The Stacks project

Lemma 58.10.2. Let $(A, \mathfrak m)$ be a local ring. Set $X = \mathop{\mathrm{Spec}}(A)$ and let $U = X \setminus \{ \mathfrak m\} $. If the punctured spectrum of the strict henselization of $A$ is connected, then

\[ \textit{FÉt}_ X \longrightarrow \textit{FÉt}_ U,\quad Y \longmapsto Y \times _ X U \]

is a fully faithful functor.

Proof. Assume $A$ is strictly henselian. In this case any finite étale cover $Y$ of $X$ is isomorphic to a finite disjoint union of copies of $X$. Thus it suffices to prove that any morphism $U \to U \amalg \ldots \amalg U$ over $U$, extends uniquely to a morphism $X \to X \amalg \ldots \amalg X$ over $X$. If $U$ is connected (in particular nonempty), then this is true.

The general case. Since the category of finite étale coverings has an internal hom (Lemma 58.5.4) it suffices to prove the following: Given $Y$ finite étale over $X$ any morphism $s : U \to Y$ over $X$ extends to a morphism $t : X \to Y$ over $X$. Let $A^{sh}$ be the strict henselization of $A$ and denote $X^{sh} = \mathop{\mathrm{Spec}}(A^{sh})$, $U^{sh} = U \times _ X X^{sh}$, $Y^{sh} = Y \times _ X X^{sh}$. By the first paragraph and our assumption on $A$, we can extend the base change $s^{sh} : U^{sh} \to Y^{sh}$ of $s$ to $t^{sh} : X^{sh} \to Y^{sh}$. Set $A' = A^{sh} \otimes _ A A^{sh}$. Then the two pullbacks $t'_1, t'_2$ of $t^{sh}$ to $X' = \mathop{\mathrm{Spec}}(A')$ are extensions of the pullback $s'$ of $s$ to $U' = U \times _ X X'$. As $A \to A'$ is flat we see that $U' \subset X'$ is (topologically) dense by going down for $A \to A'$ (Algebra, Lemma 10.39.19). Thus $t'_1 = t'_2$ by Lemma 58.10.1. Hence $t^{sh}$ descends to a morphism $t : X \to Y$ for example by Descent, Lemma 35.13.7. $\square$


Comments (2)

Comment #5350 by Jackson on

Shouldn't the extension of be a morphism over ?

There are also:

  • 2 comment(s) on Section 58.10: Local connectedness

Post a comment

Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BLQ. Beware of the difference between the letter 'O' and the digit '0'.