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29.40 Quasi-projective morphisms

The discussion in the previous section suggests the following definitions. We take our definition of quasi-projective from [EGA]. The version with the letter “H” is the definition in [H].

reference

Definition 29.40.1. Let $f : X \to S$ be a morphism of schemes.

  1. We say $f$ is quasi-projective if $f$ is of finite type and there exists an $f$-relatively ample invertible $\mathcal{O}_ X$-module.

  2. We say $f$ is H-quasi-projective if there exists a quasi-compact immersion $X \to \mathbf{P}^ n_ S$ over $S$ for some $n$.1

  3. We say $f$ is locally quasi-projective if there exists an open covering $S = \bigcup V_ j$ such that each $f^{-1}(V_ j) \to V_ j$ is quasi-projective.

As this definition suggests the property of being quasi-projective is not local on $S$. At a later stage we will be able to say more about the category of quasi-projective schemes, see More on Morphisms, Section 37.45.

Lemma 29.40.2. A base change of a quasi-projective morphism is quasi-projective.

Lemma 29.40.3. Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. If $S$ is quasi-compact and $f$ and $g$ are quasi-projective, then $g \circ f$ is quasi-projective.

Lemma 29.40.4. Let $f : X \to S$ be a morphism of schemes. If $f$ is quasi-projective, or H-quasi-projective or locally quasi-projective, then $f$ is separated of finite type.

Proof. Omitted. $\square$

Proof. Omitted. $\square$

Lemma 29.40.6. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. The morphism $f$ is locally quasi-projective.

  2. There exists an open covering $S = \bigcup V_ j$ such that each $f^{-1}(V_ j) \to V_ j$ is H-quasi-projective.

Proof. By Lemma 29.40.5 we see that (2) implies (1). Assume (1). The question is local on $S$ and hence we may assume $S$ is affine, $X$ of finite type over $S$ and $\mathcal{L}$ is a relatively ample invertible sheaf on $X/S$. By Lemma 29.39.4 we may assume $\mathcal{L}$ is ample on $X$. By Lemma 29.39.3 we see that there exists an immersion of $X$ into a projective space over $S$, i.e., $X$ is H-quasi-projective over $S$ as desired. $\square$

Proof. This follows from Lemma 29.37.6. $\square$

Lemma 29.40.8. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. If $g \circ f$ is quasi-projective and $f$ is quasi-compact2, then $f$ is quasi-projective.

Proof. Observe that $f$ is of finite type by Lemma 29.15.8. Thus the lemma follows from Lemma 29.37.10 and the definitions. $\square$

[1] This is not exactly the same as the definition in Hartshorne. Namely, the definition in Hartshorne (8th corrected printing, 1997) is that $f$ should be the composition of an open immersion followed by a H-projective morphism (see Definition 29.43.1), which does not imply $f$ is quasi-compact. See Lemma 29.43.11 for the implication in the other direction.
[2] This follows if $g$ is quasi-separated by Schemes, Lemma 26.21.14.

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