Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 58.24.1. In Situation 58.19.1 assume

  1. $A$ has a dualizing complex and is $f$-adically complete,

  2. one of the following is true

    1. $A_ f$ is $(S_2)$ and every irreducible component of $X$ not contained in $X_0$ has dimension $\geq 4$, or

    2. if $\mathfrak p \not\in V(f)$ and $V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} $, then $\text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3$.

  3. for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$.

Then the restriction functor $\textit{FÉt}_ U \to \textit{FÉt}_{U_0}$ is essentially surjective.

Proof. Let $V_0 \to U_0$ be a finite étale morphism. By Lemma 58.23.1 there exists an open $U' \subset U$ containing $U_0$ and a finite étale morphism $V' \to U$ whose base change to $U_0$ is isomorphic to $V_0 \to U_0$. Since $U' \supset U_0$ we see that $U \setminus U'$ consists of points corresponding to prime ideals $\mathfrak p_1, \ldots , \mathfrak p_ n$ as in (3). By assumption we can find finite étale morphisms $V'_ i \to \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i})$ agreeing with $V' \to U'$ over $U' \times _ U \mathop{\mathrm{Spec}}(A_{\mathfrak p_ i})$. By Limits, Lemma 32.20.1 applied $n$ times we see that $V' \to U'$ extends to a finite étale morphism $V \to U$. $\square$


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Your email address will not be published.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.