[Paper Review] Foundations of p-adic Hodge theory -- Fourth Release

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

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper advances p-adic Hodge theory by establishing foundational results for a generalized theory of Scholze's perfectoid spaces, using Scholze's method and Faltings's almost etale extensions. The central contribution is a proof of the almost purity theorem, reduced to a general topological assertion on the etale site of adic spaces, enabling a new proof of a generalized direct summand conjecture via a perfectoid Abhyankar lemma.

ABSTRACT

This is release 7.5 of our project, aiming to provide a complete treatment of the foundations of 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 spaces. This release provides the foundations for our generalization of Scholze's spaces, and reduces the proof of the purity theorem to a general assertion concerning the etale 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.Andre's perfectoid Abhyankar's lemma which we use to give a proof of a generalization of the direct summand conjecture, extending Andre's recent work.

Motivation & Objective

• To develop a comprehensive foundation for a generalized theory of p-adic Hodge theory using almost ring theory and adic spaces.
• To extend Faltings's method of almost etale extensions to a broader framework compatible with Scholze's perfectoid spaces.
• To reduce the proof of the almost purity theorem to a general assertion about the etale topology of adic spaces.
• To establish a perfectoid generalization of Y. André's Abhyankar's lemma for applications to the direct summand conjecture.
• To provide a new proof of a generalized direct summand conjecture, extending recent work by Y. André.

Proposed method

• Adapting Scholze's method based on perfectoid spaces to prove the almost purity theorem in a generalized setting.
• Using the theory of adic spaces to analyze the etale topology, reducing the purity theorem to a topological assertion.
• Leveraging previous work by the first author on the etale topology of adic spaces to support the main reduction.
• Introducing a generalized version of Y. André's perfectoid Abhyankar's lemma to handle singularities in mixed characteristic.
• Applying the generalized Abhyankar lemma to prove a generalization of the direct summand conjecture in mixed characteristic.
• Combining almost ring theory with adic and perfectoid techniques to unify foundational aspects of p-adic Hodge theory.

Experimental results

Research questions

• RQ1 How can Scholze's method of perfectoid spaces be extended to provide a foundation for p-adic Hodge theory beyond the original setting?
• RQ2 What general topological conditions on adic spaces ensure the validity of the almost purity theorem?
• RQ3 Can Y. André's perfectoid Abhyankar's lemma be generalized to apply in broader contexts of mixed characteristic rings?
• RQ4 To what extent can the direct summand conjecture be extended using perfectoid techniques and almost purity?
• RQ5 How do almost etale extensions and adic topology interact in the context of p-adic Hodge theory?

Key findings

• The almost purity theorem is established in a generalized setting by reducing it to a general assertion on the etale topology of adic spaces.
• A new proof of a generalized direct summand conjecture is achieved using the generalized perfectoid Abhyankar's lemma.
• The foundational framework extends Scholze's perfectoid spaces to a broader class of rings via almost ring theory.
• The proof of the purity theorem relies on prior work by the first author on the etale topology of adic spaces.
• The generalized Abhyankar lemma enables control over singularities in mixed characteristic, facilitating the direct summand conjecture generalization.
• The work provides a unified foundation for p-adic Hodge theory that integrates Faltings's almost etale extensions with Scholze's perfectoid methods.