Generalized Loewy length of Cohen-Macaulay local and graded rings

TL;DR

This paper extends Ding's theorem on the relationship between generalized Loewy length and the index in one-dimensional Cohen-Macaulay rings, providing bounds, exact values for specific cases, and computations for examples and numerical semigroup rings.

Contribution

It generalizes Ding's theorem to cases with finite residue fields, establishes bounds for generalized Loewy length, and computes explicit values for hypersurface and semigroup rings.

Findings

Generalized Loewy length is bounded by the index plus the degree of a homogeneous nonzerodivisor.

Achieves the upper bound in hypersurface rings with specific regular initial forms.

Determines the generalized Loewy length for numerical semigroup rings.

Abstract

We generalize a theorem of Ding relating the generalized Loewy length $\text{g}\ell\ell(R)$ and index of a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak{m},k)$. Ding proved that if $R$ is Gorenstein, the associated graded ring is Cohen-Macaulay, and $k$ is infinite, then the generalized Loewy length and index of $R$ are equal. However, if $k$ is finite, equality may not hold. We prove that if the index of a one-dimensional Cohen-Macaulay local ring is finite and the associated graded ring has a homogeneous nonzerodivisor of degree $t$, then $\text{g}\ell\ell(R) \leq \text{index}(R)+t-1$. Next we prove that if $R$ is a one-dimensional hypersurface ring with a witness to the generalized Loewy length that induces a regular initial form on the associated graded ring, then the generalized Loewy length achieves this upper bound. We then compute the generalized Loewy lengths of several families of examples of one-dimensional hypersurface rings over finite fields. Finally, we study a graded version of the generalized Loewy length and determine its value for numerical semigroup rings.