Definition 66.19.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

An

*étale neighborhood*of $\overline{x}$ of $X$ is a commutative diagram\[ \xymatrix{ & U \ar[d]^\varphi \\ {\bar x} \ar[r]^{\bar x} \ar[ur]^{\bar u} & X } \]where $\varphi $ is an étale morphism of algebraic spaces over $S$. We will use the notation $\varphi : (U, \overline{u}) \to (X, \overline{x})$ to indicate this situation.

A

*morphism of étale neighborhoods*$(U, \overline{u}) \to (U', \overline{u}')$ is an $X$-morphism $h : U \to U'$ such that $\overline{u}' = h \circ \overline{u}$.

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