Local cohomology with support in generic determinantal ideals

TL;DR

This paper computes the local cohomology modules supported on determinantal ideals in polynomial rings, providing explicit descriptions and regularity results, especially for powers of maximal minors and their products.

Contribution

It offers an explicit GL_m x GL_n-equivariant description of local cohomology modules for determinantal ideals, and characterizes when these ideals have linear resolutions.

Findings

Determined the regularity of determinantal ideals.

Identified which powers and products of minors have linear resolutions.

Provided explicit Ext modules for ideals generated by partitions.

Abstract

For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our techniques allow us to explicitly compute all the modules Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the irreducible sub-representation of S indexed by x. In particular we determine the regularity of the ideals I_x, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such powers and the maximal ideal of S.