Definition 59.29.1. Let S be a scheme.
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.
We say \overline{s} lies over s to indicate that s \in S is the image of \overline{s}.
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}).
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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