[Paper Review] Foundations for almost ring theory -- Release 7.5

Ofer Gabber, Lorenzo Ramero|arXiv (Cornell University)|Sep 29, 2004

Algebraic Geometry and Number Theory5 references57 citations

TL;DR

This paper establishes foundational tools in almost ring theory, particularly focusing on perfectoid and almost perfectoid algebras, using advanced category theory, topos theory, and homological algebra. It proves that under certain conditions, the completion of a ring along an ideal becomes almost perfectoid, extending key results in arithmetic geometry and p-adic Hodge theory.

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.

Motivation & Objective

• To develop a comprehensive framework for almost ring theory, especially in the context of perfectoid rings and algebras.
• To generalize classical results in commutative algebra and algebraic geometry to the almost setting, particularly in the presence of p-adic topology.
• To establish conditions under which completions of rings become almost perfectoid, extending the theory of perfectoid algebras.
• To provide a systematic treatment of stacks, sites, topoi, and fibrations to support the development of almost algebraic geometry.
• To prove that certain algebras associated with perfectoid Tate rings are almost isomorphic to their integral closures, ensuring flatness and compatibility under base change.

Proposed method

• Utilizes 2-category theory and Kan extensions to formalize descent and base change in higher categorical settings.
• Applies fibred sites and fibred topoi to model geometric objects over varying base categories.
• Employs homological algebra techniques, including Tor functors and flatness criteria, to analyze almost modules and algebras.
• Constructs a universal colimit of faithfully flat algebras to show that the almost version of a ring is faithfully flat over its base.
• Uses the formalism of local calculus of fractions and covering morphisms of prestacks to define and study stacks in groupoids.
• Applies the functor (−)♮ to lift structures from perfectoid rings to their integral closures, preserving almost isomorphisms.

Experimental results

Research questions

• RQ1 Under what conditions is the completion of a ring along an ideal almost perfectoid?
• RQ2 When does an almost isomorphism between algebras imply that the target is faithfully flat over the source?
• RQ3 How can the theory of stacks and topoi be extended to the almost setting, particularly in relation to descent and fibrations?
• RQ4 What is the relationship between the integral closure of a perfectoid ring and its almost completion?
• RQ5 Can the almost perfectoid property be preserved under base change and localization in the context of formal perfectoid rings?

Key findings

• The completion $ C^\wedge $ of a ring $ C $ along an ideal $ b $, equipped with the topology induced by $ b $, is shown to be an almost perfectoid basic setup, provided the Frobenius map induces an almost isomorphism on $ C/bC \to C/b^pC $.
• The algebra $ C^a $, the almost version of $ C $, is faithfully flat over $ A_0^a $, the almost version of the base ring $ A_0 $, under suitable conditions on the maps $ \Delta $, $ \pi_n $, and the multiplicative system $ S $.
• The induced morphism $ ((A^\sharp_0)^\nu)^a \to (A^\nu_0)^a $ is an isomorphism of $ (R_0, m_R)^a $-algebras, showing that integral closure and almost completion commute under the given assumptions.
• The colimit of a filtered system of faithfully flat $ A_0^a $-algebras $ D_\gamma^a $ is itself a faithfully flat $ A_0^a $-algebra, proving that the almost version of the ring $ B $ is flat over $ A_0^a $.
• The map $ \Phi_C: C/bC \to C/b^pC $ is an almost isomorphism, which is a key step in verifying the almost perfectoid condition for $ C^\wedge $.
• The construction of the $ \natural $-functor preserves almost isomorphisms and allows lifting of perfectoid structures, leading to an alternative proof of the almost flatness of $ C^a $.