Regularity under Integral Closure: A General Criterion and a Sharp Counterexample
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.
math.AC↗