arXiv subjects

Claus Michael Ringel

Publications and source records attributed to Claus Michael Ringel.

At least 19 recordsLinked to original sources

arXiv2026

Brick chain filtrations

We deal with the category of finitely generated modules over an artin algebra $A$. Recall that an object in an abelian category is said to be a brick provided its endomorphism ring is a division ring. Simple modules are, of course, bricks, but in case $A$ is connected and not local, there do exist bricks which are not simple. The aim of this survey is to focus the attention to filtrations of modules where all factors are bricks, with bricks being ordered in some definite way. In general, a module category will have many oriented cycles. Recently, Demonet has proposed to look at so-called brick chains in order to deal with a very interesting directedness feature of a module category. These are the orderings of bricks which we will use. This is a survey which relies on recent investigations by a quite large group of mathematicians. We have singled out some important observations and have reordered them in order to obtain a completely self-contained (and elementary) treatment of the relevance of bricks in a module category. (Most of the papers we rely on are devoted to what is called $τ$-tilting theory, but for the results we are interested in, there is no need to deal with $τ$-tilting, or even with the Auslander-Reiten translation $τ$).

math.RT↗

arXiv2026

The brick chain complexity of an artin algebra

We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.

math.RT↗

arXiv2025

Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle $\Bbb T(n)$

We consider the category $\mathcal S(n)$ of all pairs $X = (U,V)$, where $V$ is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n = 0$, and $U$ is a subspace of $V$ such that $T(U) \subseteq U$. Our main interest in an object $X=(U,V)$ are the three numbers $uX=\dim U$ (for the subspace), $wX=\dim V/U$ (for the factor) and $bX=\dim {\rm Ker} T$ (for the operator). Actually, instead of looking at the reference space $\Bbb R^3$ with the triples $(uX,wX,bX)$, we will focus the attention to the corresponding projective space $\Bbb T(n)$ which contains for a non-zero object $X$ the level-colevel pair {\bf pr}$X = (uX/bX,wX/bX)$ supporting the object $X$. We use $\Bbb T(n)$ to visualize part of the categorical structure of $\mathcal S(n)$: The action of the duality $D$ and the square $τ_n^2$ of the Auslander-Reiten translation are represented on $\Bbb T(n)$ by a reflection and a rotation by $120^\circ$ degrees, respectively. Moreover for $n\geq 6$, each component of the Auslander-Reiten quiver of $\mathcal S(n)$ has support either contained in the center of $\Bbb T(n)$ or with the center as its only accumulation point. We show that the only indecomposable objects $X$ in $\mathcal S(n)$ with support having boundary distance smaller than 1 are objects with $bX=1$ which lie on the boundary, whereas any rational vector in $\Bbb T(n)$ with boundary distance at least 2 supports infinitely many indecomposable objects. At present, it is not clear at all what happens for vectors with boundary distance between 1 and 2. The use of $\Bbb T(n)$ provides even in the (quite well-understood) case $n = 6$ some surprises: In particular, we will show that any indecomposable object in $\mathcal S(6)$ lies on one of 12 central lines in $\Bbb T(6)$. The paper is essentially self-contained, all prerequisites which are needed are outlined in detail.

math.RT↗

arXiv2023

The short local algebras of dimension 6 with non-projective reflexive modules

Let $A$ be a finite-dimensional local algebra over an algebraically closed field, let $J$ be the radical of $A.$ The modules we are interested in are the finitely generated left $A$-modules. Projective modules are always reflexive, and an algebra is self-injective iff all modules are reflexive. We discuss the existence of non-projective reflexive module in case $A$ is not self-injective. We assume that $A$ is short (this means that $J^3 = 0$). In a joint paper with Zhang Pu, it has been shown that 6 is the smallest possible dimension of $A$ that can occur and that in this case the following conditions have to be satisfied: $J^2$ is both the left socle and the right socle of $A$ and there is no uniform ideal of length 3. The present paper is devoted to show the converse.

math.RT↗

arXiv2022

Gorenstein-projective modules over short local algebras

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra $A$ with radical $J$ will be said to be short provided $J^3 = 0$. As in the commutative case, we show: if a short local algebra $A$ has an indecomposable non-projective Gorenstein-projective module $M$, then either $A$ is self-injective (so that all modules are Gorenstein-projective) and then $|J^2| \le 1$, or else $|J^2| = |J/J^2| - 1$ and $|JM| = |J^2||M/JM|$. More generally, we focus the attention to semi-Gorenstein-projective and $\infty$-torsionfree modules, even to $\mho$-paths of length 2, 3 and 4. In particular, we show that the existence of a non-projective reflexive module implies that $|J^2| < |J/J^2|$ and further restrictions. In addition, we consider exact complexes of projective modules with a non-projective image. Again, as in the commutative case, we see that if such a complex exists, then $A$ is self-injective or satisfies the condition $|J^2| = |J/J^2| - 1.$ Also, we show that any non-projective semi-Gorenstein-projective module $M$ satisfies $Ext^1(M,M) \neq 0$. In this way, we prove the Auslander-Reiten conjecture (one of the classical homological conjectures) for arbitrary short local algebras. Many arguments used in the commutative case actually work in general, but there are interesting differences and some of our results may be new also in the commutative case.

math.RT↗

arXiv2022

Are special biserial algebras homologically tame?

