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v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

Johannes Anschütz, Ben Heuer, Arthur-César Le Bras

TL;DR

This paper introduces a canonical geometric framework for describing continuous semilinear $C$-representations of the Galois group $G_K$ of a $p$-adic field via the Hodge--Tate locus on the Cartier--Witt stack. It constructs the HT-stack based functors $oldsymbol{eta}^+_\pi$ and $oldsymbol{eta}$ that relate HT data to Sen-type modules, yielding a full and faithful bridge from HT complexes to $K[oldsymbol{ heta}_\pi]$-modules and to $v$-vector bundles on $ ext{Spa}(K)_v$, with the essential image identified as nearly Hodge--Tate representations. A central innovation is the introduction of the period-ring–like object $B_{ ext{en}}$, which connects the HT/sen-theory perspective to Colmez' $B_{ ext{Sen}}$ and makes Galois cohomology computations tractable; the paper proves $K o R\, ext{Γ}(G_K,B_{ ext{en}})$ is an isomorphism and analyzes $H^i(G_K,B_{ ext{en}})$ in detail. The HT-stack viewpoint also yields new arithmetic insights, such as a precise short exact sequence for the $v$-Picard group of $p$-adic fields, demonstrating subtle interactions between local class field theory, Brauer groups, and the HT-Sen correspondence beyond rank-one phenomena. Overall, the work provides a canonical, functorial, and computable description of $C$-semilinear $G_K$-representations in terms of HT data and sharpened period-ring techniques, with potential applications to higher-dimensional p-adic non-abelian Hodge theory and v-descent for HT-structured objects.

Abstract

We describe the category of continuous semilinear representations and their cohomology for the Galois group of a $p$-adic field $K$ with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring $B_{\mathrm{Sen}}$ in a geometric way.

v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

TL;DR

This paper introduces a canonical geometric framework for describing continuous semilinear -representations of the Galois group of a -adic field via the Hodge--Tate locus on the Cartier--Witt stack. It constructs the HT-stack based functors and that relate HT data to Sen-type modules, yielding a full and faithful bridge from HT complexes to -modules and to -vector bundles on , with the essential image identified as nearly Hodge--Tate representations. A central innovation is the introduction of the period-ring–like object , which connects the HT/sen-theory perspective to Colmez' and makes Galois cohomology computations tractable; the paper proves is an isomorphism and analyzes in detail. The HT-stack viewpoint also yields new arithmetic insights, such as a precise short exact sequence for the -Picard group of -adic fields, demonstrating subtle interactions between local class field theory, Brauer groups, and the HT-Sen correspondence beyond rank-one phenomena. Overall, the work provides a canonical, functorial, and computable description of -semilinear -representations in terms of HT data and sharpened period-ring techniques, with potential applications to higher-dimensional p-adic non-abelian Hodge theory and v-descent for HT-structured objects.

Abstract

We describe the category of continuous semilinear representations and their cohomology for the Galois group of a -adic field with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring in a geometric way.
Paper Structure (15 sections, 28 theorems, 239 equations)

This paper contains 15 sections, 28 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. The perspective of Sen theory
  3. The perspective of the $p$-adic Simpson correspondence
  4. A new perspective via the Hodge--Tate stack
  5. Plan of the paper
  6. Complexes on the Hodge--Tate stack
  7. Recollections on the Hodge--Tate stack for $\mathcal{O}_K$
  8. From complexes on the Hodge--Tate stack to Sen modules
  9. An analytic variant
  10. Galois actions on Hodge--Tate stacks
  11. Explicit functoriality
  12. Comparison to Colmez' ring $B_{\mathrm{Sen}}$
  13. Calculation of Galois cohomology of $B_\mathrm{en}$
  14. $C$-semilinear Galois representations via Hodge--Tate stacks
  15. The $\mathrm{v}$-Picard group of $p$-adic fields

Key Result

Theorem 1.2

There is a natural equivalence of categories where $L|K$ ranges through finite Galois extensions of $K$ in $C$ and $\mathrm{Vec}$ is the category of vector bundles on the stack quotient $[\mathrm{Spf}(\mathcal{O}_L)^{\rm HT}/\mathrm{Gal}(L/K)]$.

Theorems & Definitions (71)

  • Theorem 1.2
  • Theorem 1.3: \ref{['sec:c_k-semil-galo-1-pullback-fully-faithful-up-to-quasi-isogeny']}, \ref{['sec:c_k-semil-galo-3-description-of-essential-image']}, \ref{['sec:an-analytic-variant-corollary-analytic-variant']}
  • Remark 1.4
  • Theorem 1.5
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3: bhatt2022prismatization
  • Example 2.4
  • Theorem 2.5
  • Lemma 2.6
  • ...and 61 more