The Stacks project

Remark 66.11.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of decent algebraic spaces over $S$. Let $x \in |X|$ with image $y \in |Y|$. Choose an elementary étale neighbourhood $(V, v) \to (Y, y)$ (possible by Lemma 66.11.4). Then $V \times _ Y X$ is an algebraic space étale over $X$ which has a unique point $x'$ mapping to $x$ in $X$ and to $v$ in $V$. (Details omitted; use that all points can be represented by monomorphisms from spectra of fields.) Choose an elementary étale neighbourhood $(U, u) \to (V \times _ Y X, x')$. Then we obtain the following commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}^ h) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}) \ar[r] \ar[d] & U \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}^ h) \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{V, v}) \ar[r] & V \ar[r] & Y } \]

This comes from the identifications $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$, $\mathcal{O}_{X, x}^ h = \mathcal{O}_{U, u}^ h$, $\mathcal{O}_{Y, \overline{y}} = \mathcal{O}_{V, v}^{sh}$, $\mathcal{O}_{Y, y}^ h = \mathcal{O}_{V, v}^ h$ see in Lemma 66.11.8 and Properties of Spaces, Lemma 64.22.1 and the functoriality of the (strict) henselization discussed in Algebra, Sections 10.153 and 10.154.


Comments (0)


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Your email address will not be published.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 0EPL. Beware of the difference between the letter 'O' and the digit '0'.