The Stacks project

Proof. It is immediate from the definition that (1) implies (2) and (2) implies (3). It is clear that (5) implies (4).

Assume (3) holds for the affine open covering $S = \bigcup V_ i$. We are going to show (5) holds. Since each $f^{-1}(V_ i)$ has an ample invertible sheaf we see that $f^{-1}(V_ i)$ is separated (Properties, Lemma 28.26.8). Hence $f$ is separated. By Schemes, Lemma 26.24.1 we see that $\mathcal{A} = f_*(\bigoplus _{d \geq 0} \mathcal{L}^{\otimes d})$ is a quasi-coherent graded $\mathcal{O}_ S$-algebra. Denote $\psi : f^*\mathcal{A} \to \bigoplus _{d \geq 0} \mathcal{L}^{\otimes d}$ the adjunction mapping. The description of the open $U(\psi )$ in Constructions, Section 27.19 and the definition of ampleness of $\mathcal{L}|_{f^{-1}(V_ i)}$ show that $U(\psi ) = X$. Moreover, Constructions, Lemma 27.19.1 part (3) shows that the restriction of $r_{\mathcal{L}, \psi }$ to $f^{-1}(V_ i)$ is the same as the morphism from Properties, Lemma 28.26.9 which is an open immersion according to Properties, Lemma 28.26.11. Hence (5) holds.

Let us show that (4) implies (1). Assume (4). Denote $\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ the structure morphism. Choose $V \subset S$ affine open. By Constructions, Definition 27.16.7 we see that $\pi ^{-1}(V) \subset \underline{\text{Proj}}_ S(\mathcal{A})$ is equal to $\text{Proj}(A)$ where $A = \mathcal{A}(V)$ as a graded ring. Hence $r_{\mathcal{L}, \psi }$ maps $f^{-1}(V)$ isomorphically onto a quasi-compact open of $\text{Proj}(A)$. Moreover, $\mathcal{L}^{\otimes d}$ is isomorphic to the pullback of $\mathcal{O}_{\text{Proj}(A)}(d)$ for some $d \geq 1$. (See part (3) of Constructions, Lemma 27.19.1 and the final statement of Constructions, Lemma 27.14.1.) This implies that $\mathcal{L}|_{f^{-1}(V)}$ is ample by Properties, Lemmas 28.26.12 and 28.26.2.

Assume (6). By the equivalence of (1) - (5) above we see that the property of being relatively ample on $X/S$ is local on $S$. Hence we may assume that $S$ is affine, and we have to show that $\mathcal{L}$ is ample on $X$. In this case the morphism $r_{\mathcal{L}, \psi }$ is identified with the morphism, also denoted $r_{\mathcal{L}, \psi } : X \to \text{Proj}(A)$ associated to the map $\psi : A = \mathcal{A}(V) \to \Gamma _*(X, \mathcal{L})$. (See references above.) As above we also see that $\mathcal{L}^{\otimes d}$ is the pullback of the sheaf $\mathcal{O}_{\text{Proj}(A)}(d)$ for some $d \geq 1$. Moreover, since $X$ is quasi-compact we see that $X$ gets identified with a closed subscheme of a quasi-compact open subscheme $Y \subset \text{Proj}(A)$. By Constructions, Lemma 27.10.6 (see also Properties, Lemma 28.26.12) we see that $\mathcal{O}_ Y(d')$ is an ample invertible sheaf on $Y$ for some $d' \geq 1$. Since the restriction of an ample sheaf to a closed subscheme is ample, see Properties, Lemma 28.26.3 we conclude that the pullback of $\mathcal{O}_ Y(d')$ is ample. Combining these results with Properties, Lemma 28.26.2 we conclude that $\mathcal{L}$ is ample as desired. $\square$