Lower bounds on Loewy lengths of modules of finite projective dimension

TL;DR

This paper establishes new lower bounds on the Loewy length of modules with finite projective dimension over certain local rings, confirming conjectures and strengthening previous theorems, especially for strict Cohen-Macaulay and complete intersection rings.

Contribution

It proves that the Loewy length exceeds the ring's regularity for modules over strict Cohen-Macaulay rings, confirming a conjecture and improving known bounds.

Findings

Loewy length exceeds regularity in strict Cohen-Macaulay rings.

Verification of a Lech-like conjecture for flat local extensions.

Improved lower bounds for Loewy lengths in complete intersection rings.

Abstract

This article is concerned with nonzero modules of finite length and finite projective dimension over a local ring. We show the Loewy length of such a module is larger than the regularity of the ring whenever the ring is strict Cohen-Macaulay, establishing a conjecture of Corso--Huneke--Polini--Ulrich for such rings. In fact, we show the stronger result that the Loewy length of a nonzero module of finite flat dimension is at least the regularity for strict Cohen-Macaulay rings, which significantly strengthens a theorem of Avramov--Buchweitz--Iyengar--Miller. As an application we simultaneously verify a Lech-like conjecture, comparing generalized Loewy length along flat local extensions, and a conjecture of Hanes for strict Cohen-Macaulay rings. Finally, we also give notable improvements to known lower bounds for Loewy lengths without the strict Cohen-Macaulay assumption. The strongest general bounds we achieve are over complete intersection rings.