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Crystalline representations and Wach modules in the imperfect residue field case

Abhinandan

TL;DR

This work extends Wach-module theory to the setting of an absolutely unramified base L with imperfect residue field, proving a direct equivalence between lattices inside p-adic crystalline G_L-representations and integral Wach modules for L. It introduces a Nygaard filtration framework that aligns with the filtered φ-module of the associated crystalline representation, and shows that, after inverting p, the constructions become exact, paralleling the classical perfect-residue-field theory. The strategy hinges on translating the perfect-case Wach-module machinery to the imperfect setting via (φ,Γ)-modules, overconvergent and analytic period rings, and careful control of Γ_L-actions. The results unify and extend several strands—Brinon–Trihan, Kisin-type constructions, and prismatic viewpoints—while providing explicit, constructive connections between lattices, Wach modules, and crystalline cohomology, with implications for p-adic vanishing cycles and deformation theories. These advances pave the way for a functorial deformation of D_cris in imperfect residue-field contexts and feed into subsequent work on relative Wach modules and syntomic cohomology.

Abstract

For an absolutely unramified field extension $L/\mathbb{Q}_p$ with imperfect residue field, we define and study Wach modules in the setting of $(\varphi,Γ)$-modules for $L$. Our main result establishes a direct equivalence between the category of lattices inside crystalline representations of the absolute Galois group of $L$ and the category of integral Wach modules for $L$. Moreover, we provide a direct relation between a rational Wach module equipped with the Nygaard filtration and the filtered $\varphi$-module of its associated crystalline representation.

Crystalline representations and Wach modules in the imperfect residue field case

TL;DR

This work extends Wach-module theory to the setting of an absolutely unramified base L with imperfect residue field, proving a direct equivalence between lattices inside p-adic crystalline G_L-representations and integral Wach modules for L. It introduces a Nygaard filtration framework that aligns with the filtered φ-module of the associated crystalline representation, and shows that, after inverting p, the constructions become exact, paralleling the classical perfect-residue-field theory. The strategy hinges on translating the perfect-case Wach-module machinery to the imperfect setting via (φ,Γ)-modules, overconvergent and analytic period rings, and careful control of Γ_L-actions. The results unify and extend several strands—Brinon–Trihan, Kisin-type constructions, and prismatic viewpoints—while providing explicit, constructive connections between lattices, Wach modules, and crystalline cohomology, with implications for p-adic vanishing cycles and deformation theories. These advances pave the way for a functorial deformation of D_cris in imperfect residue-field contexts and feed into subsequent work on relative Wach modules and syntomic cohomology.

Abstract

For an absolutely unramified field extension with imperfect residue field, we define and study Wach modules in the setting of -modules for . Our main result establishes a direct equivalence between the category of lattices inside crystalline representations of the absolute Galois group of and the category of integral Wach modules for . Moreover, we provide a direct relation between a rational Wach module equipped with the Nygaard filtration and the filtered -module of its associated crystalline representation.
Paper Structure (29 sections, 45 theorems, 93 equations)

This paper contains 29 sections, 45 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The classical case
  3. The imperfect residue field case
  4. Wach modules
  5. Main results
  6. Strategy for the proof of Theorem \ref{['intro_thm:crystalline_wach_equivalence']}
  7. Relation to other works
  8. Setup and notations
  9. Outline of the paper
  10. Period rings and - representations
  11. Period rings
  12. Crystalline period rings
  13. Rings of -
  14. Overconvergent rings
  15. Analytic rings
  16. ...and 14 more sections

Key Result

Theorem 1.1

The category of $G_L\textrm{-stable}$$\mathbb{Z}_p\textrm{-lattices}$ inside $p\textrm{-adic}$ crystalline representations of $G_L$ is equivalent to the category of Wach modules for $L$.

Theorems & Definitions (115)

  • Theorem 1.1: Corollary \ref{['cor:crystalline_wach_equivalence_imperfect']}
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6: Corollary \ref{['cor:crystalline_wach_equivalence_imperfect']}
  • Remark 1.7
  • Theorem 1.8: Corollary \ref{['cor:qdeformation_dcrys']}
  • Remark 1.9
  • Remark 1.10
  • ...and 105 more