On pristine morphisms

TL;DR

This paper studies a special class of flat morphisms in algebraic geometry called pristine morphisms, characterized by their relative Frobenius being an isomorphism, and explores their role in defining a Grothendieck topology for Cartier modules.

Contribution

It introduces the concept of pristine morphisms and demonstrates their connection to a new Grothendieck topology tailored for Cartier modules.

Findings

Pristine morphisms are characterized by their relative Frobenius being an isomorphism.

A natural Grothendieck topology associated with pristine morphisms is constructed.

This topology is optimized for the localization of Cartier modules.

Abstract

We investigate flat morphisms of schemes of positive characteristic whose relative Frobenius is an isomorphism, which we call pristine. We show that these give rise to a natural Grothendieck topology that is fine tuned for the localization of Cartier modules.