Ample invertible sheaves on quasi-compact schemes Ample line bundles are used most often on proper schemes, but they can be defined in much wider generality. Let
X be a scheme, and let \mathcal{L} be an invertible sheaf on
X. For each x \in X, let \mathfrak{m}_x denote the
ideal sheaf of the reduced subscheme supported only at
x. For s \in \Gamma(X, \mathcal{L}), define X_s = \{x \in X \colon s_x \not\in \mathfrak{m}_x\mathcal{L}_x\}. Equivalently, if \kappa(x) denotes the residue field at
x (considered as a skyscraper sheaf supported at
x), then X_s = \{x \in X \colon \bar s_x \neq 0 \in \kappa(x) \otimes \mathcal{L}_x\}, where \bar s_x is the image of
s in the tensor product. Fix s \in \Gamma(X, \mathcal{L}). For every
s, the restriction \mathcal{L}|_{X_s} is a free \mathcal{O}_X-module trivialized by the restriction of
s, meaning the multiplication-by-s morphism \mathcal{O}_{X_s} \to \mathcal{L}|_{X_s} is an isomorphism. The set X_s is always open, and the inclusion morphism X_s \to X is an affine morphism. Despite this, X_s need not be an affine scheme. For example, if s = 1 \in \Gamma(X, \mathcal{O}_X), then X_s = X is open in itself and affine over itself but generally not affine. Assume
X is quasi-compact. Then \mathcal{L} is
ample if, for every x \in X, there exists an n \ge 1 and an s \in \Gamma(X, \mathcal{L}^{\otimes n}) such that x \in X_s and X_s is an affine scheme. For example, the trivial line bundle \mathcal{O}_X is ample if and only if
X is
quasi-affine. In general, it is not true that every X_s is affine. For example, if X = \mathbf{P}^2 \setminus \{O\} for some point
O, and if \mathcal{L} is the restriction of \mathcal{O}_{\mathbf{P}^2}(1) to
X, then \mathcal{L} and \mathcal{O}_{\mathbf{P}^2}(1) have the same global sections, and the non-vanishing locus of a section of \mathcal{L} is affine if and only if the corresponding section of \mathcal{O}_{\mathbf{P}^2}(1) contains
O. It is necessary to allow powers of \mathcal{L} in the definition. In fact, for every
N, it is possible that X_s is non-affine for every s \in \Gamma(X, \mathcal{L}^{\otimes n}) with n \le N. Indeed, suppose
Z is a
finite set of points in \mathbf{P}^2, X = \mathbf{P}^2 \setminus Z, and \mathcal{L} = \mathcal{O}_{\mathbf{P}^2}(1)|_X. The vanishing loci of the sections of \mathcal{L}^{\otimes N} are plane curves of degree
N. By taking
Z to be a sufficiently large set of points in
general position, we may ensure that no
plane curve of degree
N (and hence any lower degree) contains all the points of
Z. In particular their non-vanishing loci are all non-affine. Define \textstyle S = \bigoplus_{n \ge 0} \Gamma(X, \mathcal{L}^{\otimes n}). Let p \colon X \to \operatorname{Spec} \mathbf{Z} denote the structural morphism. There is a natural isomorphism between \mathcal{O}_X-algebra homomorphisms \textstyle p^*(\tilde S) \to \bigoplus_{n \ge 0} \mathcal{L}^{\otimes n} and endomorphisms of the
graded ring S. The identity endomorphism of
S corresponds to a homomorphism \varepsilon. Applying the \operatorname{Proj} functor produces a morphism from an open subscheme of
X, denoted G(\varepsilon), to \operatorname{Proj} S. The basic characterization of ample invertible sheaves states that if
X is a quasi-compact quasi-separated scheme and \mathcal{L} is an invertible sheaf on
X, then the following assertions are equivalent: • \mathcal{L} is ample. • The open sets X_s, where s \in \Gamma(X, \mathcal{L}^{\otimes n}) and n \ge 0, form a basis for the topology of
X. • The open sets X_s with the property of being affine, where s \in \Gamma(X, \mathcal{L}^{\otimes n}) and n \ge 0, form a basis for the topology of
X. • G(\varepsilon) = X and the morphism G(\varepsilon) \to \operatorname{Proj} S is a dominant open immersion. • G(\varepsilon) = X and the morphism G(\varepsilon) \to \operatorname{Proj} S is a homeomorphism of the underlying topological space of
X with its image. • For every quasi-coherent sheaf \mathcal{F} on
X, the canonical map \bigoplus_{n \ge 0} \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \otimes_{\mathbf{Z}} \mathcal{L}^{\otimes{-n}} \to \mathcal{F} is surjective. • For every quasi-coherent sheaf of ideals \mathcal{J} on
X, the canonical map \bigoplus_{n \ge 0} \Gamma(X, \mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \otimes_{\mathbf{Z}} \mathcal{L}^{\otimes{-n}} \to \mathcal{J} is surjective. • For every quasi-coherent sheaf \mathcal{F} of finite type on
X, there exists an integer n_0 such that for n \ge n_0, \mathcal{F} \otimes \mathcal{L}^{\otimes n} is generated by its global sections. • For every quasi-coherent sheaf \mathcal{F} of finite type on
X, there exists integers n > 0 and k > 0 such that \mathcal{F} is isomorphic to a quotient of \mathcal{L}^{\otimes(-n)} \otimes \mathcal{O}_X^k. • For every quasi-coherent sheaf of ideals \mathcal{J} of finite type on
X, there exists integers n > 0 and k > 0 such that \mathcal{J} is isomorphic to a quotient of \mathcal{L}^{\otimes(-n)} \otimes \mathcal{O}_X^k.
On proper schemes When
X is separated and finite type over an affine scheme, an invertible sheaf \mathcal{L} is ample if and only if there exists a positive integer
r such that the tensor power \mathcal{L}^{\otimes r} is very ample. In particular, a proper scheme over
R has an ample line bundle if and only if it is projective over
R. Often, this characterization is taken as the definition of ampleness. The rest of this article will concentrate on ampleness on proper schemes over a field, as this is the most important case. An ample line bundle on a proper scheme
X over a field has positive degree on every curve in
X, by the corresponding statement for very ample line bundles. A
Cartier divisor D on a proper scheme
X over a field
k is said to be ample if the corresponding line bundle
O(
D) is ample. (For example, if
X is smooth over
k, then a Cartier divisor can be identified with a finite
linear combination of closed codimension-1 subvarieties of
X with integer coefficients.) Weakening the notion of "very ample" to "ample" gives a flexible concept with a wide variety of different characterizations. A first point is that tensoring high powers of an ample line bundle with any coherent sheaf whatsoever gives a sheaf with many global sections. More precisely, a line bundle
L on a proper scheme
X over a field (or more generally over a
Noetherian ring) is ample if and only if for every coherent sheaf
F on
X, there is an integer
s such that the sheaf F\otimes L^{\otimes r} is globally generated for all r\geq s. Here
s may depend on
F. Another characterization of ampleness, known as the
Cartan–
Serre–
Grothendieck theorem, is in terms of
coherent sheaf cohomology. Namely, a line bundle
L on a proper scheme
X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf
F on
X, there is an integer
s such that :H^i(X,F\otimes L^{\otimes r})=0 for all i>0 and all r\geq s. In particular, high powers of an ample line bundle kill cohomology in positive degrees. This implication is called the
Serre vanishing theorem, proved by
Jean-Pierre Serre in his 1955 paper
Faisceaux algébriques cohérents. ==Examples/Non-examples==