Lower bounds on Loewy lengths of modules of finite projective dimension

TL;DR

This work develops uniform lower bounds for the Loewy lengths of modules of finite length or finite projective dimension over local rings by relating these invariants to the associated graded ring $R^{g}$ and its regularity $reg(R^{g})$. The central result shows that if $R$ is strict Cohen–Macaulay and $M$ is nonzero with $\mathrm{Tor}^R_i(M,k)=0$ for some $i$, then $\ell\ell_R(M) \ge reg(R^{g})+1$, with corollaries for generalized Loewy length and reductions when residue fields are infinite. The paper also proves a Lech-type inequality for generalized Loewy length along flat local extensions with finite flat dimension, implying $g\ell\ell_R(R) \le g\ell\ell_S(S)$ in this setting and confirming Hanes’ conjecture there. For complete intersections, it establishes a sharp bound $\ell\ell_R(M) \ge \sum_{i=1}^c \mathrm{ord}(f_i) - c + 1$ in any Cohen presentation $\widehat{R} \cong Q/(f_1,\dots,f_c)$, with two proofs and a deformation–relative version, significantly strengthening prior lower bounds and connecting Loewy length to singularity measures.

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.

Theorems & Definitions (45)

• Conjecture A
• Theorem B
• Corollary C
• Corollary D
• Theorem E
• Remark 1.2
• Lemma 1.7
• proof
• Lemma 1.8
• proof