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

Xiaodong Yi

TL;DR

This work establishes finiteness of $\mathcal{D}_{X}$-module cohomology for $\mathcal{O}_{X}$-coherent modules on smooth proper schemes $X$ in characteristic $p$ by translating $\mathcal{D}_{X}$-modules into $F$-divided sheaves. It proves a cohomological boundedness theorem: for any proper $X$ over $k$ and any $F$-divided sheaf $(\mathcal{E}_{n})$ with a coherent twist $\mathcal{F}$, the values $h^{i}(X, \mathcal{E}_{n}\otimes\mathcal{F})$ are uniformly bounded in $n$ for each $i$. The projective case is established via singular Riemann–Roch and boundedness of Hilbert polynomials, then extended to general $X$ by dimension induction using Leray and Chow-type reductions. Together with an exact sequence relating $H^{i}_{\mathcal{D}_{X}}(X,\mathcal{E})$ to inverse limits of $H^{i}(X,\mathcal{E}_{n})$, the authors deduce finiteness of $H^{i}_{\mathcal{D}_{X}}(X,\mathcal{E})$ for all $i$, providing a positive-characteristic analogue of finiteness properties for $\mathcal{D}$-modules.

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.

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

TL;DR

This work establishes finiteness of -module cohomology for -coherent modules on smooth proper schemes in characteristic by translating -modules into -divided sheaves. It proves a cohomological boundedness theorem: for any proper over and any -divided sheaf with a coherent twist , the values are uniformly bounded in for each . The projective case is established via singular Riemann–Roch and boundedness of Hilbert polynomials, then extended to general by dimension induction using Leray and Chow-type reductions. Together with an exact sequence relating to inverse limits of , the authors deduce finiteness of for all , providing a positive-characteristic analogue of finiteness properties for -modules.

Abstract

Let be a smooth proper scheme over an algebraically closed field in characteristic . In this short note, by interpreting -modules as -divided sheaves and establishing a cohomological boundedness property for -divided sheaves, we prove that any -coherent -module has finite dimensional -module cohomology.

Paper Structure

This paper contains 9 sections, 18 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathcal{D}$-modules and $F$-divided sheaves
  4. Background in intersection theory
  5. Boundedness of sheaves
  6. Boundedness of sheaves and cohomology
  7. The projective case
  8. Cohomological boundedness in general
  9. The finiteness of $\mathcal{D}$-module cohomology

Key Result

Theorem 1.1

Let $X$ be a smooth proper scheme over an algebraically closed field of characteristic $p$. For any $\mathcal{O}_{X}$-coherent $\mathcal{D}_{X}$-module $\mathcal{E}$ and any cohomology degree $i$, the $\mathcal{D}_{X}$-module cohomology $H^{i}_{\mathcal{D}_{X}}(X,\mathcal{E})$ is finite dimensional

Theorems & Definitions (42)

  • Theorem 1.1: Theorem \ref{['d_mod']}
  • Theorem 1.2: Theorem \ref{['coherent']}
  • Definition 2.1
  • Lemma 2.2: Lemma 4.2 10.2748/tmj.20200727
  • proof
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 32 more