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

2026/07/20 by Dawei Shen
#math.RT

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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.

Citations

• Auslander regular algebras and Coxeter matrices
• Stable module theory

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