On the u^{\infty}-torsion submodule of prismatic cohomology

TL;DR

This paper studies the maximal finite length submodule of prismatic cohomology in algebraic geometry, revealing its role in understanding p-adic cohomology phenomena, with applications to Albanese varieties and Fontaine-Laffaille theory.

Contribution

It introduces a detailed analysis of the u^{ infty}-torsion submodule in prismatic cohomology, connecting it to geometric and arithmetic problems, and provides new examples and counterexamples.

Findings

Controlled discrepancy between Albanese varieties in characteristic p

Identified kernel of the p-adic étale cohomology specialization map

Constructed an example negating a question of Breuil

Abstract

We investigate the maximal finite length submodule of the Breuil-Kisin prismatic cohomology of a smooth proper formal scheme over a p-adic ring of integers. This submodule governs pathology phenomena in integral p-adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in characteristic p, and (2) kernel of the specialization map in p-adic \'etale cohomology. As an arithmetic application, we study the boundary case of the theory due to Fontaine-Laffaille, Fontaine-Messing, and Kato. Also included is an interesting example, generalized from a construction in Bhatt-Morrow-Scholze's work, which (1) illustrates some of our theoretical results being sharp, and (2) negates a question of Breuil.