Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians

TL;DR

This paper computes the structure of local cohomology modules supported on ideals of maximal minors and Pfaffians, providing insights into Cohen-Macaulay modules and developing new computational methods.

Contribution

It offers a GL-equivariant description of local cohomology modules for specific ideals and introduces a novel method for computing Ext modules using geometric and duality techniques.

Findings

Explicit descriptions of local cohomology modules for maximal minors and Pfaffians.

Characterization of Cohen-Macaulay modules of covariants under SL(G) action.

Development of a new computational approach for Ext modules.

Abstract

We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay modules of covariants for the action of the special linear group SL(G) on G^m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.