On the arithmetic rank and cohomological dimension of some residual intersections

Abstract

We determine the arithmetic rank and cohomological dimension of each generic residual intersection of the ideal of submaximal Pfaffians of an alternating matrix of variables. This ideal fails to be a set-theoretic complete intersection. Our methods yield several structural results: in characteristic zero, we compute the $D$-module length of the top local cohomology module supported at this ideal and show that over the integers, it admits a $\mathbb{Q}$-vector space structure. In particular, the top local cohomology module has no integer torsion. We also establish the $F$-regularity of the generic residual intersection rings considered here. Further, for each admissible height $s$, we construct $s$-residual intersections of complete intersection ideals and height three Gorenstein ideals that are set-theoretic complete intersections in any characteristic. Finally, we pose a general question of independent interest concerning the behaviour of arithmetic rank.