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

Abstract

Let S 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 I be the determinantal ideal of maximal minors or P f 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 [Adv. Math. 250 (2014), pp. 596–610] to determine the S S -module structures of E x t S j ( S / I t , S ) Ext^j_S(S/I^t, S) and E x t S j ( S / P f t , S ) Ext^j_S(S/Pf^t, S) , from which we get the degrees of generators of these E x t Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H m j ( S / I t ) H^j_\mathfrak {m}(S/I^t) .