Relative Ampleness in Rigid-analytic Geometry

1.1. Motivation. The aim of this paper is to develop a rigid-analytic theory of relative ampleness for line bundles, and to record some applications to rigid-analytic faithfully flat descent for morphisms and for proper geometric objects equipped with a relatively ample line bundle. (For coherent sheaves on rigid spaces, the theory of faithfully flat descent is established in [BG] via Raynaud’s theory of formal models [BL1].) Such a theory could have been worked out decades ago, as the tools we use have been known for a long time. Our primary motivation for working out the theory now is for applications in the context of p-adic modular forms on more general groups; as a first step in this direction, in [C2] and [C3] the theory in the present paper is used to develop a relative theory of arbitrary-level canonical subgroups in rather general rigidanalytic families of generalized elliptic curves and abelian varieties over p-adic fields. (The key point in this application is to have results not depending on the specification of discrete parameters such as the degree of a polarization.) For a proper rigid space X over a non-archimedean field k, an invertible sheaf L on X is ample if some high tensor power L ⊗N is the pullback of OPk (1) under some closed immersion j : X ↪→ P n k into a rigidanalytic projective space. The cohomological criterion for ampleness works, though the proof of the criterion in the algebraic and complex-analytic cases uses pointwise arguments and so some modifications are required in the rigid-analytic case. In [FM, §4] a few aspects of a theory of ampleness are developed for quasi-compact separated rigid spaces X, taking the cohomological criterion as the starting point, but we develop what we need for k-proper X ab ovo because our intention is to develop a relative theory (for applications in [C2] and [C3]) and so we prefer to set up the absolute case in a way that best prepares us for relativization. If f : X → S is a proper morphism between rigid spaces over k, and L is a line bundle on X, analogy with the case of proper schemes [EGA, IV3, 9.6.4] motivates the following definition: L is S-ample (or relatively ample over S) if Ls is ample on the rigid space Xs over k(s) for all s ∈ S. We adopt this as the initial definition because it is a property that can be checked in abstract situations. Does this definition satisfy all of the properties one desires? For example, is relative ampleness preserved by arbitrary extension on k (for quasi-separated S)? This would hold if the relationship between ampleness and projective embeddings relativizes, but even this relationship is not obvious: (1) Does there exist an admissible covering {Sα} of S such that each Xα = f(Sα) admits a closed Sα-immersion into some projective space Pα Sα such that some positive tensor power L ⊗Nα |Xα is isomorphic to the pullback of O(1)? (2) Is the graded OS-algebra A = ⊕n≥0f∗L ⊗n locally finitely generated? (3) Can we recover X from A (as we can in the case of schemes, by using relative Proj)? These properties of A are a trivial consequence of an affirmative answer to the question on relative projective embeddings (due to Köpf’s relative GAGA theorems over affinoids [Kö, §5]), but in fact we have to argue in reverse: relative projective embeddings (given an S-ample L ) will be constructed by using the finiteness properties of A and the reconstruction of X from A (via an analytic Proj operation). Hence, we must study A in the absence of relative projective embeddings.