Higher dimensional dominoes in de Rham-Witt cohomology

Abstract

The de Rham-Witt cohomology of a smooth proper variety in characteristic $p$ contains a canonical piece called the domino, which is not finitely generated over the Witt vectors and carries the nonzero differentials of the slope spectral sequence. Beyond dimension one, dominoes were unclassified. We classify the two-dimensional ones and put a domino of any dimension into a normal form. To each domino we attach two unipotent groups, one formal and one perfect, and prove that their isogeny partitions agree. In degree two we recover the domino of a Mazur-Ogus variety from its crystalline cohomology, compute it for two families of supersingular abelian varieties, and bound the exponent of the $p$-primary Brauer group in terms of the $a$-number, for every prime $p$. This answers a question of Grammatica-Skorobogatov-Yang.