Lemma 37.49.1. 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

1. $X \to S$ is projective,

2. $X \to S$ is H-projective,

3. $X \to S$ is quasi-projective and proper,

4. $X \to S$ is H-quasi-projective and proper,

5. $X \to S$ is proper and $X$ has an ample invertible sheaf,

6. $X \to S$ is proper and there exists an $f$-ample invertible sheaf,

7. $X \to S$ is proper and there exists an $f$-very ample invertible sheaf,

8. 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 37.48.1 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). $\square$

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