Birge Huisgen-Zimmermann calls a finite dimensional algebra homologically tame provided the little and the big finitistic dimension are equal and finite. The question formulated in the title has been discussed by her in the paper "Representation-tame algebras need not be homologically tame", by looking for any $r > 0$ at a sequence of algebras $Λ_m$ with big finitistic dimension $r+m$. As we will show, also the little finitistic dimension of $Λ_m$ is r+m. It follows that contrary to her assertion, all her algebras $Λ_m$ are homologically tame.

math.RT↗

arXiv2021

Linear Nakayama algebras which are higher Auslander algebras

An artin algebra A is said to be a higher Auslander algebra provided the global dimension and the dominant dimension coincide. We say that a linear Nakayama algebra is monotone, provided its Kupisch series first increases, then decreases. We are going to classify the monotone Nakayama algebras which are higher Auslander algebras. Let us stress that the classification strongly depends on the parity of the global dimension of A.

math.RT↗

arXiv2021

The finitistic dimension of a Nakayama algebra

If A is an artin algebra, Gélinas has introduced an interesting upper bound for the finitistic dimension of A, namely the delooping level del A. We assert that for any Nakayama algebra, its finitistic dimension is equal to del A. This yields also a new proof that the finitistic dimension of A and of its opposite algebra are equal, as shown recently by Sen. If S is a simple module, let e(S) be the minimum of the projective dimension of S and of its injective envelope (one of these numbers has to be finite); and e*(S) the minimum of the injective dimension of S and of its projective cover. Then the finitistic dimension of A is the maximum of the numbers e(S) as well as the maximum of the numbers e^*(S). Using suitable syzygy modules, we construct a permutation h of the simple modules S such that e*(h(S)) = e(S). In particular, this shows for any natural number z, that the number of simple modules S with e(S) = z is equal to the number of simple modules S' with e(S') = z.

math.RT↗

arXiv2020

On modules M such that M and M* are semi-Gorenstein-projective

Let A be an artin algebra. An A-module M is semi-Gorenstein-projective provided that Ext^i(M,A) = 0 for all i > 0. If M is Gorenstein-projective, then both M and its A-dual M* are semi-Gorenstein projective. As we have shown recently, the converse is not true, thus answering a question raised by Avramov and Martsinkovsky. The aim of the present note is to analyse in detail the modules M such that both M and M* are semi-Gorenstein-projective.

math.RT↗

arXiv2020

Simple reflexive modules over finite-dimensional algebras

Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable modules which are reflexive). In addition, we present some properties of simple reflexive modules in general. Marczinzik had motivated his question by providing large classes of algebras such that any algebra in the class which is not self-injective has simple modules which are not reflexive. However, as it turns out, most of these classes have the property that any algebra in the class which is not self-injective has simple modules which are not even torsionless.

math.RT↗

arXiv2020

Koszul modules (and the $Ω$-growth of modules) over short local algebras

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra $A$ with radical $J$ will be said to be short provided $J^3 = 0$. As in the commutative case, also in general, the asymptotic behavior of the Betti numbers of modules seems to be of interest. As we will see, there are only few possibilities for the growth of the Betti numbers of modules. We generalize results which are known for commutative algebras, but some of our results seem to be new also in the commutative case.

math.RT↗

arXiv2019

Gorenstein-projective and semi-Gorenstein-projective modules

An A-module M will be said to be semi-Gorenstein-projective provided that Ext^i(M,A) = 0 for all i > 0. All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. The aim of the paper is to provide conditions on A such that all semi-Gorenstein-projective modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then A is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra with a semi-Gorenstein-projective module M which is not torsionless (thus not Gorenstein-projective). Actually, also the dual module M* is semi-Gorenstein-projective module. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Sega. Since all the syzygy-modules of M and M* are 3-dimensional, the example can be visualized quite easily.

math.RT↗

arXiv2019

Gorenstein-projective and semi-Gorenstein-projective modules. II

Let k be a field and q a non-zero element of k. In Part I, we have exhibited a 6-dimensional k-algebra A = A(q) and we have shown that if q has infinite multiplicative order, then A has a 3-dimensional local module which is semi-Gorenstein-projective, but not torsionless, thus not Gorenstein-projective. This Part II is devoted to a detailed study of all the 3-dimensional local A-modules for this particular algebra A. If q has infinite multiplicative order, we will encounter a whole family of 3-dimensional local modules which are semi-Gorenstein-projective, but not torsionless.

