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arXiv · 2404.18368

Lower bounds on Loewy lengths of modules of finite projective dimension

Nawaj KC·Josh Pollitz
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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.

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Nawaj KC, Josh Pollitz. 2024-04-29. Lower bounds on Loewy lengths of modules of finite projective dimension. https://arxiv.org/abs/2404.18368

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