begingroup$ The nearby cycles functor admits a natural action of the Galois group of $\eta$. By descent, factoring through $X_s$ should be the same as giving a natural action of the Galois group of $s$. So we should be able to do this if and only if the canonical surjection from the Galois group of $\eta$ to the Galois group of $s$ admits a section, which it does in the case of finite residue field but I think not always. However, I don't know what such a factorization would be good for. $\endgroup$– Will SawinCommented May 6 at 10:12
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$\begingroup$ Dear @WillSawin, Thank you for your comment! How do you see that the existence of factorization implies the section of Galois groups? Chances may be that the action of $G_{\eta}$ descend to a $G_s$-action, without a section $G_s\to G_{\eta}$. $\endgroup$– Doug LiuCommented May 6 at 10:17
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$\begingroup$ No, every finite-dimensional representation of $G_{\eta}$ occurs on the stalk of a nearby cycles sheaf - just take a section $S \to X$ and pushforward along the composition of the section with the inclusion of the generic point of $S$ a locally constant sheaf corresponding to that Galois representation. $\endgroup$– Will SawinCommented May 6 at 10:32
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$\begingroup$ I see. Thanks a lot! $\endgroup$– Doug LiuCommented May 6 at 10:51
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