math.RT↗

arXiv2018

Dieter Vossieck and the Development of the Representation Theory of Artin Algebras

Text of a (pre-dinner) lecture at the Bielefeld workshop "Discrete Categories in Representation Theory", April 20 - 21, 2018. This workshop was organized in order to celebrate the 60th birthday of Dieter Vossieck: his famous paper "The algebras with discrete derived category" has to be seen as the starting point of a development which is discussed in this workshop. Dieter Vossieck has published only few papers, but his influence is much larger. We outline some of these contributions.

math.RT↗

arXiv2018

The root posets and their rich antichains

Let $Δ$ be a (connected) Dynkin diagram of rank $n\ge 2$ and $Φ_+ = Φ_+(Δ)$ the corresponding root poset (it consists of all positive roots with respect to a fixed root basis). The width of $Φ_+$ is $n$. We will show that $Φ_+$ is "conical": it is the disjoint union of $n$ solid chains. The rich antichains in $Φ_+$ are the antichains of cardinality $n-1$. It is well known that the number of rich antichains is equal to the cardinality of $Φ_+$. The set $\mathcal R(Δ)$ of rich antichains in $Φ_+$ can itself be considered as a poset which is quite similar, but not always isomorphic, to $Φ_+$. We will show that there always exists a unique rich antichain $A$ such that any rich antichain is contained in the ideal generated by $A$. For $Δ\neq \Bbb E_6$ all roots in $A$ have the same length, namely $e_2$, where $e_1 \le e_2 \le \dots \le e_n$ are the exponents of $Δ.$ For $Δ= \Bbb E_6$, the antichain $A$ consists of four roots of length $e_2 = 4$ and one root of length $5$.

math.CO↗

arXiv2018

The shift orbits of the graded Kronecker modules

The Kronecker modules (or matrix pencils) are the representations of the n-Kronecker quiver K(n) (the quiver with two vertices, namely a sink and a source, and n arrows) over some fixed field. The universal cover of K(n) is the n-regular tree with bipartite orientation. The paper deals with the representations of the n-regular tree with bipartite orientation (thus with graded Kronecker modules). The simultaneous Bernstein-Gelfand-Ponomarev reflection at all sinks will be called the shift functor, and we consider the orbits under this shift functor. Whereas the length of a regular module growths exponentially when we apply the shift functor repeatedly, the radius of such a module growths just linearly. To any regular shift orbit, we attach a positive integer r (the minimal radius of the sink modules in the orbit) and the path in T(n) which starts at the center p of the sink modules in the orbit and ends at the center q of the source modules in the orbit. If r is even, then p has to be a sink, otherwise a source. We call this path the center path of the orbit (since the center of any module in the orbit lies on this path). If the center path of a shift orbit has length b, then the orbit contains precisely b flow modules, the remaining modules are sink modules and source modules. We use this division in order to index the regular graded Kronecker modules in a coherent way. Conversely, we show that given a positive integer r and a path in T(n) (starting at a sink iff r is even), there are regular shift orbits with these invariants.

math.RT↗

arXiv2017

The eigenvector variety of a matrix pencil

Let $k$ be a field and $n,a,b$ natural numbers. A matrix pencil $P$ is given by $n$ matrices of the same size with coefficients in $k$, say by $(b\times a)$-matrices, or, equivalently, by $n$ linear transformations $α_i\:k^a \to k^b$ with $1=1,\dots,n$. We say that $P$ is reduced provided the intersection of the kernels of the linear transformations $α_i$ is zero. If $P$ is a reduced matrix pencil, a vector $v\in k^a$ will be called an eigenvector of $P$ provided the subspace $\langle α_1(v),\dots,α_n(v) \rangle$ of $k^b$ generated by the elements $α_1(v),\dots,α_n(v)$ is $1$-dimensional. Eigenvectors are called equivalent provided they are scalar multiples of each other. The set $ε(P)$ of equivalence classes of eigenvectors of $P$ is a Zariski closed subset of the projective space $\Bbb P(k^a)$, thus a projective variety. We call it the eigenvector variety of $P$. The aim of this note is to show that any projective variety arises as an eigenvector variety of some reduced matrix pencil.

math.NA↗

arXiv2017

Kronecker modules generated by modules of length 2

Let $Λ$ be a ring and $\mathcal N$ a class of $Λ$-modules. A $Λ$-module is said to be generated by $\mathcal N$ provided that it is a factor module of a direct sum of modules in $\mathcal N$. The semi-simple $Λ$-modules are just the $Λ$-modules which are generated by the $Λ$-modules of length 1. It seems that the modules which are generated by the modules of length $2$ (we call them bristled modules) have not attracted the interest they deserve. In this paper we deal with the basic case of the Kronecker modules, these are the (finite-dimensional) representations of an $n$-Kronecker quiver, where $n$ is a natural number. We show that for $n\ge 3$, there is an abundance of bristled Kronecker modules.

math.RT↗