Almost ring theory

  title={Almost ring theory},
  author={Ofer Gabber and Lorenzo Ramero},
3.1 Flat, unramified and etale morphisms 3.2 Nilpotent deformations of almost algebras and modules 3.3 Nilpotent deformations of torsors 3.4 Descent 3.5 Behaviour of etale morphisms under Frobenius 
Abstract We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is
We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients
Algebraization of a Cartier divisor
We extend to pairs classical results of R. Elkik on lifting of homomorphisms and algebraization. In particular, we establish algebraization of an affine rig-smooth formal variety with a rig-smooth
Grade of Ideals with Respect to Torsion Theories
In this article we define and compare different types of the notion of grade with respect to torsion theories over commutative rings which are not necessarily Noetherian. We do this by using
Pro-\'etale uniformisation of abelian varieties
For an abelian variety A over an algebraically closed non-archimedean field K of residue characteristic p, we show that the isomorphism class of the pro-étale perfectoid cover à = lim ←−[p] A is
Finiteness of fppf cohomology
Let R be a Henselian DVR with finite residue field. Let G be a finite type, flat R-group scheme (not necessarily commutative) with smooth generic fiber. We show that H fppf (SpecR,G) is finite. We
Non-exactness of direct products of quasi-coherent sheaves.
For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result
Prismatic cohomology of rigid analytic spaces over de Rham period ring
Inspired by Bhatt-Scholze [BS19], in this article, we introduce prismatic cohomology for rigid analytic spaces with l.c.i singularities, with coefficients over Fontaine’s de Rham period ring B dR .
$p$-adic vanishing cycles as Frobenius-fixed points
Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.
Logarithmic adic spaces: some foundational results
We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer \'etale and pro-Kummer \'etale topology for such spaces. We also establish the


Étale Cohomology of Rigid Analytic Varieties and Adic Spaces
Summary of the results on the etale cohomology of rigid analytic varieties - Adic spaces - The etale site of a rigid analytic variety and an adic space - Comparison theorems - Base change theorems -
Commutative Formal Groups
Formal varieties.- Formal groups and buds.- The general equivalence of categories.- The special equivalences of categories.- The structure theorem and its consequences.- On formal groups in
On totally ordered groups, and K0
Some results are described concerning totally ordered abelian groups. These can be interpreted, via the functor K0, as classification results for certain noncommutative rings, for which K0 as an
Localisation de la lissite formelle
A formally smooth morphism A→B between local noetherian rings gives a formlly smooth morphism AP→BQ for any prime ideal Q if this property holds for B=Â. Regular excellent rings of characteristic p>0
Commutative Ring Theory
Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7.
Spectral Theory and Analytic Geometry over Non-Archimedean Fields
The spectrum of a commutative Banach ring Affinoid spaces Analytic spaces Analytic curves Analytic groups and buildings The homotopy type of certain analytic spaces Spectral theory Perturbation
p -Divisible Groups
After a brief review of facts about finite locally free commutative group schemes in § 1, we define p-divisible groups in § 2, and discuss their relation to formal Lie groups. The § 3 contains some
A counterexample to a 1961 “theorem” in homological algebra
Abstract.In 1961, Jan-Erik Roos published a “theorem”, which says that in an [AB4*] abelian category, lim1 vanishes on Mittag–Leffler sequences. See Propositions 1 and 5 in [4]. This is a “theorem”
Un lemme de descente
Soient A un anneau, f un element simplifiable de A, A le separe complete de A pour la topologie (f)-adique. Nous prouvons que la donnee d'un fibre vectoriel sur Spec (A) equivaut a celle d'un fibre
Irreducible components of rigid spaces
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