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$h$-topology for rigid spaces and an application to $p$-adic Simpson correspondence

Hanlin Cai, Zeyu Liu

TL;DR

This work develops the $h$-topology for rigid spaces and uses it to extend the $p$-adic Simpson correspondence to singular proper rigid spaces over $C$. By proving foundational flatness results, a rigid-analytic platification by blowups, and Zariski-type refinements of fppf covers, it builds a robust descent framework for vector bundles and Higgs bundles. Through simplicial/hypercover techniques, it reduces singular cases to smooth ones and establishes an equivalence between pro-étale vector bundles and Higgs bundles on the $h$-site, together with a cohomology comparison that aligns pro-étale and Higgs cohomology. The results significantly generalize Heuer’s correspondence and open avenues for non-smooth non-abelian $p$-adic Hodge theory, including a structured cohomological dimension and descent theory for Higgs data on the $h$-site.

Abstract

In this paper, we study the $h$-topology for rigid spaces. Along the way, we establish several foundational results on morphisms of rigid spaces: we prove generic flatness and openness of the flat locus in the rigid setting, and we show that (for affinoid rigid spaces) strict transforms become flat after a blowup. Moreover, we show that any fppf cover admits a quasi-finite refinement and prove a version of Zariski's main theorem for rigid spaces. As an application, we deduce that for a proper rigid space $X$ over $C$, the category of pro-étale vector bundles on $X$ is equivalent to the category of Higgs bundles on the $h$-site of $X$, thereby generalizing Heuer's results to the singular setting.

$h$-topology for rigid spaces and an application to $p$-adic Simpson correspondence

TL;DR

This work develops the -topology for rigid spaces and uses it to extend the -adic Simpson correspondence to singular proper rigid spaces over . By proving foundational flatness results, a rigid-analytic platification by blowups, and Zariski-type refinements of fppf covers, it builds a robust descent framework for vector bundles and Higgs bundles. Through simplicial/hypercover techniques, it reduces singular cases to smooth ones and establishes an equivalence between pro-étale vector bundles and Higgs bundles on the -site, together with a cohomology comparison that aligns pro-étale and Higgs cohomology. The results significantly generalize Heuer’s correspondence and open avenues for non-smooth non-abelian -adic Hodge theory, including a structured cohomological dimension and descent theory for Higgs data on the -site.

Abstract

In this paper, we study the -topology for rigid spaces. Along the way, we establish several foundational results on morphisms of rigid spaces: we prove generic flatness and openness of the flat locus in the rigid setting, and we show that (for affinoid rigid spaces) strict transforms become flat after a blowup. Moreover, we show that any fppf cover admits a quasi-finite refinement and prove a version of Zariski's main theorem for rigid spaces. As an application, we deduce that for a proper rigid space over , the category of pro-étale vector bundles on is equivalent to the category of Higgs bundles on the -site of , thereby generalizing Heuer's results to the singular setting.
Paper Structure (17 sections, 62 theorems, 103 equations)

This paper contains 17 sections, 62 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. $h$-topology for rigid spaces
  4. Flat morphisms
  5. Quasi-finite morphisms
  6. $h$-topology
  7. $h$-descent of vector bundles and Higgs bundles
  8. Descent of vector bundles
  9. Descent of differentials
  10. Descent of Higgs bundles
  11. Simplicial rigid spaces and $p$-adic Simpson correspondence
  12. Generalities on simplicial rigid spaces
  13. Simplicial Higgs bundles
  14. $p$-adic Simpson correspondence for smooth proper simplicial rigid spaces
  15. $p$-adic Simpson correspondence for proper rigid spaces
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X$ be a proper rigid space over $C$. Fix a $1$-truncated smooth proper $h$-hypercover of $X$ with a lift to $B^+_{\mathrm{dR}}/\xi^2$ and an exponential map $\exp:C\to 1+\mathfrak{m}$. Then there exists an equivalence of symmetric monoidal categories:

Theorems & Definitions (147)

  • Theorem 1.1: \ref{['thm. p-adic simpson for proper rigid space']}
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: \ref{['thm. cohomology comparison']}
  • Remark 1.5
  • Definition 1.6: \ref{['def. h topology']}
  • Remark 1.7
  • Theorem 1.8: \ref{['thm.flat on dense open']}, \ref{['prop.openness of flat locus']}
  • Remark 1.9
  • Proposition 1.10: \ref{['prop.platification by blowup']}
  • ...and 137 more