Absolute calculus and prismatic crystals on cyclotomic rings

Gros, Michel; Le Stum, Bernard; Quirós, Adolfo

doi:10.2140/ant.2026.20.1301

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Absolute calculus and prismatic crystals on cyclotomic rings

Michel Gros, Bernard Le Stum and Adolfo Quirós

Vol. 20 (2026), No. 7, 1301–1384

DOI: 10.2140/ant.2026.20.1301

Abstract

Let $p$ be a prime, $W$ the ring of Witt vectors of a perfect field $k$ of characteristic $p$ and $ζ$ a primitive $p$ -th root of unity. We introduce a new notion of calculus over $W$ , which we call absolute calculus. It may be seen as a singular version of the $q$ -calculus used in previous work, in the sense that the role of the coordinate is now played by $q$ itself. We show that what we call a weakly nilpotent $Δ$ -connection on a finite free module is equivalent to a prismatic vector bundle on $W [ζ]$ . As a corollary of a theorem of Bhatt and Scholze, we finally obtain that a $Δ$ -connection with a frobenius structure on a finite free module is equivalent to a lattice in a crystalline representation. We also consider the case of de Rham prismatic crystals as well as Hodge–Tate prismatic crystals.

Keywords

prismatic crystal, q-analog of differential operators, twisted connection, frobenius structure, crystalline representation, de Rham cohomology

Mathematical Subject Classification

Primary: 14F30, 14F40

Milestones

Received: 7 November 2023

Revised: 21 May 2025

Accepted: 27 June 2025

Published: 14 August 2026

Authors

Michel Gros
IRMAR, Université de Rennes Campus de Beaulieu Rennes France

Bernard Le Stum
IRMAR, Université de Rennes Campus de Beaulieu Rennes France

Adolfo Quirós
Departamento de Matemáticas, Facultad de Ciencias Universidad Autónoma de Madrid Madrid Spain

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Vol. 20, No. 7, 2026