Descent for Non-archimedean Analytic Spaces

In the theory of schemes, faithfully flat descent is a very powerful tool. One wants a descent theory not only for quasi-coherent sheaves and morphisms of schemes (which is rather elementary), but also for geometric objects and properties of morphisms between them. In rigid-analytic geometry, descent theory for coherent sheaves was worked out by Bosch and Görtz [BG, 3.1] under some quasi-compactness hypotheses by using Raynaud’s theory of formal models, and their result can be generalized [C, 4.2.8] to avoid quasicompactness assumptions (as is necessary to include analytifications of faithfully flat maps arising from algebraic geometry [CT, §2.1]). Similarly, faithfully flat descent for morphisms, admissible open sets, and standard properties of morphisms works out nicely in the rigid-analytic category [C, §4.2]. In Berkovich’s theory of k-analytic spaces, one can ask if there are similar results. The theory of flatness in k-analytic geometry is more subtle than in the case of schemes or complex-analytic spaces, ultimately because morphisms of k-affinoid spaces generally have non-empty relative boundary (in the sense of [Ber1, 2.5.7]). In the case of quasi-finite morphisms [Ber2, §3.1], which are maps that are finite locally on the source and target, it is not difficult to set up a satisfactory theory of flatness [Ber2, §3.2]. The appendix to this paper (by Ducros) develops a more general theory of flatness for k-analytic maps with empty relative boundary; this includes flat quasi-finite maps, smooth maps, and (relative) analytifications of flat maps between schemes locally of finite type over a k-affinoid algebra. Let f : X → Y be a map of k-analytic spaces, and let Y ′ → Y be a surjective flat map. For various properties P of morphisms that are preserved by base change (proper, finite, separated, closed immersion, etc.) we say that P is local for the flat topology if f satisfies P precisely when the base change f ′ : X ′ → Y ′ does. For example, consider the property of a morphism f : X → Y being without boundary in the sense that for any k-affinoid W and morphism W → Y , the base change X ×Y W is a good k-analytic space (i.e., each point has a k-affinoid neighborhood) and the morphism of good spaces X ×Y W → W has empty relative boundary in the sense of [Ber1, p. 49]. This property is preserved by k-analytic base change, but it is not at all obvious from the definitions if it is local for the flat topology. Similarly, if Y is a k-analytic space and it has a flat quasi-finite cover Y ′ → Y such that Y ′ is good then it is natural to expect that Y is good but this does not seem to follow easily from the definitions since the target of a finite surjective morphism with affinoid source can be non-affinoid [Liu]. It is also natural to ask if goodness descends even when Y ′ → Y is merely a flat cover. Finally, one can also ask for analogous descent results with respect to extension of the ground field. That is, if f : Y ′ → Y is a map of k-analytic spaces and if K/k is an arbitrary analytic field extension then we ask if f satisfies a property P precisely when fK : Y ′ K → YK satisfies this same property. Likewise, if YK is good then is Y good (the converse being obvious)? This latter question seems to be very non-trivial, and in general the problem of descent through a field extension is much harder than descent through flat surjections. The purpose of this paper is to apply the theory of reduction of germs (as developed in [T2]) to provide affirmative answers to all of the above descent questions.