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Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ and $S$ be two smooth varieties over $k$ and $\mathcal F$ a constructible \'etale sheaf of $\mathbb F_\ell$-modules on $X$ ($\ell\neq p$). Let $\pi\colon X\rightarrow S$ be a smooth morphism.

Let $x\in X$ be a closed point and $s=\pi(x)$. Let $f\colon X\rightarrow \mathbb A_S^1$ be a $S$-morphism.

Assume that:

(1) $\pi$ is universally locally acyclic relative to $\mathcal F$.

(2) $f_s\colon X_s\rightarrow \mathbb A_s^1$ is universally locally acyclic relative to $\mathcal F|_{X_s}$ for all $s\in S$.

My question is that:

Is $f$ locally acyclic relative to $\mathcal F$? Can you prove this or give a counterexample.

How about this simple case: $S=\mathrm{Spec}~k[T_0]$ and $X=\mathrm{Spec}~k[T_0,T_1]$ or $X=\mathrm{Spec}~k[T_0,T_1,T_2]$?

For the definition of locally acyclic we refer to SGA 4.5, P. Deligne, Theoremes de finitude en cohomologie $\ell$-adique.

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  • $\begingroup$ This is not true. Take $S=\mathbb A_k^1, X=\mathbb A_S^1$ and $f=identity$. Let $\mathcal F$ is Artin-Schreier sheaf on $X$.... $\endgroup$
    – ely
    Commented Oct 14, 2015 at 14:21
  • $\begingroup$ Dear @ely, An Artin-Schreier sheaf is a lisse sheaf. Isn't every lisse sheaf universally locally acyclic relative to the identity? $\endgroup$
    – Doug Liu
    Commented Oct 1 at 5:19

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