具有有限射影维数的Cohen-Macaulay模的Hilbert系数界
Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension
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中文总结 AI 辅助
本文研究Cohen-Macaulay模的Hilbert系数下界，将已知下界推广至Cohen-Macaulay环，并给出第二系数的上界及等号情形的深度性质，最后推广至严格完全交环。
中文摘要 AI 辅助
设$(A,\mathfrak{m})$为Gorenstein局部环，且$G(A)$为Cohen-Macaulay，设$M$为有限射影维数的Cohen-Macaulay $A$-模。在文献\cite{Quasipure}中，作者证明了$e_1(M)\geq \binom{c+1}{2}$，其中$c=\operatorname{reg}G(A)$，$e_i(M)$为$M$的第$i$个Hilbert系数。我们首先证明当$A$为Cohen-Macaulay时该界仍然成立。然后，当$e_1(M)=\binom{c+1}{2}+i$（$i=1,2$）时，我们研究$e_2(M)$的上界，并考察等号成立时的推论。特别地，我们得到$G(M)$的$h$-多项式的深度性质和显式描述。最后，我们将这些结果推广到严格完全交环，而不假设$M$具有有限射影维数。
英文摘要
Let $(A,\mathfrak{m})$ be a Gorenstein local ring with $G(A)$ Cohen-Macaulay, and let $M$ be a Cohen-Macaulay $A$-module of finite projective dimension. In \cite{Quasipure}, the authors proved that $e_1(M)\geq \binom{c+1}{2}$, where $c=\operatorname{reg}G(A)$ and $e_i(M)$ is the $i$th Hilbert coefficient of $M$. We first show that this bound remains valid when $A$ is Cohen-Macaulay. We then study upper bounds for $e_2(M)$ when $e_1(M)=\binom{c+1}{2}+i$ for $i=1,2$, and investigate the consequences of equality. In particular, we obtain depth properties and explicit descriptions of the $h$-polynomial of $G(M)$. Finally, we extend these results to strict complete intersection rings without assuming that $M$ has finite projective dimension.
发表机构
- Indian Institute of Technology Dharwad（达瓦德印度理工学院）
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