arXiv subjects
Publications and source records attributed to Viktória Klász.
We give a new characterisation of Auslander-Gorenstein finite dimensional algebras by showing that they are exactly the finite dimensional algebras with a well-defined Auslander-Reiten bijection. This proves a conjecture of Marczinzik. We use this to give a new proof that a finite lattice with an Auslander-Gorenstein incidence algebra has to be distributive.
We show that Iyama's grade bijection for Auslander-Gorenstein algebras coincides with the bijection introduced by Auslander-Reiten. This result uses a new characterisation of Auslander-Gorenstein algebras. Furthermore, we show that the grade bijection of an Auslander regular algebra coincides with the permutation matrix P in the Bruhat factorisation of the Coxeter matrix. This gives a new, purely linear algebraic interpretation of the grade bijection and allows us to calculate it in a much quicker way than was previously known. We give several applications of our main results. First, we show that the permanent of the Coxeter matrix of an Auslander regular algebra is either 1 or -1. Second, we obtain a new combinatorial characterisation of distributive lattices among the class of finite lattices. Explicitly, a lattice is distributive if and only if its Coxeter matrix can be written as PU where P is a permutation matrix and U is an upper triangular matrix. Other applications include new homological results about modules in blocks of category $\mathcal{O}$ of semisimple Lie algebras.
We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular if and only if its Coxeter matrix $C$ has a Bruhat factorisation $U_1 P U_2$ with $U_1$ the identity matrix. In particular, this holds without restrictions for linear Nakayama algebras and we use the Bruhat decomposition to answer a question raised by Ringel by showing that his homological permutation coincides with the permutation coming from the Bruhat factorisation of the Coxeter matrix. We also use our methods to show that general Auslander regular acyclic quiver algebras are echelon-independent, proving a conjecture of Defant-Jiang-Marczinzik-Segovia-Speyer-Thomas-Williams, and we answer another question by Ringel on the delooping level of simple modules over Nakayama algebras.
In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a completion classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution. We use our results to answer a question raised by Ajitabh, Smith and Zhang by showing that there exist Auslander-Gorenstein polynomial identity rings without a pure injective coresolution.
We give a survey on Auslander-Gorenstein algebras with a focus on finite-dimensional algebras. We put an emphasis on recent classification results for special classes of algebras and the newly discovered interactions of the Auslander-Reiten bijection with other well studied bijections in the literature.
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 $n\ge 1.$
We study the global dimension of Nakayama algebras. In the case of linear Nakayama algebras, which are in canonical bijection to Dyck paths, we show that the global dimension has the same distribution as the height of Dyck paths. For cyclic Nakayama algebras an explicit classification of finite global dimension is not known. However, we show that in certain special cases cyclic Nakayama algebras with finite global dimension can again be interpreted as Dyck paths. In particular, we show that there is a natural bijection between sincere Nakayama algebras and Dyck paths. In this case, we find that the global dimension is in fact twice the bounce count of the corresponding Dyck path.
We give a combinatorial classification of Nakayama algebras of small homological dimension using the Krattenthaler bijection between Dyck paths and 132-avoiding permutations.
Let $KG$ be a group algebra with $G$ a finite group and $K$ a field and $M$ an indecomposable $KG$-module. We pose the question, whether $Ext_{KG}^1(M,M) \neq 0$ implies that $Ext_{KG}^i(M,M) \neq 0$ for all $i \geq 1$. We give a positive answer in several important special cases such as for periodic groups and give a positive answer also for all Nakayama algebras, which allows us to improve a classical result of Gustafson. We then specialise the question to the case where the module $M$ is simple, where we obtain a positive answer also for all tame blocks of group algebras. For simple modules $M$, the appendix provides a Magma program that gives strong evidence for a positive answer to this question for groups of small order.