The Stacks project

Proof. This follows from Properties, Lemma 28.26.2. $\square$

Proof. Being separated is local on the base (see Schemes, Lemma 26.21.7 for example; it also follows easily from the definition). Hence we may assume $S$ is affine and $X$ has an ample invertible sheaf. In this case the result follows from Properties, Lemma 28.26.8. $\square$

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. $\square$

Proof. Immediate from Lemma 29.37.4 and the definitions. $\square$

Proof. Follows from Properties, Lemma 28.27.1 and the definitions. $\square$

Proof. Assume $\mathcal{L}$ is $f$-ample and $\mathcal{M}$ ample. By assumption $Y$ and $f$ are quasi-compact (see Definition 29.37.1 and Properties, Definition 28.26.1). Hence $X$ is quasi-compact. By Properties, Lemma 28.26.8 the scheme $Y$ is separated and by Lemma 29.37.3 the morphism $f$ is separated. Hence $X$ is separated by Schemes, Lemma 26.21.12. Pick $x \in X$. We can choose $m \geq 1$ and $t \in \Gamma (Y, \mathcal{M}^{\otimes m})$ such that $Y_ t$ is affine and $f(x) \in Y_ t$. Since $\mathcal{L}$ restricts to an ample invertible sheaf on $f^{-1}(Y_ t) = X_{f^*t}$ we can choose $n \geq 1$ and $s \in \Gamma (X_{f^*t}, \mathcal{L}^{\otimes n})$ with $x \in (X_{f^*t})_ s$ with $(X_{f^*t})_ s$ affine. By Properties, Lemma 28.17.2 part (2) whose assumptions are satisfied by the above, there exists an integer $e \geq 1$ and a section $s' \in \Gamma (X, \mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes em})$ which restricts to $s(f^*t)^ e$ on $X_{f^*t}$. For any $b > 0$ consider the section $s'' = s'(f^*t)^ b$ of $\mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes (e + b)m}$. Then $X_{s''} = (X_{f^*t})_ s$ is an affine open of $X$ containing $x$. Picking $b$ such that $n$ divides $e + b$ we see $\mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes (e + b)m}$ is the $n$th power of $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ for some $a$ and we can get any $a$ divisible by $m$ and big enough. Since $X$ is quasi-compact a finite number of these affine opens cover $X$. We conclude that for some $a$ sufficiently divisible and large enough the invertible sheaf $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $X$. On the other hand, we know that $\mathcal{M}^{\otimes c}$ (and hence its pullback to $X$) is globally generated for all $c \gg 0$ by Properties, Proposition 28.26.13. Thus $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a + c}$ is ample (Properties, Lemma 28.26.5) for $c \gg 0$ and (1) is proved.

Part (2) follows from Lemma 29.37.6, Properties, Lemma 28.26.2, and part (1). $\square$

Proof. Let $S = \bigcup _{i = 1, \ldots , n} V_ i$ be a finite affine open covering. By Lemma 29.37.4 it suffices to prove that $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $(g \circ f)^{-1}(V_ i)$ for $i = 1, \ldots , n$. Thus the lemma follows from Lemma 29.37.7. $\square$

Proof. By Lemma 29.37.4 it suffices to find an affine open covering $S' = \bigcup U'_ i$ such that $\mathcal{L}'$ restricts to an ample invertible sheaf on $(f')^{-1}(U_ i')$ for all $i$. We may choose $U'_ i$ mapping into an affine open $U_ i \subset S$. In this case the morphism $(f')^{-1}(U'_ i) \to f^{-1}(U_ i)$ is affine as a base change of the affine morphism $U'_ i \to U_ i$ (Lemma 29.11.8). Thus $\mathcal{L}'|_{(f')^{-1}(U'_ i)}$ is ample by Lemma 29.37.7. $\square$

Proof. Assume $f$ is quasi-compact and $\mathcal{L}$ is $g \circ f$-ample. Let $U \subset S$ be an affine open and let $V \subset Y$ be an affine open with $g(V) \subset U$. Then $\mathcal{L}|_{(g \circ f)^{-1}(U)}$ is ample on $(g \circ f)^{-1}(U)$ by assumption. Since $f^{-1}(V) \subset (g \circ f)^{-1}(U)$ we see that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)$ by Properties, Lemma 28.26.14. Namely, $f^{-1}(V) \to (g \circ f)^{-1}(U)$ is a quasi-compact open immersion by Schemes, Lemma 26.21.14 as $(g \circ f)^{-1}(U)$ is separated (Properties, Lemma 28.26.8) and $f^{-1}(V)$ is quasi-compact (as $f$ is quasi-compact). Thus we conclude that $\mathcal{L}$ is $f$-ample by Lemma 29.37.4. $\square$