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Sheared displays and $p$-divisible groups

Manuel Hoff, Eike Lau

TL;DR

This work develops a Dieudonné theory for $p$-divisible groups via sheared Witt vectors ${}^sW(R)$, proving an equivalence between all $p$-divisible groups and plain sheared displays and thereby extending classical displays to the full BT category. It builds a comprehensive framework of frames and windows in the sheared setting, constructs a Tate-module functor and a crystalline Dieudonné functor ${}^{s\!}D_R$, and establishes descent results across fpqc, syntomic, and $p^\infty$-root topologies. The approach yields a robust, mostly non-higher-categorical path to the Dieudonné theory in the sheared context, aligns with known results in the perfect-reduction case, and provides a platform to address Drinfeld’s conjectures and related prismatic perspectives, including $p=2$ considerations via crystalline techniques. Overall, the paper unifies displays and BT groups through sheared structures, enabling explicit constructions, deformation analyses, and dualities that underpin a full Dieudonné equivalence in broad generality.

Abstract

We develop a Dieudonné theory for $p$-divisible groups using sheared Witt vectors.

Sheared displays and $p$-divisible groups

TL;DR

This work develops a Dieudonné theory for -divisible groups via sheared Witt vectors , proving an equivalence between all -divisible groups and plain sheared displays and thereby extending classical displays to the full BT category. It builds a comprehensive framework of frames and windows in the sheared setting, constructs a Tate-module functor and a crystalline Dieudonné functor , and establishes descent results across fpqc, syntomic, and -root topologies. The approach yields a robust, mostly non-higher-categorical path to the Dieudonné theory in the sheared context, aligns with known results in the perfect-reduction case, and provides a platform to address Drinfeld’s conjectures and related prismatic perspectives, including considerations via crystalline techniques. Overall, the paper unifies displays and BT groups through sheared structures, enabling explicit constructions, deformation analyses, and dualities that underpin a full Dieudonné equivalence in broad generality.

Abstract

We develop a Dieudonné theory for -divisible groups using sheared Witt vectors.
Paper Structure (73 sections, 109 theorems, 106 equations)

This paper contains 73 sections, 109 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Displays and $p$-divisible groups
  3. Sheared Witt vectors
  4. Sheared displays
  5. Sheared functors
  6. Outline
  7. Details on the Tate module functor
  8. Details on the sheared display functor
  9. Comment on descent
  10. Acknowledgements
  11. Notation
  12. Sheared Witt vectors
  13. First flat cohomology of $\hat{W}$
  14. Definitions and basic properties
  15. Rings with perfect reduction
  16. ...and 58 more sections

Key Result

Theorem A

There is a functor ${}^s\mathfrak{Z}_R\colon \mathop{\mathrm{{}^sDisp}}\nolimits(R)\to\mathop{\mathrm{BT}}\nolimits(R)$ which is determined by the following property. For $G={}^s\mathfrak{Z}_R(M)$ there is a natural exact sequence of fpqc sheaves on $\mathop{\mathrm{Spec}}\nolimits R$ where the sheaves $\tilde{M}_i$ are defined by base change of sheared displays. The functor ${}^s\mathfrak{Z}_R$ i

Theorems & Definitions (297)

  • Theorem A
  • Theorem B
  • Proposition C
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 287 more