The paper serves as an introduction to etale cohomology of rigid analytic spaces. A number of basic results are proved, e.g. concerning cohomological dimension, base change, invariance for change of base elds, the homotopy axiom and comparison for etale cohomology of algebraic varieties. The methods are those of classical rigid analytic geometry and along the way a number of known results on ri...

In this paper we construct a natural category ~r of locally and topologically ringed spaces which contains both the category of locally noetherian formal schemes and the category of rigid analytic varieties as full subcategories. This category has applications in algebraic geometry and rigid analytic geometry. The idea of the definition of the category ~r is the following. From a formal point o...

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

We give a rigid analytic version of Hironaka’s Embedded Resolution of Singularities over an algebraically closed field of characteristic zero, complete with respect to a non-archimedean norm. This resolution is local with respect to the Grothendieck topology. The proof uses Hironaka’s original result, together with an application of our analytization functor. 0. Introduction and preliminaries.

Let K be an algebraically closed field endowed with a complete nonarchimedean norm with valuation ring R. Let f : Y → X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y ) ⊂ X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function ...

We prove a version of faithfully flat descent in rigid analytic geometry, for almost perfect complexes and without finiteness assumptions on the rings involved. This extends results Drinfeld vector bundles.

the aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. in [1], ungar and chen showed that the algebra of the group sl(2,c) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the lorentz group and its underlying hyperbolic geometry. they defined the chen addition and then chen model of hyperbolic geometry. in this paper,...

We describe the homotopy type of the analytic Milnor fiber in terms of the special fiber of a strictly semi-stable reduction, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. In the appendix, we consider the de Rham cohomology of the analytic Milnor fiber of an isolated singularity, and we re-interpret some class...

In [7], Coleman and Mazur construct a rigid analytic space E that parameterizes overconvergent and therefore classical modular eigenforms of finite slope. The geometry of E is at present poorly understood, and seems quite complicated, especially over the centre of weight space. Recently, some progress has been made in understanding the geometry of E in certain examples (see for example [3],[4])...