On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

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The finiteness of the Gorenstein dimension for Artin algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501123 ◽

2019 ◽

Vol 18 (06) ◽

pp. 1950112

Author(s):

René Marczinzik

Keyword(s):

Projective Dimension ◽

Indecomposable Module ◽

Global Dimension ◽

Injective Dimension ◽

Artin Algebra ◽

Artinian Rings ◽

Gorenstein Dimension ◽

Finite Global Dimension ◽

Artin Algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

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On the structure of indecomposable modules over Artin algebras

Colloquium Mathematicum ◽

10.4064/cm-48-1-17-34 ◽

1984 ◽

Vol 48 (1) ◽

pp. 17-34 ◽

Cited By ~ 1

Author(s):

Andrzej Skowroński

Keyword(s):

Indecomposable Modules ◽

Artin Algebras

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The Inductive Step of the Second Brauer-Thrall Conjecture

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-026-0 ◽

1980 ◽

Vol 32 (2) ◽

pp. 342-349 ◽

Cited By ~ 9

Author(s):

Sverre O. Smalø

Keyword(s):

Representation Theory ◽

Finitely Generated ◽

Inductive Step ◽

Artin Algebra ◽

Indecomposable Modules ◽

Infinite Type ◽

Categorical Methods ◽

Artin Algebras ◽

Almost Split Sequences ◽

Finitely Generated Modules

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.Section 1 is devoted to giving the necessary background in the theory of almost split sequences.As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

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Minimal representation-infinite artin algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100072546 ◽

1994 ◽

Vol 116 (2) ◽

pp. 229-243 ◽

Cited By ~ 23

Author(s):

Andrzej Skowroński

Keyword(s):

Jacobson Radical ◽

Minimal Representation ◽

Finitely Generated ◽

Artin Algebra ◽

Indecomposable Modules ◽

Artin Ring ◽

Artin Algebras ◽

Infinite Power

Let A be an artin algebra over a commutative artin ring R, mod A be the category of finitely generated right A-modules, and rad∞ (modA) be the infinite power of the Jacobson radical rad(modA) of modA. Recall that A is said to be representation-finite if mod A admits only finitely many non-isomorphic indecomposable modules. It is known that A is representation-finite if and only if rad∞ (mod A) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [26, 2] we know that A is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod A.

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Composition of irreducible morphisms in quasi-tubes

Journal of Algebra and Its Applications ◽

10.1142/s0219498817500712 ◽

2017 ◽

Vol 16 (04) ◽

pp. 1750071 ◽

Cited By ~ 2

Author(s):

Claudia Chaio ◽

Piotr Malicki

Keyword(s):

Module Category ◽

Indecomposable Modules ◽

Artin Algebras ◽

Irreducible Morphisms

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander–Reiten quivers of artin algebras in relation with the powers of the radical of their module category.

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Habitat Association of Klebsiella Species

Infection Control ◽

10.1017/s0195941700062603 ◽

1985 ◽

Vol 6 (2) ◽

pp. 52-58 ◽

Cited By ~ 111

Author(s):

Susan T. Bagley

Keyword(s):

Drinking Water ◽

Gastrointestinal Tract ◽

Surface Waters ◽

Industrial Effluents ◽

Opportunistic Pathogen ◽

Habitat Associations ◽

Habitat Association ◽

Klebsiella Species ◽

Almost All ◽

Polluted Waters

AbstractThe genus Klebsiella is seemingly ubiquitous in terms of its habitat associations. Klebsiella is a common opportunistic pathogen for humans and other animals, as well as being resident or transient flora (particularly in the gastrointestinal tract). Other habitats include sewage, drinking water, soils, surface waters, industrial effluents, and vegetation. Until recently, almost all these Klebsiella have been identified as one species, ie, K. pneumoniae. However, phenotypic and genotypic studies have shown that “K. pneumoniae” actually consists of at least four species, all with distinct characteristics and habitats. General habitat associations of Klebsiella species are as follows: K. pneumoniae—humans, animals, sewage, and polluted waters and soils; K. oxytoca—frequent association with most habitats; K. terrigena— unpolluted surface waters and soils, drinking water, and vegetation; K. planticola—sewage, polluted surface waters, soils, and vegetation; and K. ozaenae/K. rhinoscleromatis—infrequently detected (primarily with humans).

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SEM Cold Stage Techniques for the Observation of Vegetative and Floral Apices of Oat (Avena sativa L.) Plants

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100080377 ◽

1977 ◽

Vol 35 ◽

pp. 590-591

Author(s):

B. K. Kirchoff ◽

L.F. Allard ◽

W.C. Bigelow

Keyword(s):

Critical Point ◽

Avena Sativa ◽

Substantial Improvement ◽

Fresh Tissue ◽

Cold Stage ◽

Critical Point Drying ◽

Almost All ◽

Cell Collapse ◽

Conductive Paint ◽

Carbon Based

In attempting to use the SEM to investigate the transition from the vegetative to the floral state in oat (Avena sativa L.) it was discovered that the procedures of fixation and critical point drying (CPD), and fresh tissue examination of the specimens gave unsatisfactory results. In most cases, by using these techniques, cells of the tissue were collapsed or otherwise visibly distorted. Figure 1 shows the results of fixation with 4.5% formaldehyde-gluteraldehyde followed by CPD. Almost all cellular detail has been obscured by the resulting shrinkage distortions. The larger cracks seen on the left of the picture may be due to dissection damage, rather than CPD. The results of observation of fresh tissue are seen in Fig. 2. Although there is a substantial improvement over CPD, some cell collapse still occurs.Due to these difficulties, it was decided to experiment with cold stage techniques. The specimens to be observed were dissected out and attached to the sample stub using a carbon based conductive paint in acetone.

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