Higher dimensional dominoes in de Rham-Witt cohomology

2026/07/28 by Yuanning Zhang
Mathematics · #math.AG #msc:14F22 #msc:14F30 #msc:14K05 #msc:14L05

paper · pdf

50 pages

arxiv created 2026/07/28 · arxiv updated 2026/07/30

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.

Citations

• Brauer groups of abelian varieties over fields of finite characteristic
• Boundedness of the p-primary torsion of the Brauer group of products of varieties
• The prismatization of p-adic formal schemes
• Representability of cohomology of finite flat abelian group schemes
• Prismatization
• Counterexamples to Hochschild--Kostant--Rosenberg in characteristic p
• Derived invariants from topological Hochschild homology
• Topological Hochschild homology and integral p-adic Hodge theory
• The Brauer group of Kummer surfaces and torsion of elliptic curves

Related

• Arbitrarily large jumps in the de Rham and Hodge cohomology of families in characteristic p
• Derived invariants from topological Hochschild homology
• Dimensional vanishing of the saturated de Rham-Witt complex
• On p-torsions of geometric Brauer groups
• Crystalline cohomology and de Rham cohomology