Generalised representations as q-connections in integral $p$-adic Hodge theory

Abstract

We relate various approaches to coefficient systems in relative integral $p$-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over $A_{\text{inf}}$ inspired by Faltings, modules with q-connection in the sense of q-de Rham cohomology, crystals on the prismatic site of Bhatt--Scholze, and q-deformations of Higgs bundles.

Version 2 with new material: global comparison between prismatic F-crystals and relative Breuil--Kisin--Fargues modules, and details of the Dieudonné theory. Also minor changes and corrections

References in corpus (3)

• $q$-crystals and $q$-connections
• Prismatic and $q$-crystalline sites of higher level
• Revisiting derived crystalline cohomology

Cited by in corpus (3)

• Prismatic and $q$-crystalline sites of higher level
• Crystalline representations and Wach modules in the relative case
• Relative $(\phi ,\Gamma )$-modules and prismatic $F$-crystals