Iterated local cohomology groups and Lyubeznik numbers for determinantal rings

TL;DR

This paper provides an explicit method to compute iterated local cohomology groups and Lyubeznik numbers for determinantal rings, revealing their structure in characteristic zero and addressing a longstanding question.

Contribution

It introduces a new explicit recipe for calculating local cohomology groups and Lyubeznik numbers of determinantal rings, especially distinguishing between square and non-square matrices.

Findings

Explicit computation method for local cohomology groups.

Determination of Lyubeznik numbers for all generic determinantal rings.

Identification of indecomposable D-modules in the structure.

Abstract

We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable D-modules. For non-square matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster.