$D$-module and $F$-module length of local cohomology modules

TL;DR

This paper investigates the lengths of local cohomology modules over polynomial or power series rings in the categories of D-modules and F-modules, providing bounds and examples that reveal their algebraic complexity.

Contribution

It establishes polynomial bounds on D-module length and bounds on F-module length in terms of Frobenius stable parts, highlighting differences between D- and F-module lengths.

Findings

D-module length is polynomially bounded by generator degrees

F-module length bounds depend on Frobenius stable parts and special primes

Examples show D- and F-module lengths can differ significantly

Abstract

Let $R$ be a polynomial or power series ring over a field $k$. We study the length of local cohomology modules $H^j_I(R)$ in the category of $D$-modules and $F$-modules. We show that the $D$-module length of $H^j_I(R)$ is bounded by a polynomial in the degree of the generators of $I$. In characteristic $p>0$ we obtain upper and lower bounds on the $F$-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of $R/I$. The obtained upper bound is sharp if $R/I$ is an isolated singularity, and the lower bound is sharp when $R/I$ is Gorenstein and $F$-pure. We also give an example of a local cohomology module that has different $D$-module and $F$-module lengths.