## 29.37 Relatively ample sheaves

Let $X$ be a scheme and $\mathcal{L}$ an invertible sheaf on $X$. Then $\mathcal{L}$ is ample on $X$ if $X$ is quasi-compact and every point of $X$ is contained in an affine open of the form $X_ s$, where $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ and $n \geq 1$, see Properties, Definition 28.26.1. We turn this into a relative notion as follows.

Definition 29.37.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. We say $\mathcal{L}$ is relatively ample, or $f$-relatively ample, or ample on $X/S$, or $f$-ample if $f : X \to S$ is quasi-compact, and if for every affine open $V \subset S$ the restriction of $\mathcal{L}$ to the open subscheme $f^{-1}(V)$ of $X$ is ample.

We note that the existence of a relatively ample sheaf on $X$ does not force the morphism $X \to S$ to be of finite type.

Lemma 29.37.2. Let $X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $n \geq 1$. Then $\mathcal{L}$ is $f$-ample if and only if $\mathcal{L}^{\otimes n}$ is $f$-ample.

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

Lemma 29.37.3. Let $f : X \to S$ be a morphism of schemes. If there exists an $f$-ample invertible sheaf, then $f$ is separated.

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. $\square$

There are many ways to characterize relatively ample invertible sheaves, analogous to the equivalent conditions in Properties, Proposition 28.26.13. We will add these here as needed.

Lemma 29.37.4. Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. The following are equivalent:

1. The invertible sheaf $\mathcal{L}$ is $f$-ample.

2. There exists an open covering $S = \bigcup V_ i$ such that each $\mathcal{L}|_{f^{-1}(V_ i)}$ is ample relative to $f^{-1}(V_ i) \to V_ i$.

3. There exists an affine open covering $S = \bigcup V_ i$ such that each $\mathcal{L}|_{f^{-1}(V_ i)}$ is ample.

4. There exists a quasi-coherent graded $\mathcal{O}_ S$-algebra $\mathcal{A}$ and a map of graded $\mathcal{O}_ X$-algebras $\psi : f^*\mathcal{A} \to \bigoplus _{d \geq 0} \mathcal{L}^{\otimes d}$ such that $U(\psi ) = X$ and

$r_{\mathcal{L}, \psi } : X \longrightarrow \underline{\text{Proj}}_ S(\mathcal{A})$

is an open immersion (see Constructions, Lemma 27.19.1 for notation).

5. The morphism $f$ is quasi-separated and part (4) above holds with $\mathcal{A} = f_*(\bigoplus _{d \geq 0} \mathcal{L}^{\otimes d})$ and $\psi$ the adjunction mapping.

6. Same as (4) but just requiring $r_{\mathcal{L}, \psi }$ to be an immersion.

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. $\square$

Lemma 29.37.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $S$ affine. Then $\mathcal{L}$ is $f$-relatively ample if and only if $\mathcal{L}$ is ample on $X$.

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

Lemma 29.37.6. Let $f : X \to S$ be a morphism of schemes. Then $f$ is quasi-affine if and only if $\mathcal{O}_ X$ is $f$-relatively ample.

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

Lemma 29.37.7. Let $f : X \to Y$ be a morphism of schemes, $\mathcal{M}$ an invertible $\mathcal{O}_ Y$-module, and $\mathcal{L}$ an invertible $\mathcal{O}_ X$-module.

1. If $\mathcal{L}$ is $f$-ample and $\mathcal{M}$ is ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample for $a \gg 0$.

2. If $\mathcal{M}$ is ample and $f$ quasi-affine, then $f^*\mathcal{M}$ is ample.

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. 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 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). $\square$

Lemma 29.37.8. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. Let $\mathcal{M}$ be an invertible $\mathcal{O}_ Y$-module. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $S$ is quasi-compact, $\mathcal{M}$ is $g$-ample, and $\mathcal{L}$ is $f$-ample, then $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is $g \circ f$-ample for $a \gg 0$.

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. $\square$

Lemma 29.37.9. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $S' \to S$ be a morphism of schemes. Let $f' : X' \to S'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$. If $\mathcal{L}$ is $f$-ample, then $\mathcal{L}'$ is $f'$-ample.

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. $\square$

Lemma 29.37.10. Let $g : Y \to S$ and $f : X \to Y$ be morphisms of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $\mathcal{L}$ is $g \circ f$-ample and $f$ is quasi-compact1 then $\mathcal{L}$ is $f$-ample.

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. $\square$

[1] This follows if $g$ is quasi-separated by Schemes, Lemma 26.21.14.

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