The Stacks project

Remark 32.12.2. In the situation of Chow's Lemma 32.12.1:

  1. The morphism $\pi $ is actually H-projective (hence projective, see Morphisms, Lemma 29.43.3) since the morphism $X' \to \mathbf{P}^ n_ S \times _ S X = \mathbf{P}^ n_ X$ is a closed immersion (use the fact that $\pi $ is proper, see Morphisms, Lemma 29.41.7).

  2. We may assume that $X'$ is reduced as we can replace $X'$ by its reduction without changing the other assertions of the lemma.

  3. We may assume that $X' \to X$ is of finite presentation without changing the other assertions of the lemma. This can be deduced from the proof of Lemma 32.12.1 but we can also prove this directly as follows. By (1) we have a closed immersion $X' \to \mathbf{P}^ n_ X$. By Lemma 32.9.4 we can write $X' = \mathop{\mathrm{lim}}\nolimits X'_ i$ where $X'_ i \to \mathbf{P}^ n_ X$ is a closed immersion of finite presentation. In particular $X'_ i \to X$ is of finite presentation, proper, and surjective. For large enough $i$ the morphism $X'_ i \to \mathbf{P}^ n_ S$ is an immersion by Lemma 32.4.16. Replacing $X'$ by $X'_ i$ we get what we want.

Of course in general we can't simultaneously achieve both (2) and (3).

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