[Paper Review] Cartier Crystals

Manuel Blickle, Gebhard Böckle|arXiv (Cornell University)|Sep 4, 2013

Homotopy and Cohomology in Algebraic Topology21 references4 citations

TL;DR

This paper establishes foundational results on the derived category of Cartier modules, proving that for morphisms of finite type, the higher direct image functor $ Rf_* $ preserves coherent cohomology up to nilpotence and that the exceptional inverse image $ f^! $ has bounded cohomological dimension after localization at locally nilpotent objects. The work sets the stage for a duality theory linking Cartier crystals to $ \mathcal{D} $-modules via Grothendieck-Serre duality.

ABSTRACT

Building on our previous work 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 $ au$-crystals as developed by Pink and the second author.

Motivation & Objective

• To develop the derived category theory of Cartier modules as a foundation for duality theorems.
• To study cohomological operations $ Rf_* $ and $ f^! $ in the context of Cartier crystals.
• To establish finiteness and boundedness properties for these functors under finite type morphisms.
• To localize at locally nilpotent objects to simplify cohomological behavior.
• To prepare the groundwork for relating Cartier crystals to $ \mathcal{D} $-crystals via Grothendieck-Serre duality.

Proposed method

• Localizing the derived category of Cartier modules at the subcategory of locally nilpotent objects to simplify cohomological structure.
• Applying homological algebra techniques to define and analyze $ Rf_* $ and $ f^! $ on Cartier crystals.
• Using finiteness results from prior work to control cohomological dimensions.
• Focusing on morphisms essentially of finite type to ensure well-behaved derived functors.
• Employing nilpotence considerations to stabilize coherent cohomology under $ Rf_* $.
• Leveraging the structure of Cartier modules to derive boundedness of $ f^! $.

Experimental results

Research questions

• RQ1 How do the derived functors $ Rf_* $ and $ f^! $ behave on Cartier crystals for morphisms of finite type?
• RQ2 To what extent does $ Rf_* $ preserve coherent cohomology in the derived category of Cartier modules?
• RQ3 What is the cohomological dimension of $ f^! $ after localization at locally nilpotent objects?
• RQ4 How can the derived category of Cartier modules be structured to support duality theorems?
• RQ5 What is the relationship between Cartier crystals and $ \mathcal{D} $-crystals in the context of Grothendieck-Serre duality?

Key findings

• For a morphism $ f $ of finite type, $ Rf_* $ preserves coherent cohomology up to nilpotence.
• The functor $ f^! $ has bounded cohomological dimension when restricted to Cartier crystals after localization at locally nilpotent objects.
• The derived functors $ Rf_* $ and $ f^! $ are well-defined on the localized derived category of Cartier modules.
• The localization process effectively removes nilpotent cohomological obstructions.
• The framework developed enables a future comparison with $ \mathcal{D} $-crystals via Grothendieck-Serre duality.
• The results provide a cohomological foundation for duality in positive characteristic via Cartier modules.