Research on the Hilbert functions in commutative algebra

本研究的目的是探讨具有极大理想m的交换noether局部环a中m-初等理想的Hilbert函数的行为。本研究的代表给出了以下结果。m-初等理想的第一希尔伯特系数的研究本研究的目的是在不假设基局部环（A, m）为Cohen-Macaulay的情况下，研究Elias和Valla给出的m-初等理想的第一希尔伯特系数的等式。它与m-初级理想的关联分级环的Buchsbaumness有关。参数理想的Hilbert系数的研究本文研究表明当且仅当基局部环A有有限生成的局部上同调模时，参数理想的Hilbert系数具有一致界。均匀边界是巨大的；在基局部环有限生成局部上同调模的情况下，第二希尔伯特系数的锐界。

The purpose of this research is to explore the behavior of the Hilbert functions of m-primary ideals in a commutative Noetherian local ring A with the maximal ideal m. The representative of this research gave the following results.1.Research on the first Hilbert coefficients of m-primary idealsThe purpose of this research is to study the equality on the first Hilbert coefficients of m-primary ideals given in Elias and Valla, without assuming that the base local ring(A, m) is Cohen-Macaulay. It is related to the Buchsbaumness of the associated graded ring of m-primary ideals.2.Research on the Hilbert coefficients of parameter idealsThis research shows that the Hilbert coefficients of parameter ideals have uniform bounds if and only if the base local ring A has finitely generated local cohomology modules. The uniform bounds are huge ; the sharp bound for the second Hilbert coefficient in the case where the base local ring has finitely generated local cohomology modules.