The Stacks project

Proof. Combine Lemma 29.10.2 with the remark that $X \to S$ is separated if and only if the image of $\Delta _{X/S}$ is closed in $X \times _ S X$, see Schemes, Definition 26.21.3 and the discussion following it. $\square$

Comments (3)

Comment #7350 by Hao Peng on

I believe more general is true that any injective morphism is surjective. I am not sure if this is stated somewhere?

Comment #7351 by Hao Peng on

The last comment contains a mistake, surjective should be separated.

Comment #7358 by Zhiyu Z on

For Comment #7350-#7351: any injective morphism of schemes is separated, see 26.23.6.

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