Supergeometry and Arithmetic Geometry

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 $R$ as a functor on a subcategory of the category of supercommutative $R$-algebras. As an application the notion of a $p$-adic superspace is introduced and used to give a transparent construction of the Frobenius map on $p$-adic cohomology of a smooth projective variety over the ring of $p$-adic integers.