The Stacks project

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

  1. $f$ is locally acyclic at $\overline{x}$ relative to $K$ and

  2. $g$ 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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