Ample Sheaves and Ample Families

1 Ample Sheaves Let X be a scheme and L an invertible sheaf. Given a global section f ∈ Γ(X,L ) the set Xf = {x ∈ X | germxf / ∈ mxLx} is open (MOS,Lemma 29). The inclusion Xf −→ X is affine (RAS,Lemma 6) and in particular if X is an affine scheme then Xf is itself affine. Given a sequence of global sections f1, . . . , fn the open sets Xfi cover X if and only if the fi generate L (MOS,Lemma 32). Lemma 1. Let (X,OX) be a quasi-compact ringed space and F a sheaf of modules of finite type. If F is generated by global sections then it can be generated by a finite number of global sections. Proof. See (MOS,Definition 2) for the definition of a sheaf of modules of finite type. Let {si}i∈I be a nonempty family of global sections of F which generate. Let Λ be the set of all finite subsets of I and for each λ ∈ Λ let Fλ be the submodule of F generated by the si belonging to λ. This is a direct family of submodules and clearly F = lim −→λ Fλ, so it follows from (MOS,Lemma 57) that F = Fλ for some λ. In other words, F can be generated by a finite number of global sections. Lemma 2. Let (X,OX) be a quasi-noetherian ringed space, F a sheaf of modules on X and {Fα}α∈Λ a direct family of submodules of F . If U ⊆ X is quasi-compact then