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$G$-torsors on perfectoid spaces

Ben Heuer

TL;DR

The paper addresses the problem of comparing non-abelian G-torsors on adic spaces in étale and v-topologies within p-adic geometry. It develops an approximation framework for non-abelian cohomology, leverages the $p$-adic exponential and Lie algebra to reduce torsors to open subgroups, and proves an equivalence of $G$-torsors on perfectoid spaces for arbitrary rigid $G$; it also provides integral and generalized representation results connecting to vector bundles. Key contributions include the reduction-of-structure-group to open subgroups, the integral analogue for $\mathcal{O}^+$-modules (and generalized representations), and the application to the $p$-adic Simpson correspondence via $v$-vector bundles. The work significantly advances non-abelian p-adic Hodge theory by showing that, on perfectoid bases, non-abelian torsor data is robust across topologies, with broad implications for moduli and analytic representation theory in p-adic geometry.

Abstract

For any rigid analytic group variety $G$ over a non-archimedean field $K$ over $\mathbb Q_p$, we study $G$-torsors on adic spaces over $K$ in the $v$-topology. Our main result is that on perfectoid spaces, $G$-torsors in the étale and $v$-topology are equivalent. This generalises the known cases of $G=\mathbb G_a$ and $G=\mathrm{GL}_n$ due to Scholze and Kedlaya--Liu. On a general adic space $X$ over $K$, where there can be more $v$-topological $G$-torsors than étale ones, we show that for any open subgroup $U\subseteq G$, any $G$-torsor on $X_v$ admits a reduction of structure group to $U$ étale-locally on $X$. This has applications in the context of the $p$-adic Simpson correspondence: For example, we use it to show that on any adic space, generalised $\mathbb Q_p$-representations are equivalent to $v$-vector bundles.

$G$-torsors on perfectoid spaces

TL;DR

The paper addresses the problem of comparing non-abelian G-torsors on adic spaces in étale and v-topologies within p-adic geometry. It develops an approximation framework for non-abelian cohomology, leverages the -adic exponential and Lie algebra to reduce torsors to open subgroups, and proves an equivalence of -torsors on perfectoid spaces for arbitrary rigid ; it also provides integral and generalized representation results connecting to vector bundles. Key contributions include the reduction-of-structure-group to open subgroups, the integral analogue for -modules (and generalized representations), and the application to the -adic Simpson correspondence via -vector bundles. The work significantly advances non-abelian p-adic Hodge theory by showing that, on perfectoid bases, non-abelian torsor data is robust across topologies, with broad implications for moduli and analytic representation theory in p-adic geometry.

Abstract

For any rigid analytic group variety over a non-archimedean field over , we study -torsors on adic spaces over in the -topology. Our main result is that on perfectoid spaces, -torsors in the étale and -topology are equivalent. This generalises the known cases of and due to Scholze and Kedlaya--Liu. On a general adic space over , where there can be more -topological -torsors than étale ones, we show that for any open subgroup , any -torsor on admits a reduction of structure group to étale-locally on . This has applications in the context of the -adic Simpson correspondence: For example, we use it to show that on any adic space, generalised -representations are equivalent to -vector bundles.
Paper Structure (15 sections, 51 theorems, 86 equations)

This paper contains 15 sections, 51 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. $\boldsymbol{G}$-torsors on rigid spaces
  3. Reduction of structure group
  4. Generalised representations on rigid spaces
  5. Generalised representations
  6. Approximation of non-abelian sheaf cohomology
  7. Generalised representations on perfectoid spaces
  8. G-torsors for rigid groups G on adic spaces
  9. The correspondence between rigid groups and Lie algebras
  10. The $\boldsymbol{p}$-adic exponential of a rigid group $\boldsymbol{G}$
  11. $\boldsymbol{G}$-torsors in the étale and $\boldsymbol{v}$-topology
  12. Reduction of structure group to open subgroups
  13. Approximation for quotient sheaves
  14. $\boldsymbol{p}$-adic integral subgroups for rigid groups of good reduction
  15. $\boldsymbol{G}$-torsors on perfectoid spaces

Key Result

Theorem 1.1

Let $X$ be a perfectoid space, and let $G$ be a rigid group over $K$. The functor is an equivalence of categories. If $G$ is commutative, then we more generally have $R\nu_{\ast}G=G$.

Theorems & Definitions (127)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • ...and 117 more