Espaces rigides quasi-séparés

On higher direct images of convergent isocrystals

“…We denote by the category whose objects are adic formal -schemes of finite type [Abb10, 2.3.13] and morphisms are adic morphisms [Abb10, 2.2.7]. By [EGAI, 6.1.5(v)], morphisms of are of finite type.…”

“…Let be an object of and a coherent -module. Since is quasi-compact, by Gabber–Bosch–Görtz’s fppf descent for coherent -modules [Abb10, 5.11.11], the presheaf on is a sheaf for the fppf topology. In particular, defines a sheaf of rings of that we still denote by .…”

“…Let be an object of and an open ideal of finite type of containing [Abb10, 2.1.19]. We denote by the admissible blow-up of in [Abb10, 3.1.2].…”

“…Let be an object of and an open ideal of finite type of containing [Abb10, 2.1.19]. We denote by the admissible blow-up of in [Abb10, 3.1.2]. The ideal is invertible [Abb10, 3.1.4(i)] and is flat over [Abb10, 3.1.4(ii)].…”

“…Using Dwork’s trick and Theorem 1.5, we deduce Theorem 1.9 in the case where is smooth over (Theorem 5.10). In § 6, we briefly review Raynaud’s approach to rigid geometry following Abbes’ book [Abb10]. Section 7 is devoted to a modification of convergent topos which allows us to apply the faithfully flat descent in rigid geometry in the full extent.…”