A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras

Dawei Shen

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arXiv · 2607.17497

A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras

Dawei Shen

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Abstract

Let $R$ be a left and right Noetherian ring. We introduce a Serre-type condition $(G_n)$, formulated in terms of the first occurrence and the flat dimension of indecomposable injective modules, and prove that $R$ is $n$-Gorenstein if and only if it satisfies $(G_n)$. We then apply the criterion to Nakayama algebras via syzygy filtration. It is shown that syzygy filtration preserves the Auslander-Gorenstein property and reflects it within the class of $2$-Gorenstein Nakayama algebras. Combined with a result of Klász, Kleinau and Marczinzik, this shows that simple modules of odd grade over Auslander-Gorenstein Nakayama algebras are regular in their grade, thereby settling their conjecture.

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Dawei Shen. 2026-08-03. A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras. https://arxiv.org/abs/2607.17497

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