The Stacks project

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

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). $\square$