Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S = \textsf{k}[x_1, \ldots , x_n]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mi mathvariant="sans-serif">k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, I be an ideal of S, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\bar{I}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi>I</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation> denote its integral closure. A conjecture of Küronya and Pintye states that for any homogeneous ideal I of S, the inequality <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {reg}(\bar{I}) \le \operatorname {reg}(I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>reg</mo> <mo stretchy="false">(</mo> <mover accent="true"> <mrow> <mi>I</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">)</mo> <mo>≤</mo> <mo>reg</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> holds, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\operatorname {reg}(\_)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>reg</mo> <mo stretchy="false">(</mo> <mi>_</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the Castelnuovo–Mumford regularity. In this article, we prove the conjecture for certain classes of monomial ideals.

A comparison of the regularity of certain classes of monomial ideals and their integral closure

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2026年3月14日

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Omkar Javadekar

摘要

Let $S = \textsf{k}[x_1, \ldots , x_n]$ $S = k [x_{1}, \dots, x_{n}]$ , I be an ideal of S, and $\bar{I}$ $\bar{I}$ denote its integral closure. A conjecture of Küronya and Pintye states that for any homogeneous ideal I of S, the inequality $\operatorname {reg}(\bar{I}) \le \operatorname {reg}(I)$ $reg (\bar{I}) \leq reg (I)$ holds, where $\operatorname {reg}(\_)$ $reg (_)$ denotes the Castelnuovo–Mumford regularity. In this article, we prove the conjecture for certain classes of monomial ideals.