Supergeometry and Arithmetic Geometry
A. Schwarz, I. Shapiro

TL;DR
This paper introduces supergeometry concepts over rings, including p-adic superspaces, and applies them to construct the Frobenius map in p-adic cohomology of smooth projective varieties.
Contribution
It defines superspaces over rings and introduces p-adic superspaces, providing a new framework for understanding Frobenius maps in p-adic cohomology.
Findings
Defined superspaces as functors on supercommutative algebras.
Introduced p-adic superspaces for the first time.
Constructed Frobenius map transparently using supergeometry.
Abstract
We define a superspace over a ring as a functor on a subcategory of the category of supercommutative -algebras. As an application the notion of a -adic superspace is introduced and used to give a transparent construction of the Frobenius map on -adic cohomology of a smooth projective variety over the ring of -adic integers.
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