Vol. 14, No. 9, 2020

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Iterated local cohomology groups and Lyubeznik numbers for determinantal rings

András C. Lőrincz and Claudiu Raicu

Vol. 14 (2020), No. 9, 2533–2569
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 nonsquare 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.

Keywords
determinantal varieties, local cohomology, Lyubeznik numbers, equivariant D-modules
Mathematical Subject Classification 2010
Primary: 13D45
Secondary: 13D07, 14M12
Milestones
Received: 20 November 2019
Revised: 27 March 2020
Accepted: 29 April 2020
Published: 13 October 2020
Authors
András C. Lőrincz
Department of Mathematics
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
Claudiu Raicu
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States
Institute of Mathematics “Simion Stoilow” of the Romanian Academy
Bucharest
Romania