Arizona Winter School Project on Analytifications

There is a natural analytification functor from the category of locally separated algebraic spaces locally of finite type over C to the category of complex-analytic spaces [Kn, Ch. I, 5.17ff]. (Recall that a map of algebraic spaces X → S is locally separated if the diagonal ∆X/S : X → X ×S X is an immersion. We require algebraic spaces to have quasi-compact diagonal over SpecZ.) It is natural to ask if a similar theory works over a non-archimedean base field k. This is a non-trivial question because the construction of analytifications in the complex-analytic case is so local that any attempt to carry out the same method in the rigid setting seems to get stuck on unpleasant admissibility issues. The contrast is perhaps better appreciated in view of the surprise that there are counterexamples showing that local separatedness is not sufficient for analytifiability of an algebraic space locally of finite type over k; see Example 3.1. Put in more concrete terms, there are locally separated algebraic spaces of finite type over Q with dimension 2 such that the k-fiber does not admit an analytification for any non-archimedean field k of characteristic 0. Roughly speaking, this dichotomy between the archimedean and non-archimedean worlds is explained by the lack of a Gelfand–Mazur theorem over non-archimedean fields. (That is, any non-archimedean field k admits non-trivial non-archimedean extension fields with a compatible absolute value, even if k is algebraically closed.) Over C, analytification is defined in terms of a quotient process, and it follows from [SGA1, XII, 3.2(iv)] that an algebraic space locally of finite type over C admits an analytification if and only if it is locally separated over Spec(C). It is unclear if local separatedness is a necessary condition for the existence of analytifications in the non-archimedean case (for reasons that we explain above Lemma 2.16). Since local separatedness fails to be a sufficient criterion for the existence of non-archimedean analytification, it is natural to seek a reasonable salvage of the situation. It turns out that separatedness suffices. These notes are extracted from a paper [CT] in preparation by the two authors, and omitted details are left as exercises for the Winter School. These notes explain how the general problem is reduced to the special case of finite free actions by a finite group in the context of Berkovich spaces, but this special case is not addressed because it requires an entirely different viewpoint (Temkin’s theory of reduction of germs [T]) for which there will likely not be time to be discussed at the Winter School. We systematically develop the basic definitions and prove some of the basic lemmas that we need, and by stating other results without proof (left as exercises) we hope this will provide a structured framework for the project. An existence theorem for analytifications in the separated case is discussed in §4. Throughout these notes, it is tacitly understood that “algebraic space” means “algebraic space locally of finite type over k” unless we explicitly say otherwise.