A note on cohomological boundedness for $F$-divided sheaves and $\mathcal{D}$-modules

TL;DR

This paper proves that on smooth proper schemes over algebraically closed fields of characteristic p, all coherent D-modules have finite-dimensional cohomology by linking D-modules to F-divided sheaves and establishing cohomological boundedness.

Contribution

It introduces a new interpretation of D-modules as F-divided sheaves and proves a cohomological boundedness property leading to finite-dimensional cohomology.

Findings

Finite-dimensional cohomology for coherent D-modules.

Cohomological boundedness property for F-divided sheaves.

Interpretation of D-modules as F-divided sheaves.

Abstract

Let $X$ be a smooth proper scheme over an algebraically closed field $k$ in characteristic $p$. In this short note, by interpreting $\mathcal{D}_{X}$-modules as $F$-divided sheaves and establishing a cohomological boundedness property for $F$-divided sheaves, we prove that any $\mathcal{O}_{X}$-coherent $\mathcal{D}_{X}$-module has finite dimensional $\mathcal{D}_{X}$-module cohomology.