The Stacks project

Proposition 41.6.1. Sections of unramified morphisms.

  1. Any section of an unramified morphism is an open immersion.

  2. Any section of a separated morphism is a closed immersion.

  3. Any section of an unramified separated morphism is open and closed.

Proof. Fix a base scheme $S$. If $f : X' \to X$ is any $S$-morphism, then the graph $\Gamma _ f : X' \to X' \times _ S X$ is obtained as the base change of the diagonal $\Delta _{X/S} : X \to X \times _ S X$ via the projection $X' \times _ S X \to X \times _ S X$. If $g : X \to S$ is separated (resp. unramified) then the diagonal is a closed immersion (resp. open immersion) by Schemes, Definition 26.21.3 (resp. Morphisms, Lemma 29.34.13). Hence so is the graph as a base change (by Schemes, Lemma 26.18.2). In the special case $X' = S$, we obtain (1), resp. (2). Part (3) follows on combining (1) and (2). $\square$


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