The Stacks project

Remark 59.29.3. Let $S$ be a scheme and $s \in S$ a point. In More on Morphisms, Definition 37.35.1 we defined the notion of an étale neighbourhood $(U, u) \to (S, s)$ of $(S, s)$. If $\overline{s}$ is a geometric point of $S$ lying over $s$, then any étale neighbourhood $(U, \overline{u}) \to (S, \overline{s})$ gives rise to an étale neighbourhood $(U, u)$ of $(S, s)$ by taking $u \in U$ to be the unique point of $U$ such that $\overline{u}$ lies over $u$. Conversely, given an étale neighbourhood $(U, u)$ of $(S, s)$ the residue field extension $\kappa (u)/\kappa (s)$ is finite separable (see Proposition 59.26.2) and hence we can find an embedding $\kappa (u) \subset \kappa (\overline{s})$ over $\kappa (s)$. In other words, we can find a geometric point $\overline{u}$ of $U$ lying over $u$ such that $(U, \overline{u})$ is an étale neighbourhood of $(S, \overline{s})$. We will use these observations to go between the two types of étale neighbourhoods.