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A class of quasitilted rings that are not tilted

Riccardo Colpi, Kent R. Fuller, Enrico Gregorio (2006)

Colloquium Mathematicae

Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.

A duality result for almost split sequences

Lidia Hügel, Helmut Valenta (1999)

Colloquium Mathematicae

Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.

A family of noetherian rings with their finite length modules under control

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

We investigate the category mod Λ of finite length modules over the ring Λ = A k Σ , where Σ is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k -algebra. Each simple module E j gives rise to a quasiprogenerator P j = A E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that mod Λ j b a d h b o x P j . As a consequence we can (1) construct for each elementary k -algebra A over a finite field k a nonartinian noetherian ring Λ such that mod A mod Λ , (2) find twisted...

A note on quasitilted algebras

Andrzej Skowroński, Adam Skowyrski (2014)

Colloquium Mathematicae

We provide a characterization of artin algebras without chains of nonzero homomorphisms between indecomposable finitely generated modules starting with an injective module and ending with a projective module.

A note on tilting sequences

Clezio Braga, Flávio Coelho (2008)

Open Mathematics

We discuss the existence of tilting modules which are direct limits of finitely generated tilting modules over tilted algebras.

Algebras standardly stratified in all orders

Fidel Hernández Advíncula, Eduardo do Nascimento Marcos (2007)

Colloquium Mathematicae

The aim of this work is to characterize the algebras which are standardly stratified with respect to any order of the simple modules. We show that such algebras are exactly the algebras with all idempotent ideals projective. We also deduce as a corollary a characterization of hereditary algebras, originally due to Dlab and Ringel.

Almost split sequences for non-regular modules

S. Liu (1993)

Fundamenta Mathematicae

Let A be an Artin algebra and let 0 X i = 1 r Y i Z 0 be an almost split sequence of A-modules with the Y i indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver Γ A of A. Then r ≤ 4, and r = 4 implies that one of the Y i is projective-injective. Moreover, if X j = 1 t Y j is a source map with the Y j indecomposable and X on an oriented cycle in Γ A , then t ≤ 4 and at most three of the Y j are not projective. The dual statement for a sink map holds. Finally, if an arrow...

Bar-invariant bases of the quantum cluster algebra of type A 2 ( 2 )

Xueqing Chen, Ming Ding, Jie Sheng (2011)

Czechoslovak Mathematical Journal

We construct bar-invariant [ q ± 1 / 2 ] -bases of the quantum cluster algebra of the valued quiver A 2 ( 2 ) , one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Calabi-Yau stable module categories of finite type

Jerzy Białkowski, Andrzej Skowroński (2007)

Colloquium Mathematicae

We describe the stable module categories of the self-injective finite-dimensional algebras of finite representation type over an algebraically closed field which are Calabi-Yau (in the sense of Kontsevich).

Combinatorial topology and the global dimension of algebras arising in combinatorics

Stuart Margolis, Franco Saliola, Benjamin Steinberg (2015)

Journal of the European Mathematical Society

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the...

Cycle-finite algebras of semiregular type

Jerzy Białkowski, Andrzej Skowroński, Adam Skowyrski, Paweł Wiśniewski (2012)

Colloquium Mathematicae

We describe the structure of artin algebras for which all cycles of indecomposable finitely generated modules are finite and all Auslander-Reiten components are semiregular.

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