Non-archimedean Analytification of Algebraic Spaces

1.1. Motivation. This paper is largely concerned with constructing quotients by étale equivalence relations. We are inspired by questions in classical rigid geometry, but to give satisfactory answers in that category we have to first solve quotient problems within the framework of Berkovich’s k-analytic spaces. One source of motivation is the relationship between algebraic spaces and analytic spaces over C, as follows. If X is a reduced and irreducible proper complex-analytic space then the meromorphic functions on X form a field M (X) and this field is finitely generated over C with transcendence degree at most dimX [CAS, 8.1.3, 9.1.2, 10.6.7]. A proper complex-analytic space X is called Moishezon if trdegC(M (Xi)) = dimXi for all irreducible components Xi of X (endowed with the reduced structure). Examples of such spaces are analytifications of proper C-schemes, but Moishezon found more examples, and Artin found “all” examples by analytifying proper algebraic spaces over C. To be precise, the analytification X an of an algebraic space X locally of finite type over C [Kn, Ch. I, 5.17ff] is the unique solution to an étale quotient problem that admits a solution if and only if X is locally separated over C (in the sense that ∆X /C is an immersion). The functor X X an from the category of proper algebraic spaces over C to the category of Moishezon spaces is fully faithful, and it is a beautiful theorem of Artin [A2, Thm. 7.3] that this is an equivalence of categories. It is natural to ask if a similar theory works over a non-archimedean base field k (i.e., a field k that is complete with respect to a fixed nontrivial non-archimedean absolute value). This is a surprisingly nontrivial question. One can carry over the definition of analytification of locally finite type algebraic spaces X over k in terms of uniquely solving a rigid-analytic étale quotient problem; when the quotient problem has a solution we say that X is analytifiable in the sense of rigid geometry. (See §2 for a general discussion of definitions, elementary results, and functorial properties of analytification over such k.) In Theorem 2.3.4 we show that local separatedness is a necessary condition for analytifiability over non-archimedean base fields, but in contrast with the complex-analytic case it is not sufficient; there are smooth 2-dimensional counterexamples over any k (even arising from algebraic spaces over the prime field), as we shall explain in Example 3.1.1. In concrete terms, the surprising dichotomy between the archimedean and non-archimedean worlds is due to the lack of a Gelfand–Mazur theorem over non-archimedean fields. (That is, any nonarchimedean field k admits nontrivial non-archimedean extension fields with a compatible absolute value, even if k is algebraically closed.) Since local separatedness fails to be a sufficient criterion for the existence of non-archimedean analytification of an algebraic space, it is natural to seek a reasonable salvage of the situation. We view separatedness as a reasonable additional hypothesis to impose on the algebraic space.