Partially proper sites of rigid analytic varieties and adic spaces

Abstract

Let X be a rigid analytic variety. The category underlying the site of X (resp. étale site of X) is the category of all open embeddings (resp. étale morphisms) U → X. Considering only those open embeddings (resp. étale morphisms) U → X which are partially proper ((1.3.19.i)), we obtain two sites Xp.p and Xét.p.p which we call the partially proper sites of X. In the first two paragraphs of this chapter we will study the relation between the toposes X~ and X ~p,p (resp. X ~ét and X ~ét.p.p ). In the third paragraph we will compare the hausdorff strictly analytic spaces defined by Berkovich in [Berl] with rigid analytic varieties. For a hausdorff strictly analytic space Y and associated rigid analytic variety s(Y), it is easily seen that the étale site Yét.s which is given by the étale morphisms of hausdorff strictly analytic spaces V → Y is equivalent to the partially proper étale site s(Y)ét.p.p of s(Y). Hence the result of §2 concerning the relation between the étale topos and the partially proper étale topos of a rigid analytic variety gives a comparison theorem between Y ~ét and s(Y) ~ét .KeywordsOpen SubsetValuation RingNatural MorphismOpen EmbeddingProper MorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.