Proposition 41.6.1. Sections of unramified morphisms.

Any section of an unramified morphism is an open immersion.

Any section of a separated morphism is a closed immersion.

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

Proposition 41.6.1. Sections of unramified morphisms.

Any section of an unramified morphism is an open immersion.

Any section of a separated morphism is a closed immersion.

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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