A counterexample to K\"othe's conjecture and a question of Rowen
We construct a counterexample to K\"othe's conjecture and use this to also answer a question due to Rowen.
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arXiv subjects
Rene Marczinzik
Publications and source records attributed to Rene Marczinzik.
Tor and Ext vanishing results for commutative Artinian rings
We give a negative answer to a question of Avramov, Buchweitz and \c{S}ega by constructing a commutative local finite-dimensional non-Gorenstein algebra $R$ with $\operatorname{Ext}_R^1(D(R),R)=0$; this question is related to the first Tachikawa conjecture. We also give a counterexample to a conjecture of Huneke, \c{S}ega and Vraciu on Tor vanishing over commutative finite-dimensional algebras. Finally, we construct a finite-dimensional commutative local self-injective algebra $R$ over $\mathbb{F}_2$ and an indecomposable non-projective $R$-module $M$ such that $\operatorname{Ext}_R^1(M,M)=\operatorname{Ext}_R^2(M,M)=0$, related to the second Tachikawa conjecture and answering a question of Dao.
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A new characterisation of Auslander-Gorenstein algebras
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.
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Preprojective algebras and generalisations: A short survey
The preprojective algebra of a hereditary algebra $H$ can be defined as a certain orbit construction of the regular representation generated by the Auslander-Reiten translation. In this short survey, we will look at two important generalisations, namely, the contracted preprojective algebra and the total preprojective algebra. We will include several open problems and questions motivated by examples in the hope to stimulate future research on general orbit algebras of $H$-modules.
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Fractionally Calabi-Yau algebras and cluster tilting
We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of our main result stating that an algebra $A$ of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists $i$ such that the replicated algebra $A^{(i)}$ is a higher Auslander algebra if and only if there exist infinitely many $i$ such that $A^{(i)}$ is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted $\frac{n}{2}$-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the $\mathbb{Z}$-graded stable module category of an associated higher preprojective algebra.
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On a question of Auslander and Bridger on 2-reflexive modules
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
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Pure minimal injective resolutions and perfect modules for lattices
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.
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A survey on Auslander-Gorenstein algebras
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.
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Rowmotion and Echelonmotion
Given a linear extension $\sigma$ of a finite poset $R$, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $\sigma$. This yields a bijection $\mathrm{Ech}_\sigma\colon R\to R$ that we call echelonmotion; it is the inverse of the Coxeter permutation studied by Kl\'asz, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when $R$ is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions.
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Cluster tilting modules for local algebras
We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package QPA. We discuss related problems and conjectures for local finite-dimensional algebras.
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Total preprojective algebras
We introduce total preprojective algebras $\Psi$ of path algebras of Dynkin quivers $kQ$, and prove that they are isomorphic to $2$-Auslander algebras of preprojective algebras $\Pi$ of $kQ$. In particular, $\Psi$ has global dimension $3$ and dominant dimension $3$. We also describe $\Psi$ as a tensor algebra of a certain explicit bimodule over the Auslander algebra of $kQ$. As an application, we give a presentation of $\Psi$ by explicit quivers with relations. More generally, we introduce total $(d+1)$-preprojective algebras of $d$-representation finite algebras, and give all the corresponding results.
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Auslander regular algebras and Coxeter matrices
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.
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Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras
Auslander and Reiten called a finite dimensional algebra $A$ over a field Cohen-Macaulay if there is an $A$-bimodule $W$ which gives an equivalence between the category of finitely generated $A$-modules of finite projective dimension and the category of finitely generated $A$-modules of finite injective dimension. For example, Iwanaga-Gorenstein algebras and algebras with finitistic dimension zero on both sides are Cohen-Macaulay, and tensor products of Cohen-Macaulay algebras are again Cohen-Macaulay. They seem to be all of the known examples of Cohen-Macaulay algebras. In this paper, we give the first non-trivial class of Cohen-Macaulay algebras by showing that all contracted preprojective algebras of Dynkin type are Cohen-Macaulay. As a consequence, for each simple singularity $R$ and a maximal Cohen-Macaulay $R$-module $M$, the stable endomorphism algebra $\underline{End}_R(M)$ is Cohen-Macaulay. We also give a negative answer to a question of Auslander-Reiten asking whether the category $CM A$ of Cohen-Macaulay $A$-modules coincides with the category of $d$-th syzygies, where $d\ge1$ is the injective dimension of $W$. In fact, if $A$ is a Cohen-Macaulay algebra that is additionally $d$-Gorenstein in the sense of Auslander, then $CM A$ always coincides with the category of $d$-th syzygies.
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Selfextensions of modules over group algebras
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.
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On self-orthogonal modules in Iwanaga-Gorenstein rings
Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal Cohen-Macaulay module is periodic. This answers a question of Enomoto and shows the conjecture for monomial quiver algebras and hypersurface rings.
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Dominant Auslander-Gorenstein algebras and mixed cluster tilting
We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra $T(A)$ of a $d$-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that $d$-representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.
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On total stability conditions for Dynkin quivers
We show that for a Dynkin quiver $Q$ of type $E_7$ with a specific orientation, the path algebra $KQ$ has no slope function of the form $μ=\fracθ{\dim}$ that defines a total stability condition. This gives a counterexample to a conjecture of Reineke.
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Homological algebra of Nakayama algebras and 321-avoiding permutations
Linear Nakayama algebras over a field $K$ are in natural bijection to Dyck paths and Dyck paths are in natural bijection to 321-avoiding bijections via the Billey-Jockusch-Stanley bijection. Thus to every 321-avoiding permutation $\pi$ we can associate in a natural way a linear Nakayama algebra $A_{\pi}$. We give a homological interpretation of the fixed points statistic of 321-avoiding permutations using Nakayama algebras with a linear quiver. We furthermore show that the space of self-extension for the Jacobson radical of a linear Nakayama algebra $A_{\pi}$ is isomorphic to $K^{\mathfrak{s}(\pi)}$, where $\mathfrak{s}(\pi)$ is defined as the cardinality $k$ such that $\pi$ is the minimal product of transpositions of the form $s_i=(i,i+1)$ and $k$ is the number of distinct $s_i$ that appear.
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