关于某些单项式理想类的门德斯 - 平托 - 比利亚雷亚尔猜想
The Mendez-Pinto-Villarreal Conjecture for some classes of monomial ideals
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中文总结 AI 辅助
研究交换代数中刻画理想符号幂与普通幂重合问题（西米斯理想），通过研究门德斯 - 平托 - 比利亚雷亚尔猜想，利用支撑集条件证明相关结论，建立两类单项式理想的猜想并研究其科恩 - 麦考利性质。
中文摘要 AI 辅助
刻画理想的符号幂与普通幂何时重合是交换代数中的核心问题，满足此性质的理想称为西米斯理想。本文通过研究门德斯、平托和比利亚雷亚尔关于具有极小不可约分解的单项式理想的最新猜想，来研究单项式理想的西米斯性质。设\(I\)为单项式理想，\(\mathcal{F}(I)\)表示\(I\)的极小生成元的支撑集。假设\(\mathcal{F}(I)=\mathcal{F}(\sqrt{I})\)，证明若\(I\)有多个具有固定支撑的极小生成元，则它不是西米斯理想。利用此约化，建立了两类单项式理想的门德斯 - 平托 - 比利亚雷亚尔猜想，即支撑 - 3单项式理想和其关联单纯复形为单纯森林的单项式理想。最后研究了其关联单纯复形为嫁接且满足\(\mathcal{F}(I)=\mathcal{F}(\sqrt{I})\)的单项式理想的科恩 - 麦考利性质。
英文摘要
Characterizing when the symbolic and ordinary powers of an ideal coincide is a central problem in commutative algebra, and ideals satisfying this property are called Simis ideals. In this article, we investigate the Simis property of monomial ideals by studying the recent conjecture of Mendez, Pinto, and Villarreal on monomial ideals with minimal irreducible decomposition. Let $I$ be a monomial ideal, and let $\mathcal{F}(I)$ denote the collection of supports of the minimal generators of $I$. Assuming that $\mathcal{F}(I)=\mathcal{F}(\sqrt{I})$, we prove that if $I$ admits more than one minimal generator with a fixed support, then it is not Simis. Using this reduction, we establish the Mendez-Pinto-Villarreal conjecture for two broad classes of monomial ideals, namely support-$3$ monomial ideals and monomial ideals whose associated simplicial complexes are simplicial forests. Finally, we study the Cohen-Macaulay property of monomial ideals whose associated simplicial complexes are grafted and satisfy $\mathcal{F}(I)=\mathcal{F}(\sqrt{I})$.
