Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.06957

The Auslander-Gorenstein condition for monomial algebras

Viktória Klász
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper investigates the Auslander-Gorenstein property for monomial algebras. First, we prove that every Auslander-Gorenstein monomial algebra is a string algebra and present a simple combinatorial classification of Auslander-Gorenstein gentle algebras. Furthermore, we describe a procedure to transform any 2-Gorenstein monomial algebra into a Nakayama algebra, thereby reducing the classification of Auslander-Gorenstein monomial algebras to that of Auslander-Gorenstein Nakayama algebras. As an application of this reduction method, we prove that every monomial algebra satisfies a stronger version of the Auslander-Reiten Conjecture. Our second main result establishes that a monomial algebra is Auslander-Gorenstein if and only if it has a well-defined, bijective Auslander-Reiten map, confirming a conjecture of Marczinzik for monomial algebras. This yields a new homological characterisation of the Auslander-Gorenstein property. Additionally, we provide an explicit description of the Auslander-Reiten bijection in the case of gentle algebras. Along the way, we also generalise a result of Iwanaga and Fuller: We show that every $2n$-Gorenstein monomial algebra is also $(2n+1)$-Gorenstein for every $n\ge 1.$

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Viktória Klász. 2025-08-09. The Auslander-Gorenstein condition for monomial algebras. https://arxiv.org/abs/2508.06957

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

KEEP EXPLORING

Related papers

arXiv2026

A Course on Lie algebras and Chevalley groups

These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~ -- ~from the simple Lie algebras over $\C$ occurring in the Cartan--Killing classification. Together with subsequent variations, Chevalley's work had a profound and long-lasting impact on group theory and Lie theory in general. Classical, and widely used references are the lectures notes by Steinberg (1967) and the monograph by Carter (1972). Our aim here is to present a self-contained introduction to the theory of Chevalley groups, based on recent simplifications arising from Lusztig's fundamental theory of ``canonical bases''. A further feature of our text is that we explicitly incorporate algorithmic methods in our treatment, both for the handling of substantial examples and regarding some aspects of the general theory. Eventually, this may turn into a book project.

math.RT↗
arXiv2026

On the p-part of the conductor of a generalised character

We show that the $p$-part of the conductor of a generalised character of a finite group is equal to the conductor of its generalised decomposition numbers. We use this to show that $p$-parts of conductors of irreducible characters are preserved under isotypies and perfect isometries that arise in the context of stable equivalences of Morita type with endopermutation source. We apply this to blocks with abelian defect and Frobenius inertial quotient.

math.RT↗
arXiv2026

On the triviality of inhomogeneous deformations of $\mathfrak{osp}(1|2n)$

We specify a symmetrized mixed-oscillator deformation family of $B(0,n)=\operatorname{osp}(1|2n)$, with even mixed coefficients and one odd square-zero parameter. For every $n\geq1$, we derive its bracket from a faithful oscillator realization and exhibit an odd cochain whose coboundary is the recovered deformation coefficient. The resulting even change of generators is an exact isomorphism over the exterior parameter algebra. For $n=1$, the cochain agrees with the normalization of Bakalov-Sullivan. We give the source relations and the even-central specialization explicitly, together with a Lean 4 formalization.

math.RT↗