Iterated local cohomology groups and Lyubeznik numbers for determinantal rings

Lőrincz, András C.; Raicu, Claudiu

doi:10.2140/ant.2020.14.2533

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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

DOI: 10.2140/ant.2020.14.2533

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

Vol. 14, No. 9, 2020