On the arithmetic rank and cohomological dimension of some residual intersections

@misc{batavia2026arithmetic,
  title = {{On the arithmetic rank and cohomological dimension of some residual intersections}},
  author = {Manav Batavia and Vaibhav Pandey and Uli Walther},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25680},
  eprint = {2610.25680},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25680},
}

TY  - UNPB
TI  - On the arithmetic rank and cohomological dimension of some residual intersections
AU  - Manav Batavia
AU  - Vaibhav Pandey
AU  - Uli Walther
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.25680
UR  - https://arxiv.org/abs/2610.25680
AB  - 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 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.
ER  -