Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.17497

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

Dawei Shen
View original source ↗Read source document ↗Export metadata ↓

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 Kl\'asz, 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.

Explore related subjects

math.RT

Keep this discovery

BibTeXRIS

Dawei Shen. 2026-07-20. A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras. https://arxiv.org/abs/2607.17497

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗
arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗
arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