Higher displays arising from filtered de Rham-Witt complexes

2017/11/27 by Gregory, Oliver, Langer, Andreas
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1711.09940

Abstract

For a smooth projective scheme X over a ring R on which p is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine's strongly divisible lattices. Frobenius-divisibilty is induced by the Nygaard filtration on the relative de Rham-Witt complex. For a nilpotent PD-thickening S/R we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham-Witt complex associated to a lifting of X over S. We prove that there is a crystal of relative displays if moreover the mod p reduction of X has a smooth and versal deformation space.

Related

• Hodge-Witt decomposition of relative crystalline cohomology
• The filtered Poincaré lemma in higher level (with applications to algebraic groups)
• G-displays of Hodge type and formal p-divisible groups
• Crystalline prisms: Reflections and diffractions, present and past
• Prismatic crystals for smooth schemes in characteristic p with Frobenius lifting mod p²