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Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces

Johannes Anschütz, Ben Heuer, Arthur-César Le Bras

TL;DR

The paper develops a derived, non-abelian p-adic Hodge-theory framework for v-perfect complexes on rigid spaces by connecting Perf(X^HT) with Perf(X_v) via a fully faithful α_X^* up to isogeny. It provides a Higgs–Sen (and Hitchin-small) description of HT-data, enabling a derived p-adic Simpson functor that unifies local and global Simpson theories in smoothoid and arithmetic settings. The approach leverages prismatization and the Hodge–Tate stack to translate HT–Higgs structures into v-perspectives, and globalizes the construction through square-zero lifts, yielding a global Simpson correspondence in favorable cases. The results extend Faltings–Liu–Zhu–Wang paradigms to derived categories of perfect complexes, clarify Galois/cohomological obstructions via explicit period-ring cohomology, and offer a uniform framework for future extensions to principal bundles and imperfect residue fields, with broad potential impact on p-adic non-abelian Hodge theory.

Abstract

Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that is smooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring of integers of a $p$-adic field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfect complexes on the v-site of the generic fibre of $X$. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adic Simpson functor.

Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces

TL;DR

The paper develops a derived, non-abelian p-adic Hodge-theory framework for v-perfect complexes on rigid spaces by connecting Perf(X^HT) with Perf(X_v) via a fully faithful α_X^* up to isogeny. It provides a Higgs–Sen (and Hitchin-small) description of HT-data, enabling a derived p-adic Simpson functor that unifies local and global Simpson theories in smoothoid and arithmetic settings. The approach leverages prismatization and the Hodge–Tate stack to translate HT–Higgs structures into v-perspectives, and globalizes the construction through square-zero lifts, yielding a global Simpson correspondence in favorable cases. The results extend Faltings–Liu–Zhu–Wang paradigms to derived categories of perfect complexes, clarify Galois/cohomological obstructions via explicit period-ring cohomology, and offer a uniform framework for future extensions to principal bundles and imperfect residue fields, with broad potential impact on p-adic non-abelian Hodge theory.

Abstract

Let be a quasi-compact quasi-separated -adic formal scheme that is smooth either over a perfectoid -algebra or over some ring of integers of a -adic field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of up to isogeny to perfect complexes on the v-site of the generic fibre of . Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived -adic Simpson functor.
Paper Structure (34 sections, 69 theorems, 269 equations)

This paper contains 34 sections, 69 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. v-perfect complexes via the Hodge--Tate stack
  3. The smoothoid case
  4. Globalization
  5. A derived $p$-adic Simpson correspondence
  6. The arithmetic case
  7. Relation to previous work
  8. Outlook
  9. Plan of the paper
  10. Notations
  11. A criterion for fully faithfulness
  12. Explicit naturality for the prismatization
  13. Explicit naturality
  14. Automorphisms of $\overline{\rho_A}$
  15. Application to group actions
  16. ...and 19 more sections

Key Result

Theorem 1.2

If $X$ is qcqs smoothoid as in (1) or arithmetic as in (2), then $\alpha_X^\ast$ is fully faithful.

Theorems & Definitions (177)

  • Theorem 1.2: \ref{['sec:tori-over-perfectoid']}, \ref{['sec:smooth-form-schem-galois-cohomo-in-arithmetic-case']}
  • Theorem 1.3: \ref{['sec:smoothoid-case-1-complexes-on-ht-in-split-case']}
  • Theorem 1.4: \ref{['sec:appl-Hodge--Tate-functor-for-x-lift']}, \ref{['sec:appl-Hodge--Tate-embedding-from-colimit']}
  • Theorem 1.5: \ref{['t:local-p-adic-Simpson-functor-geometric']}
  • Theorem 1.6: \ref{['sec:smoothoid-case-1-derived-local-p-adic-simpson-correspondence']}
  • Theorem 1.7: \ref{['sec:autom-overl-1-description-of-g-a']}, \ref{['sec:appl-Hodge--Tate-structure-of-x-ht-for-some-prismatic-lift']}
  • Theorem 1.8: \ref{['sec:arithmetic-case-1-derived-local-p-adic-simpson-in-arithmetic-case-plus-description-of-perf-on-x-ht']}
  • Lemma 2.1
  • proof
  • Proposition 2.3
  • ...and 167 more