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

Qixiang Wang

TL;DR

The article develops a comprehensive relative GAGA framework for non-archimedean analytic geometry, establishing equivalences between algebraic and analytic categories for perfect and pseudo-coherent complexes over Fredholm bounded analytic rings. Central to the approach are the categorical Künneth formula and an enhanced GAGA result, which together enable precise comparisons of derived categories and, in turn, mapping stacks. A key application shows that, for X proper and Y a perfectly Tannakian Artin stack of geometric nature, the analytic mapping stack Map(X^{an}, Y^{an}) is equivalent to the analytification of the algebraic mapping stack Map(X, Y), i.e., Map(X^{an}, Y^{an}) ≃ Map(X, Y)^{an}. The paper also develops the analytic cotangent theory, demonstrates the behavior of analytification under Tannaka duality, and provides a robust framework for studying cotangent complexes in the analytic setting, with potential applications to p-adic geometric Langlands and moduli problems. All mathematical notation is presented within $...$ delimiters throughout.

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 \(\mathrm{Map}(X^{\mathrm{an}},Y^{\mathrm{an}})\).

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

TL;DR

The article develops a comprehensive relative GAGA framework for non-archimedean analytic geometry, establishing equivalences between algebraic and analytic categories for perfect and pseudo-coherent complexes over Fredholm bounded analytic rings. Central to the approach are the categorical Künneth formula and an enhanced GAGA result, which together enable precise comparisons of derived categories and, in turn, mapping stacks. A key application shows that, for X proper and Y a perfectly Tannakian Artin stack of geometric nature, the analytic mapping stack Map(X^{an}, Y^{an}) is equivalent to the analytification of the algebraic mapping stack Map(X, Y), i.e., Map(X^{an}, Y^{an}) ≃ Map(X, Y)^{an}. The paper also develops the analytic cotangent theory, demonstrates the behavior of analytification under Tannaka duality, and provides a robust framework for studying cotangent complexes in the analytic setting, with potential applications to p-adic geometric Langlands and moduli problems. All mathematical notation is presented within delimiters throughout.

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 and an Artin stack with suitable conditions, the analytification of the algebraic mapping stack \(\mathrm{Map}(X,Y)\) agrees with the intrinsic analytic mapping stack \(\mathrm{Map}(X^{\mathrm{an}},Y^{\mathrm{an}})\).
Paper Structure (23 sections, 85 theorems, 135 equations)

This paper contains 23 sections, 85 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Analytification of mapping stacks
  3. Relative GAGA theorem
  4. Recollection on $\mathrm{Pr}^L$
  5. Lurie tensor product
  6. Small stable categories
  7. Recollection on analytic geometry
  8. Non-archimedean analytic stacks
  9. Realization functors
  10. Algebraic realization
  11. Analytification
  12. GAGA theorem revisited
  13. Enhanced GAGA theorem
  14. Categorical Künneth formula
  15. Relative GAGA theorem for perfect complexes
  16. ...and 8 more sections

Key Result

Theorem 1

Let $X$ be a proper scheme over $K$, and let $Y$ be a perfectly TannakianArtin stack of geometric nature (perfectly tannakian). Suppose that $\underline{\mathrm{Map}} (X, Y)$ is an Artin stack. Then we have:

Theorems & Definitions (200)

  • Theorem 1: \ref{['analytification']}
  • Definition 2
  • Theorem 3: \ref{['main']}
  • Theorem 4: kesting2025categoricalkunnethformulasanalytic
  • Remark 5
  • Example 6
  • Theorem 7: Enhanced GAGA
  • Theorem 8: MR422671, AIF_2006__56_4_1049_0, kedlaya2015relativepadichodgetheory*Theorem 2.7.7, andreychev2021pseudocoherentperfectcomplexesvector*Theorem 1.4
  • Theorem 9: \ref{['GAGA of pseudo']}
  • Remark
  • ...and 190 more