In a local Cohen-Macaulay ring (A, m), we study the Hilbert function of an m-primary ideal I whose reduction number is two. It is a continuous work of the papers of Huneke, Ooishi, Sally, and Goto-Nishida-Ozeki. With some conditions, we show the inequality e₁(I) ≥ e₀(I) − ℓ_A(A/I) + e₂(I) of the Hilbert coefficients, which is the converse inequality of Sally and Itoh. We also study relations between the Hilbert coefficients and the depth of the associated graded ring.

中文翻译：

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减少第二个理想

在局部Cohen-Macaulay环（A，m）中，我们研究了约数为2的m初等理想I的希尔伯特函数。这是Huneke，Ooishi，Sally和Goto-Nishida-Ozeki论文的连续著作。对于某些条件下，我们示出了不等式ë ₁（我）≥È ₀（我） - ℓ_甲（A / I）+ E ₂（我）的希尔伯特系数，其是Sally和伊藤的逆不等式的。我们还研究了希尔伯特系数与相关渐变环的深度之间的关系。