The idea is simple: we want to develop a theory of analytic manifolds and spaces over fields equipped with an arbitrary complete valuation. Of course, it is a standard fact that such a field must be either R, C, or a field with a nonarchimedean valuation, so what we really mean is that we want to develop a theory of nonarchimedean analytic spaces. Doing this näıvely (i.e., defining manifolds in...

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic information underlying classical moduli space. The main tool in our construction is representability theorem geometry, which has been established previous work. provides us with sufficient and necessary conditions for an functor to possess structure stack. In order verify ...

These notes are intended to be a short course in rigid analytic geometry, without, however, providing always proofs. The excellent book [4] by Bosch, Güntzer and Remmert is an extensive introduction into rigid analytic geometry, and includes all the proofs I have omitted here.

The purpose of these notes is to introduce the basic notions of rigid analytic geometry, with the aim of discussing the non-archimedean uniformizations of certain abelian varieties.

1.1. Motivation. The aim of this paper is to develop a rigid-analytic theory of relative ampleness for line bundles, and to record some applications to rigid-analytic faithfully flat descent for morphisms and for proper geometric objects equipped with a relatively ample line bundle. (For coherent sheaves on rigid spaces, the theory of faithfully flat descent is established in [BG] via Raynaud’s...

We define the concept of blowing up map in rigid analytic geometry and show that such maps exist in full generality by giving an explicit construction. We then derive some elementary properties of blowing up maps, similar to those in the classical case. 0 Introduction and preliminaries 0.