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On the Unipotent $p$-adic Simpson Correspondence

Thiago Solovera e Nery

Abstract

The goal of this paper is to show a (derived) $p$-adic Simpson correspondence for (locally) unipotent coefficients on smooth rigid-analytic varieties. Our results depend on a deformation to $\mathbf{B}_\mathtt{dr}^+/ξ^2$, and not on a choice of exponential (as required for more general coefficients). Our methods are inherently higher categorical, hinging on the theory of modules over $\mathbf{E}_\infty$-algebras.

On the Unipotent $p$-adic Simpson Correspondence

Abstract

The goal of this paper is to show a (derived) -adic Simpson correspondence for (locally) unipotent coefficients on smooth rigid-analytic varieties. Our results depend on a deformation to , and not on a choice of exponential (as required for more general coefficients). Our methods are inherently higher categorical, hinging on the theory of modules over -algebras.
Paper Structure (16 sections, 52 theorems, 131 equations)

This paper contains 16 sections, 52 theorems, 131 equations.

Table of Contents

  1. Pro-étale and $\mathtt{v}$-vector bundles
  2. Pro-étale and $\mathtt{v}$ vector bundles; local systems
  3. Unipotent bundles and ${\operatorname{\mathit{R}}} \nu_* \widehat{\mathcal{O}}_X$-modules.
  4. Higgs bundles
  5. Higgs bundles
  6. Derived Higgs bundles
  7. The symmetric monoidal structure, and the cohomology of Higgs bundles
  8. Unipotent Higgs bundles
  9. Locally unipotent Higgs bundles
  10. The correspondence
  11. The Hodge-Tate filtration
  12. The lift to $\mathbf{B}_{\mathtt{dR}}^+/\xi^2$
  13. Splitting the Hodge-Tate filtration
  14. Finishing the proof
  15. The cotangent complex and deformation theory
  16. ...and 1 more sections

Key Result

Theorem 1

Let $X$ be a smooth rigid-analytic space defined over a closed and complete $p$-adic field $C$ (or mixed characteristic perfectoid with all $p$-power roots of unity) endowed with a (flat) deformation to $\mathbf{B}_{\mathtt{dR}}^+/\xi^2$. Then there is an equivalence of symmetric monoidal abelian ca between pro-étale vector bundles and unipotent Higgs bundles on $X$. A derived analogue of this sta

Theorems & Definitions (142)

  • Theorem : \ref{['main_theorem']}
  • Corollary
  • Theorem 1.1.1
  • Remark 1.1.2
  • proof
  • Definition 1.1.3
  • Lemma 1.1.4
  • proof
  • Proposition 1.1.5
  • proof
  • ...and 132 more