Reduction numbers for witnesses to the generalized Loewy length

Abstract

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 $\text{g}\ell\ell(R)-e(R)$.