AI 中文总结
本文针对一维Cohen-Macaulay局部环的超曲面族，研究广义Loewy长度见证元对应的理想约化数，给出差值实例并证明其可独立于环的重数相关量变化。
AI 中文摘要
设$(R,\mathfrak{m})$为一维Cohen-Macaulay局部环，本文针对无穷多个超曲面族$\{(R,\mathfrak{m})\}$，求广义Loewy长度$\text{g}\ell\ell(R)$的见证元$z\in \mathfrak{m}^d \setminus \mathfrak{m}^{d+1}$对应的$\boldsymbol{\text{m}^d}$的约化数$r_z(\boldsymbol{\text{m}^d})$；对$\boldsymbol{\text{m}^d}$的每个主约化元$w$，有$\text{g}\ell\ell(R) \leq d(r_w(\boldsymbol{\text{m}^d})+1)$，给出满足$d(r_z(\boldsymbol{\text{m}^d})+1)-\text{g}\ell\ell(R)=0$和$1$的族例子，还证明该差值可独立于$\text{g}\ell\ell(R)-e(R)$变化。
英文摘要
Let $(R,\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number $r_{z}(\mathfrak{m}^d)$ of $\mathfrak{m}^d$ with respect to a witness $z\in \mathfrak{m}^d \setminus \mathfrak{m}^{d+1}$ to the generalized Loewy length $\text{g}\ell\ell(R)$ for several infinite families of hypersurfaces $\left\{(R,\mathfrak{m}) \right\}$. For every principal reduction $w$ of $\mathfrak{m}^d$, we have $\text{g}\ell\ell(R) \leq d(r_{w}(\mathfrak{m}^d)+1)$. We give examples of families $\left\{ (R,\mathfrak{m}) \right\}$ such that $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=0$ and $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=1$. In particular, we prove that for every prime $p$, there are local hypersurfaces $R$ over $\mathbb{F}_p$ such that $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=0$ and $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)=1.$ We also prove that the difference $d(r_{z}(\mathfrak{m}^d)+1)-\text{g}\ell\ell(R)$ can vary independently of $\text{g}\ell\ell(R)-e(R)$.
CommentsAdded Theorem 2.16
