The relative GAGA Theorem and an application to the analytic mapping stacks

TL;DR

This paper establishes a relative GAGA theorem in non-archimedean analytic geometry, extending the understanding of analytic and algebraic mapping stacks, with applications to proper schemes and Artin stacks.

Contribution

It proves a new relative GAGA theorem for complexes over non-archimedean bases, including perfectoid spaces, and applies it to relate algebraic and analytic mapping stacks.

Findings

Analytification of algebraic mapping stacks matches the intrinsic analytic stacks.

Answers a previously open question in p-adic non-abelian Hodge theory.

Extends GAGA principles to bases from Fredholm analytic rings.

Abstract

We prove a relative GAGA theorem for perfect and pseudo-coherent complexes in non-archimedean analytic geometry, allowing bases given by Fredholm analytic rings, including those associated from affinoid perfectoid spaces. This answers a question raised in \cite{heuer2024padicnonabelianhodgetheory}. As an application, we show that for a proper scheme \(X\) and an Artin stack \(Y\) with suitable conditions, the analytification of the algebraic mapping stack \(\mathrm{Map}(X,Y)\) agrees with the intrinsic analytic mapping stack \(\mathrm{Map}(X^{\mathrm{an}},Y^{\mathrm{an}})\).