Higher Displays Arising from Filtered de Rham–Witt Complexes

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-divisibility 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.