[Paper Review] GAGA theorems

Jack Hall|arXiv (Cornell University)|Apr 5, 2018

Algebraic Geometry and Number Theory17 references4 citations

TL;DR

This paper presents a unified GAGA theorem that extends analytic and formal GAGA results to relative and non-noetherian settings using a novel method applicable to both GAGA and Lefschetz-type theorems. The approach provides a general framework valid for the Fargues-Fontaine curve, unifying existing results and enabling new comparison theorems.

ABSTRACT

We prove a new and unified GAGA theorem. This recovers all analytic and formal GAGA results in the literature, and is also valid in the relative and non-noetherian setting. Our method can also be used to establish various Lefschetz theorems and comparison results for the Fargues-Fontaine curve.

Motivation & Objective

• To generalize existing GAGA theorems to non-noetherian and relative settings.
• To unify analytic and formal GAGA results under a single framework.
• To develop a method applicable to both GAGA and Lefschetz-type theorems.
• To establish new comparison results for the Fargues-Fontaine curve.
• To provide a systematic approach for relating algebraic and analytic structures in arithmetic geometry.

Proposed method

• Introduce a new general framework for GAGA theorems based on a unified analytic-algebraic correspondence.
• Apply the method to relative and non-noetherian schemes, extending classical GAGA to broader geometric contexts.
• Use the framework to derive comparison theorems between algebraic and analytic objects on the Fargues-Fontaine curve.
• Leverage the method to prove Lefschetz-type theorems in the context of perfectoid spaces and curves.
• Utilize advanced techniques in p-adic geometry and formal schemes to ensure applicability across diverse settings.
• Establish compatibility with existing results while generalizing their scope through structural coherence.

Experimental results

Research questions

• RQ1 Can a single framework unify existing GAGA theorems across analytic, formal, and relative settings?
• RQ2 How can GAGA theorems be extended to non-noetherian schemes?
• RQ3 What structural conditions allow the method to apply to the Fargues-Fontaine curve?
• RQ4 To what extent can the method yield Lefschetz-type theorems in p-adic geometry?
• RQ5 How does the new approach preserve and generalize classical comparison results between algebraic and analytic objects?

Key findings

• The paper establishes a new GAGA theorem that unifies and generalizes all known analytic and formal GAGA results.
• The method applies to relative and non-noetherian settings, extending the scope of classical GAGA theorems.
• The framework enables new comparison theorems for the Fargues-Fontaine curve, linking algebraic and analytic structures.
• The approach yields Lefschetz-type theorems in the context of perfectoid geometry and p-adic curves.
• The results are derived through a coherent method that maintains compatibility with existing theorems while broadening their applicability.
• The framework provides a systematic tool for studying cohomological and geometric properties in arithmetic geometry.