The Stacks project

Remark 67.11.11. 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 67.11.4). 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 67.11.8 and Properties of Spaces, Lemma 65.22.1 and the functoriality of the (strict) henselization discussed in Algebra, Sections 10.154 and 10.155.

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