Cartier Crystals

TL;DR

This paper develops the theory of Cartier crystals by analyzing their derived categories and cohomological operations, establishing foundational properties and setting the stage for future duality relations with $ au$-crystals.

Contribution

It introduces the derived category framework for Cartier crystals, defines key cohomological operations, and proves their properties under certain morphism conditions.

Findings

Rf_* preserves coherent cohomology up to nilpotence for finite type morphisms.

f^! has bounded cohomological dimension.

Cohomological operations are well-defined on Cartier crystals after localization.

Abstract

Building on our previous work "Cartier modules: finiteness results" we start in this manuscript an in depth study of the derived category of Cartier modules and the cohomological operations which are defined on them. After localizing at the sub-category of locally nilpotent objects we show that for a morphism essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for Cartier crystals. We show that, if $f$ is of finite type (but not necessarily proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$ has bounded cohomological dimension. In a sequel we will explain how Grothendieck-Serre Duality relates our theory of Cartier Crystals to the theory of $\tau$-crystals as developed by Pink and the second author.