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Tag 03FR

Chapter 54: Properties of Algebraic Spaces > Section 54.15: Étale morphisms of algebraic spaces

Definition 54.15.2. Let $S$ be a scheme. A morphism $f : X \to Y$ between algebraic spaces over $S$ is called étale if and only if for every étale morphism $\varphi : U \to X$ where $U$ is a scheme, the composition $f \circ \varphi$ is étale also.

    The code snippet corresponding to this tag is a part of the file spaces-properties.tex and is located in lines 2012–2019 (see updates for more information).

    \begin{definition}
    \label{definition-etale}
    Let $S$ be a scheme.
    A morphism $f : X \to Y$ between algebraic spaces over $S$ is
    called {\it \'etale} if and only if for every \'etale morphism
    $\varphi : U \to X$ where $U$ is a scheme, the composition
    $f \circ \varphi$ is \'etale also.
    \end{definition}

    Comments (2)

    Comment #2317 by Nithi Rungtanapirom on December 7, 2016 a 4:45 pm UTC

    In the last clause, the composition $\varphi\circ f$ should read $f\circ\varphi$.

    Comment #2393 by Johan (site) on February 17, 2017 a 1:08 am UTC

    Thanks. Fixed here.

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