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 $\text{g}\ell \ell (R)\le d(r_w({\mathfrak{m}}^d)+1)$. 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 $\text{g}\ell \ell (R)-e(R)$.

arXiv commentAdded Theorem 2.16