Lemma 59.93.4. Let
\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S }
be a cartesian diagram of schemes. Let K be an object of D(X_{\acute{e}tale}). Let \overline{x}' be a geometric point of X' with image \overline{x} in X. If
f is locally acyclic at \overline{x} relative to K and
g is locally quasi-finite, or S' = \mathop{\mathrm{lim}}\nolimits S_ i is a directed inverse limit of schemes locally quasi-finite over S with affine transition morphisms, or g : S' \to S is integral,
then f' locally acyclic at \overline{x}' relative to (g')^{-1}K.
Proof.
Denote \overline{s}' and \overline{s} the images of \overline{x}' and \overline{x} in S' and S. Let \overline{t}' be a geometric point of the spectrum of \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S', \overline{s}'}) and denote \overline{t} the image in \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}). By Algebra, Lemma 10.156.6 and our assumptions on g we have
\mathcal{O}^{sh}_{X, \overline{x}} \otimes _{\mathcal{O}^{sh}_{S, \overline{s}}} \mathcal{O}^{sh}_{S', \overline{s}'} \longrightarrow \mathcal{O}^{sh}_{X', \overline{x}'}
is an isomorphism. Since by our conventions \kappa (\overline{t}) = \kappa (\overline{t}') we conclude that
F_{\overline{x}', \overline{t}'} = \mathop{\mathrm{Spec}}\left( \mathcal{O}^{sh}_{X', \overline{x}'} \otimes _{\mathcal{O}^{sh}_{S', \overline{s}'}} \kappa (\overline{t}')\right) = \mathop{\mathrm{Spec}}\left( \mathcal{O}^{sh}_{X, \overline{x}} \otimes _{\mathcal{O}^{sh}_{S, \overline{s}}} \kappa (\overline{t})\right) = F_{\overline{x}, \overline{t}}
In other words, the varieties of vanishing cycles of f' at \overline{x}' are examples of varieties of vanishing cycles of f at \overline{x}. The lemma follows immediately from this and the definitions.
\square
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