满足$e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)$的整闭理想
Integrally closed ideals with $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)$
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中文总结 AI 辅助
该研究针对Cohen-Macaulay局部环的整闭$\mathfrak{m}$-准素理想，建立了广义型的最优下界，分析了相伴分次环的性质，还探讨了Buchsbaum局部环的Hilbert系数界及相关极端情形的结论。
中文摘要 AI 辅助
设$(A,\mathfrak{m})$是维数为$d$的Cohen-Macaulay局部环，我们引入并研究关于$\mathfrak{m}$-准素理想$I$的$A$的广义型概念，记为$\operatorname{type}_I(A)$。令$e_i(I)$表示$A$关于$I$的第$i$个Hilbert系数。假设$I$是整闭理想且满足$e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I) \neq 0$，我们建立了用重数及与$I$相关的某些长度表示的$\operatorname{type}_I(A)$的最优下界，还证明当该下界达到时，相伴分次环$G(I)$是Cohen-Macaulay的。对于维数为$d$且深度至少为$d-1$的Buchsbaum局部环，我们利用$S_2$-fication技术得到了$e_{2}(\mathfrak{m})$的最优下界。此外，对于整闭的$\mathfrak{m}$-准素理想$I$，我们还研究了第二种极端情形$e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)+1$及其对$G(I)$的影响，同时也探讨了$e_3(I)$的界，并在$d=3$时研究这些界达到时的相关结论。
英文摘要
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d.$ We introduce and study the notion of the generalized type of $A$ with respect to an $\mathfrak{m}$-primary ideal $I$ denoted by $\operatorname{type}_I(A).$ Let $e_i(I)$ denote $i$th Hilbert coefficients of $A$ w.r.t. $I$. Assuming $I$ is integrally closed and $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I) \neq 0$, we establish a sharp lower bound for $\operatorname{type}_I(A)$ in terms of the multiplicity and certain lengths associated to $I.$ We further show that when this lower bound is attained, the associated graded ring $G(I)$, is Cohen Macaulay. In the case of Buchsbaum local rings of dimension $d$ and depth at least $d-1$, we obtain an optimal lower bound for $e_{2}(\mathfrak{m})$ using the technique of $S_{2}$-fication. Additionally, for an integrally closed $\mathfrak{m}$-primary ideal $I,$ we also study the second extremal case $e_{2}(I)=e_{1}(I)-e_{0}(I)+λ(A/I)+1$ and its consequences on $G(I).$ We also investigate bounds on $e_3(I)$ and for $d=3,$ we study the consequences when these bounds are attained for.
发表机构
- Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
- Indian Institute of Technology Dharwad（印度理工学院达尔瓦德分校）
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