Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

TL;DR

This paper determines the structure and socle degrees of local cohomology modules related to powers of determinantal and Pfaffian ideals in polynomial rings, using desingularizations and representation theory.

Contribution

It extends previous work by explicitly computing Ext module structures and socle degrees for local cohomology of thickenings of determinantal and Pfaffian ideals.

Findings

Explicit descriptions of Ext modules for powers of determinantal and Pfaffian ideals.

Determination of socle degrees of local cohomology modules for these ideals.

Answering a question of Wenliang Zhang on socle degrees.

Abstract

Let $S$ be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let $I$ be the determinantal ideal of maximal minors or $\operatorname{Pf}$ the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu--Weyman--Witt to determine the $S$-module structures of $\operatorname{Ext}^j_S(S/I^t, S)$ and $\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S)$, from which we get the degrees of generators of these $\operatorname{Ext}$ modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang on the socle degrees of local cohomology modules of the form $H^j_\mathfrak{m}(S/I^t)$.