  ## 28.35 Relatively ample sheaves

Let be a scheme and an invertible sheaf on . Then is ample on if is quasi-compact and every point of is contained in an affine open of the form , where and , see Properties, Definition 27.26.1. We turn this into a relative notion as follows.

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

Lemma 28.35.2. Let be a morphism of schemes. Let be an invertible -module. Let . Then is -ample if and only if is -ample.

Lemma 28.35.3. Let be a morphism of schemes. If there exists an -ample invertible sheaf, then is separated.

Proof. Being separated is local on the base (see Schemes, Lemma 25.21.7 for example; it also follows easily from the definition). Hence we may assume is affine and has an ample invertible sheaf. In this case the result follows from Properties, Lemma 27.26.8.

There are many ways to characterize relatively ample invertible sheaves, analogous to the equivalent conditions in Properties, Proposition 27.26.13. We will add these here as needed.

Lemma 28.35.4. Let be a quasi-compact morphism of schemes. Let be an invertible sheaf on . The following are equivalent:

1. The invertible sheaf is Norwell Men's Carolina Limited Andrew-ample.

2. There exists an open covering such that each is ample relative to .

3. There exists an affine open covering such that each is ample.

4. There exists a quasi-coherent graded -algebra and a map of graded -algebras such that and

is an open immersion (see Constructions, Lemma 26.19.1 for notation).

5. The morphism is quasi-separated and part (4) above holds with and the adjunction mapping.

6. Same as (4) but just requiring 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 . We are going to show (5) holds. Since each has an ample invertible sheaf we see that is separated (Properties, Lemma 27.26.8). Hence is separated. By Schemes, Lemma 25.24.1 we see that is a quasi-coherent graded -algebra. Denote the adjunction mapping. The description of the open in Constructions, Section 26.19 and the definition of ampleness of show that . Moreover, Constructions, Lemma 26.19.1 part (3) shows that the restriction of to is the same as the morphism from Properties, Lemma 27.26.9 which is an open immersion according to Properties, Lemma 27.26.11. Hence (5) holds.

Let us show that (4) implies (1). Assume (4). Denote the structure morphism. Choose affine open. By Constructions, Definition 26.16.7 we see that is equal to where as a graded ring. Hence maps - Sullivan Jerseys John Apparel Jersey Shop Vikings isomorphically onto a quasi-compact open of . Moreover, is isomorphic to the pullback of for some . (See part (3) of Constructions, Lemma 26.19.1 and the final statement of Constructions, Lemma 26.14.1.) This implies that is ample by Properties, Lemmas 27.26.12 and Selanne Anaheim Jersey Ducks Teemu.

Assume (6). By the equivalence of (1) - (5) above we see that the property of being relatively ample on is local on . Hence we may assume that is affine, and we have to show that is ample on . In this case the morphism is identified with the morphism, also denoted associated to the map . (See references above.) As above we also see that is the pullback of the sheaf for some . Moreover, since is quasi-compact we see that gets identified with a closed subscheme of a quasi-compact open subscheme 2017 Durant Jersey Durant Usa Usa Jersey. By Constructions, Lemma 26.10.6 (see also Properties, Lemma 27.26.12) we see that is an ample invertible sheaf on for some . Since the restriction of an ample sheaf to a closed subscheme is ample, see Properties, Lemma Black Shirts Malcolm Jerseys Eagles Jersey Jenkins Authentic we conclude that the pullback of is ample. Combining these results with Properties, Lemma Selanne Anaheim Jersey Ducks Teemu we conclude that is ample as desired.

Lemma 28.35.5. Let be a morphism of schemes. Let be an invertible -module. Assume affine. Then is -relatively ample if and only if is ample on .

Cheap Fashion Black Flag Usa 4 Adidas Jersey Anaheim Ducks Cam Nhl Women's Authentic Fowler Let be a morphism of schemes. Then is quasi-affine if and only if is -relatively ample.

Proof. Follows from Properties, Lemma 27.27.1 and the definitions.

Lemma 28.35.7. Let be a morphism of schemes, an invertible -module, and an invertible -module.

1. If is -ample and is ample, then is ample for .

2. If is ample and quasi-affine, then is ample.

Proof. Assume is -ample and ample. By assumption and are quasi-compact (see Definition 28.35.1 and Properties, Definition 27.26.1). Hence is quasi-compact. Pick . We can choose and such that is affine and . Since restricts to an ample invertible sheaf on we can choose and with with affine. By Properties, Lemma 27.17.2 there exists an integer and a section which restricts to on . For any consider the section of . Then is an affine open of containing . Picking such that divides we see is the th power of for some and we can get any divisible by and big enough. Since is quasi-compact a finite number of these affine opens cover . We conclude that for some sufficiently divisible and large enough the invertible sheaf is ample on . On the other hand, we know that (and hence its pullback to ) is globally generated for all by Properties, Proposition 27.26.13. Thus is ample (Properties, Lemma 27.26.5) for and (1) is proved.

Part (2) follows from Lemma 28.35.6, Properties, Lemma Selanne Anaheim Jersey Ducks Teemu, and part (1).

Lemma 28.35.8. Let and be morphisms of schemes. Let be an invertible -module. Let be an invertible -module. If is quasi-compact, is -ample, and is -ample, then is -ample for .

Team Sleeve Green Jersey Half Color National Football Cameroon Fit Dry it suffices to prove that is ample on for . Thus the lemma follows from Lemma 28.35.7.

Lemma 28.35.9. Let be a morphism of schemes. Let be an invertible -module. Let be a morphism of schemes. Let be the base change of and denote the pullback of to . If is -ample, then is -ample.

Proof. Assume is quasi-compact and is -ample. Let be an affine open and let be an affine open with . Then is ample on by assumption. Since we see that is ample on by Properties, Lemma 27.26.14. Namely, is a quasi-compact open immersion by Schemes, Lemma 25.21.14 as is separated (Properties, Lemma 27.26.8) and is quasi-compact (as is quasi-compact). Thus we conclude that is -ample by Lemma Wholesale Hockey Jerseys Cheap Shop Online Sports.

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