The Stacks project

Definition 67.11.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in X$ be a point. An elementary étale neighbourhood is an étale morphism $(U, u) \to (X, x)$ where $U$ is a scheme, $u \in U$ is a point mapping to $x$, and $\kappa (x) \to \kappa (u)$ is an isomorphism. A morphism of elementary étale neighbourhoods $(U, u) \to (U', u')$ is defined as a morphism $U \to U'$ over $X$ mapping $u$ to $u'$.

Comments (2)

Comment #7786 by Laurent Moret-Bailly on

The problem with this phrasing is that does not make sense unless is a "good" point. In fact, it suffices to assume that is represented by a monomorphism; then the rest of the section remains valid.

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