Foundations for almost ring theory -- Release 7.5

Abstract

This is release 7.5 of our project, aiming to provide a complete treatment of the foundations of almost ring theory, following and extending Faltings's method of "almost etale extensions". The central result is the "almost purity theorem", for whose proof we adapt Scholze's method, based on his perfectoid spaces. This release provides the foundations for our generalization of Scholze's perfectoid spaces, and reduces the proof of the almost purity theorem to a general assertion concerning the étale topology of adic spaces, whose proof uses previous work by the first author. As usual, this new release is a mix of corrections and various improvements, with a final chapter dedicated to applications; notably, we include a generalization of Y.André's "perfectoid Abhyankar's lemma" which we use to give a proof of a generalization of the "direct summand conjecture", extending André's recent work.

1650 pages, written in AMS-LaTeX. Updates and corrections shall also be posted on my web page : http://math.univ-lille1.fr/~ramero

Cited by in corpus (33)

• Relative p-adic Hodge theory: Foundations
• Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras
• An application of the almost purity theorem to the homological conjectures
• Integral perfectoid big Cohen-Macaulay algebras via André's theorem
• $K(\pi ,1)$-neighborhoods and comparison theorems
• Logarithmic adic spaces: some foundational results
• An analog of adjoint ideals and PLT singularities in mixed characteristic
• Derived Logarithmic Geometry I
• Some recent advances in topological Hochschild homology
• Log smoothness and polystability over valuation rings
• Gluing pseudo functors via $n$-fold categories
• Logarithmic good reduction, monodromy and the rational volume
• Frobenius-Witt differentials and regularity
• A Jacobian criterion for nonsingularity in mixed characteristic
• Parallel transport and the p-adic Simpson correspondence
• The root closure of a ring of mixed characteristic
• The geometry of degenerations of Hilbert schemes of points
• Excellence, F-singularities, and solidity
• Galois representations, $(\phi ,\Gamma )$-modules and prismatic F-crystals
• Logarithmic good reduction and the index
• On commutative nonarchimedean Banach fields
• Relative p-adic Hodge theory, II: (phi, Gamma)-modules
• Homological aspects of perfect algebras
• (Non)Vanishing results on local cohomology of valuation rings
• Cohomological Descent for Faltings' $p$-adic Hodge Theory and Applications
• Reified valuations and adic spectra
• Local Cohomology of Module of Differentials of integral extensions
• Toroidal compactifications of the moduli spaces of Drinfeld modules
• Motivic zeta function of the Hilbert schemes of points on a surface
• Tropical compactification in log-regular varieties
• Finite étale extension of Tate rings and decompletion of perfectoid algebras
• Topics on Geometric and Representation Theoretic Aspects of Period Rings I
• Almost Cohen-Macaulay algebras in mixed characteristic via Fontaine rings