37.50 Projective schemes
This section is the analogue of Section 37.49 for projective morphisms.
Lemma 37.50.1. Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \to S$ be a morphism of schemes. The following are equivalent
$X \to S$ is projective,
$X \to S$ is H-projective,
$X \to S$ is quasi-projective and proper,
$X \to S$ is H-quasi-projective and proper,
$X \to S$ is proper and $X$ has an ample invertible sheaf,
$X \to S$ is proper and there exists an $f$-ample invertible sheaf,
$X \to S$ is proper and there exists an $f$-very ample invertible sheaf,
there is a quasi-coherent graded $\mathcal{O}_ S$-algebra $\mathcal{A}$ generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ with $\mathcal{A}_1$ a finite type $\mathcal{O}_ S$-module such that $X = \underline{\text{Proj}}_ S(\mathcal{A})$.
Proof.
Observe first that in each case the morphism $f$ is proper, see Morphisms, Lemmas 29.43.3 and 29.43.5. Hence it suffices to prove the equivalence of the notions in case $f$ is a proper morphism. We will use this without further mention in the following.
The equivalences (1) $\Leftrightarrow $ (3) and (2) $\Leftrightarrow $ (4) are Morphisms, Lemma 29.43.13.
The implication (2) $\Rightarrow $ (1) is Morphisms, Lemma 29.43.3.
The implications (1) $\Rightarrow $ (2) and (3) $\Rightarrow $ (4) are Morphisms, Lemma 29.43.16.
The implication (1) $\Rightarrow $ (7) is immediate from Morphisms, Definitions 29.43.1 and 29.38.1.
The conditions (3) and (6) are equivalent by Morphisms, Definition 29.40.1.
Thus (1) – (4), (6) are equivalent and imply (7). By Lemma 37.49.1 conditions (3), (5), and (7) are equivalent. Thus we see that (1) – (7) are equivalent.
By Divisors, Lemma 31.30.5 we see that (8) implies (1). Conversely, if (2) holds, then we can choose a closed immersion
\[ i : X \longrightarrow \mathbf{P}^ n_ S = \underline{\text{Proj}}_ S(\mathcal{O}_ S[T_0, \ldots , T_ n]). \]
See Constructions, Lemma 27.21.5 for the equality. By Divisors, Lemma 31.31.1 we see that $X$ is the relative Proj of a quasi-coherent graded quotient algebra $\mathcal{A}$ of $\mathcal{O}_ S[T_0, \ldots , T_ n]$. Then $\mathcal{A}$ satisfies the conditions of (8).
$\square$
Lemma 37.50.2. Let $S$ be a scheme which has an ample invertible sheaf. Let $\text{P}_ S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma 37.50.1.
if $S' \to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\text{P}_ S \to \text{P}_{S'}$,
if $X \in \text{P}_ S$ and $Y \in \text{P}_ X$, then $Y \in \text{P}_ S$,
the category $\text{P}_ S$ is closed under fibre products,
the category $\text{P}_ S$ is closed under finite disjoint unions,
if $X \to S$ is finite, then $X$ is in $\text{P}_ S$,
add more here.
Proof.
Part (1) follows from Morphisms, Lemma 29.43.9.
Part (2) follows from the fifth characterization of Lemma 37.50.1 and the fact that compositions of proper morphisms are proper (Morphisms, Lemma 29.41.4).
If $X \to S$ and $Y \to S$ are projective, then $X \times _ S Y \to Y$ is projective by Morphisms, Lemma 29.43.9. Hence (3) follows from (2).
If $X = Y \amalg Z$ is a disjoint union of schemes and $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module such that $\mathcal{L}|_ Y$ and $\mathcal{L}|_ Z$ are ample, then $\mathcal{L}$ is ample (details omitted). Thus part (4) follows from the fifth characterization of Lemma 37.50.1.
Part (5) follows from Morphisms, Lemma 29.44.16.
$\square$
Here is a slightly different type of result.
reference
Lemma 37.50.3. Let $f : X \to Y$ be a proper morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_ y$ is ample on $X_ y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.
Proof.
We may assume $Y$ is affine. Then we find a directed set $I$ and an inverse system of morphisms $X_ i \to Y_ i$ of schemes with $Y_ i$ of finite type over $\mathbf{Z}$, with affine transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$, with $X_ i \to Y_ i$ proper, such that $X \to Y = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$. See Limits, Lemma 32.13.3. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_ i}$-modules $\mathcal{L}_ i$ pulling back to $\mathcal{L}$, see Limits, Lemma 32.10.3. Let $y_ i \in Y_ i$ be the image of $y$. Then $\kappa (y) = \mathop{\mathrm{colim}}\nolimits \kappa (y_ i)$. Hence for some $i$ we have $\mathcal{L}_{i, y_ i}$ is ample on $X_{i, y_ i}$ by Limits, Lemma 32.4.15. By Cohomology of Schemes, Lemma 30.21.4 we find an open neighbourhood $V_ i \subset Y_ i$ of $y_ i$ such that $\mathcal{L}_ i$ restricted to $f_ i^{-1}(V_ i)$ is ample relative to $V_ i$. Letting $V \subset Y$ be the inverse image of $V_ i$ finishes the proof (hints: use Morphisms, Lemma 29.37.9 and the fact that $X \to Y \times _{Y_ i} X_ i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 29.37.7).
$\square$
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