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

Shizhang Li; Tong Liu

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arXiv · 2204.03411

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

Shizhang Li·Tong Liu

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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.

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Shizhang Li, Tong Liu. 2022-04-07. On the u^{\infty}-torsion submodule of prismatic cohomology. https://arxiv.org/abs/2204.03411

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