Definition 59.29.1. Let $S$ be a scheme.

1. A geometric point of $S$ is a morphism $\mathop{\mathrm{Spec}}(k) \to S$ where $k$ is algebraically closed. Such a point is usually denoted $\overline{s}$, i.e., by an overlined small case letter. We often use $\overline{s}$ to denote the scheme $\mathop{\mathrm{Spec}}(k)$ as well as the morphism, and we use $\kappa (\overline{s})$ to denote $k$.

2. We say $\overline{s}$ lies over $s$ to indicate that $s \in S$ is the image of $\overline{s}$.

3. An étale neighborhood of a geometric point $\overline{s}$ of $S$ is a commutative diagram

$\xymatrix{ & U \ar[d]^\varphi \\ {\overline{s}} \ar[r]^{\overline{s}} \ar[ur]^{\bar u} & S }$

where $\varphi$ is an étale morphism of schemes. We write $(U, \overline{u}) \to (S, \overline{s})$.

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

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