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arXiv · 2105.07186

Integrally closed ideals of reduction number three

Shinya Kumashiro
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Abstract

In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge \mathrm{e}_0(I) - \mathrm{e}_1(I) + \dfrac{\mathrm{e}_2(I) + \ell_A(I^2/QI)}{2}$, where $\mathrm{e}_i(I)$ denotes the $i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of $I$ is larger than or equal to $\dim A-1$. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

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Shinya Kumashiro. 2021-05-15. Integrally closed ideals of reduction number three. https://arxiv.org/abs/2105.07186

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