Prismatic G-displays and descent theory

Ito, Kazuhiro

doi:10.2140/ant.2025.19.1685

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Prismatic $G$-displays and descent theory

Kazuhiro Ito

Vol. 19 (2025), No. 9, 1685–1770

DOI: 10.2140/ant.2025.19.1685

Abstract

For a smooth affine group scheme $G$ over the ring of $p$ -adic integers $ℤ_{p}$ and a cocharacter $μ$ of $G$ , we study $G$ - $μ$ -displays over the prismatic site of Bhatt and Scholze. In particular, we obtain several descent results for them. If $G = {GL}_{n}$ , then our $G$ - $μ$ -displays can be thought of as Breuil–Kisin modules with some additional conditions. The relation between our $G$ - $μ$ -displays and prismatic $F$ -gauges introduced by Drinfeld and Bhatt–Lurie is also discussed.

In fact, our main results are formulated and proved for smooth affine group schemes over the ring of integers $𝒪_{E}$ of any finite extension $E$ of $ℚ_{p}$ by using $𝒪_{E}$ -prisms, which are $𝒪_{E}$ -analogues of prisms.

Keywords

prisms, displays, $p$-divisible groups, prismatic $F$-gauges

Mathematical Subject Classification

Primary: 14F30

Secondary: 14G45, 14L05

Milestones

Received: 11 June 2023

Revised: 31 August 2024

Accepted: 18 October 2024

Published: 21 July 2025

Authors

Kazuhiro Ito
Mathematical Institute Tohoku University Sendai Japan

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Vol. 19, No. 9, 2025