Lemma 29.21.12. Let $f : X \to Y$ be a morphism of schemes with diagonal $\Delta : X \to X \times _ Y X$. If $f$ is locally of finite type then $\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta$ is of finite presentation.

Proof. Note that $\Delta$ is a morphism of schemes over $X$ (via the second projection $X \times _ Y X \to X$). Assume $f$ is locally of finite type. Note that $X$ is of finite presentation over $X$ and $X \times _ Y X$ is locally of finite type over $X$ (by Lemma 29.15.4). Thus the first statement holds by Lemma 29.21.11. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is a monomorphism, hence separated (Schemes, Lemma 26.23.3). $\square$

## Comments (2)

Comment #891 by Matthew Emerton on

At the end of the third sentence, I think it should state that $X\times_Y X$ is locally of finite type over $X$.

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