Lower bounds on Loewy lengths of modules of finite projective dimension
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913801535750144 |
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| author | KC, Nawaj Pollitz, Josh |
| author_facet | KC, Nawaj Pollitz, Josh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds on Loewy lengths of modules of finite projective dimension KC, Nawaj Pollitz, Josh Commutative Algebra 13C14 (primary), 13B10, 13D05, 13H10, 13H15 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. |
| title | Lower bounds on Loewy lengths of modules of finite projective dimension |
| topic | Commutative Algebra 13C14 (primary), 13B10, 13D05, 13H10, 13H15 |
| url | https://arxiv.org/abs/2404.18368 |