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28.35 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 2017 Series Stadium Jersey 2017 Series Stadium Jersey n \geq 1, see Properties, Definition 27.26.1. We turn this into a relative notion as follows.

Definition 28.35.1.reference Let f : X \to STennessee Titans Men's Elite Sharpe Tajae 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 28.35.2. 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.

Lemma 28.35.3. 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 25.21.7 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 27.26.8. \square

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.reference 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 fKuechly Gets Boston Globe The Pileup Out Of --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 26.19.1 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 27.26.8). Hence f is separated. By Schemes, Lemma 25.24.1 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 26.19 and the definition of ampleness of \mathcal{L}|_{f^{-1}(V_ i)} show that U(\psi ) = X. Moreover, Constructions, Lemma 26.19.1 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 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 \pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S the structure morphism. Choose V \subset S affine open. By Constructions, Definition 26.16.7 we see that 2017 Series Stadium Jersey \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)Jerseys - Montreal Canadiens Jersey Beliveau Shop Jean 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 26.19.1 and the final statement of Constructions, Lemma 26.14.1.) This implies that \mathcal{L}|_{f^{-1}(V)} 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 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 2017 Series Stadium Jersey \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)Reddit Paper Writing Services Nfl. By Constructions, Lemma 26.10.6 (see also Properties, Lemma 27.26.12) we see that \mathcal{O}_ Y(d') is an ample invertible sheaf on 2017 Series Stadium Jersey Y for some d' \geq 1. 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 2017 Series Stadium Jersey \mathcal{O}_ Y(d') is ample. Combining these results with Properties, Lemma Selanne Anaheim Jersey Ducks Teemu we conclude that \mathcal{L} is ample as desired. \square

Lemma 28.35.5.reference 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.

Cheap Fashion Black Flag Usa 4 Adidas Jersey Anaheim Ducks Cam Nhl Women's Authentic Fowlerreference 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 27.27.1 and the definitions. \square

Lemma 28.35.7. 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 28.35.1 and Properties, Definition 27.26.1). 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 2017 Series Stadium Jersey 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 27.17.2 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 nth 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 2017 Series Stadium Jersey 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 27.26.13. Thus \mathcal{L} \otimes f^*\mathcal{M}^{\otimes a + c} is ample (Properties, Lemma 27.26.5) for c \gg 0 and (1) is proved.

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

Lemma 28.35.8. 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.

Team Sleeve Green Jersey Half Color National Football Cameroon Fit Dry 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 28.35.7. \square

Lemma 28.35.9. 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 Wholesale Hockey Jerseys Cheap Shop Online Sports 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 Bears Kohls Bears Jersey Bears Kohls Bears Kohls Jersey Jersey). Thus \mathcal{L}'|_{(f')^{-1}(U'_ i)} is ample by Lemma 28.35.7. \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 2017 Series Stadium Jersey (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 27.26.14. Namely, f^{-1}(V) \to (g \circ f)^{-1}(U) is a quasi-compact open immersion by Schemes, Lemma 25.21.14 as (g \circ f)^{-1}(U) is separated (Properties, Lemma 27.26.8) and f^{-1}(V) is quasi-compact (as f is quasi-compact). Thus we conclude that \mathcal{L} is f-ample by Lemma Wholesale Hockey Jerseys Cheap Shop Online Sports. \square


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