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Regularity under Integral Closure: A General Criterion and a Sharp Counterexample

  • Soumyadeep Misra
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Abstract

We study the behavior of Castelnuovo--Mumford regularity under integral closure. We prove a general cohomological criterion for the inequality reg(J̅)≤ reg(J), and in particular obtain it for every homogeneous ideal in a polynomial ring in at most three variables. The proof uses a high-degree comparison between J̅ and its saturation. We also give a saturated monomial counterexample in four variables, showing that the three-variable result is sharp.

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Cite

BibTeXRIS
BibTeX
@misc{misra2026regularity,
  title = {{Regularity under Integral Closure: A General Criterion and a Sharp Counterexample}},
  author = {Soumyadeep Misra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.13879},
  eprint = {2605.13879},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.13879},
}
RIS
TY  - UNPB
TI  - Regularity under Integral Closure: A General Criterion and a Sharp Counterexample
AU  - Soumyadeep Misra
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2605.13879
UR  - https://arxiv.org/abs/2605.13879
AB  - We study the behavior of Castelnuovo--Mumford regularity under integral closure. We prove a general cohomological criterion for the inequality reg(J̅)≤ reg(J), and in particular obtain it for every homogeneous ideal in a polynomial ring in at most three variables. The proof uses a high-degree comparison between J̅ and its saturation. We also give a saturated monomial counterexample in four variables, showing that the three-variable result is sharp.
ER  -