The Greenberg Functor is Site Cocontinuous

TL;DR

This paper establishes a new topology on formal schemes over p-adic integers that makes the Greenberg functor site cocontinuous, enabling a geometric interpretation of p-adic quasicharacters via local systems.

Contribution

It introduces a topology on formal schemes over p-adic integers that ensures the Greenberg functor's adjoint is site cocontinuous, linking fundamental groups and quasicharacters.

Findings

Defined a topology making the Greenberg functor site cocontinuous.

Established an isomorphism between fundamental groups of related schemes.

Geometrized quasicharacters of p-adic tori as local systems.

Abstract

In this paper we show that it is possible to define a topology on the category of formal schemes over a ring of $p$-adic integers such that the left adjoint of the Greenberg Transform is a site cocontinuous functor when we equip the category of schemes over the residue field with the \'etale topology. We show furthermore that this topology allows us to give an isomorphism between the corresponding fundamental groups, and use this isomorphism to show that it is possible to geometrize the quasicharacters of a $p$-adic torus by a local system on a formal scheme over the ring of $p$-adic integers.