Loading [MathJax]/extensions/tex2jax.js

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. $\square$


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.