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

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 étale 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. Contents1. Introduction 2. Various modules and their Galois representations 2.1. Kisin modules 2.2. Breuil modules 2.3. Fontaine-Laffaille modules 2.4. Relations to Galois representations 3. Boundary degree prismatic cohomology 3.1. Structure of u ∞ -torsion 3.2. Comparing Frobenius and Verschiebung 3.3. Induced Nygaard filtration 4. Geometric applications 4.1. The discrepancy of Albanese varieties 4.2. The p-adic specialization maps 4.3. Revisiting the integral Hodge-de Rham spectral sequence 5. Crystalline cohomology in boundary degree 5.1. Understand filtrations 5.2. Compute divided Frobenius 5.3. The connection 5.4. Fontaine-Laffaille and Breuil modules 5.5. Comparison to étale cohomology 6. An example 6.1. Raynaud's theorem on prolongations References