2022/10/23 by Christopher Dodd, Dodd, Christopher
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2210.12611
openalex publication_date 2022/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this paper is to develop a new theory of gauges in mixed characteristic. Namely, let k be a perfect field of characteristic p>0 and W(k) the p-typical Witt vectors. Making use of Berthelot's arithmetic differential operators, we define for a smooth formal scheme \mathfrakX over W(k), a new sheaf of algebras \widehatD_\mathfrakX(0,1) which can be considered a higher dimensional analogue of the (commutative) Dieudonne ring. Modules over this sheaf of algebras can be considered the analogue (over \mathfrakX) of the gauges of Ekedahl and Fontain-Jannsen. We show that modules over \widehatD_\mathfrakX(0,1) admit all of the usual D-module operations, and we prove a robust generalization of Mazur's theorem in this context. Finally, we show that an integral form of a mixed Hodge module of geometric origin admits, after a suitable p-adic completion, the structure of a module over \widehatD_\mathfrakX(0,1). This allows us to prove a version of Mazur's theorem for the intersection cohomology and the ordinary cohomology of an arbitrary quasiprojective variety defined over a number field.