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 X_p.p and X_ét.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 [Ber_l] 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 Y_ét.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 .