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arXiv · 2605.13879

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 $\operatorname{reg}(\overline J)\leq \operatorname{reg(J)}$, 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 $\overline J$ 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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Soumyadeep Misra. 2026-05-11. Regularity under Integral Closure: A General Criterion and a Sharp Counterexample. https://arxiv.org/abs/2605.13879

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