Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Batavia, Manav, Pandey, Vaibhav, Walther, Uli
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7223032543