The construction of almost split sequences, III: Modules over two classes of tame local algebras

Butler, M. C. R.; Shahzamanian, M. H.

doi:10.1007/bf01364137

Cited by 15 publications

(10 citation statements)

References 7 publications

(22 reference statements)

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“…For instance, one example of a selfinjective artin algebra having a ZA ∞ ∞ component which contains modules of finite complexity is the group algebra of the dihedral 2-group of order 8 over a field of characteristic 2, [7], and an example where a ZÃ 1,2 component appears can be found in [6,14]. Also, using Tachikawa's theorem [15], one can see that a trivial extension of a hereditary algebra of typeÃ n has a component of type ZÃ n .…”

Section: Corollary 33 Let Be a Self Injective Artin Algebra And Letmentioning

confidence: 99%

On Modules of Finite Complexity Over Selfinjective Artin Algebras

Green

Zacharia

2010

Algebr Represent Theor

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In this paper we study Auslander-Reiten sequences of modules with finite complexity over selfinjective artin algebras. In particular, we show that for all eventually -perfect modules of finite complexity, the number of indecomposable non projective summands of the middle term of such sequences is bounded by 4. We also describe situations in which all non projective modules in a connected component of the Auslander-Reiten quiver are eventually -perfect.

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Section: Corollary 33 Let Be a Self Injective Artin Algebra And Letmentioning

confidence: 99%

On Modules of Finite Complexity Over Selfinjective Artin Algebras

Green

Zacharia

2010

Algebr Represent Theor

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“…We assume throughout that k is an algebraically closed field of characteristic 2 and that G is a finite group with a dihedral Sylow 2-subgroup having order at least 8. The methods we will use apply in this generality: if we were to assume that G is a dihedral 2-group the string and band module methods of [7] would become available to us, resulting in some easier arguments.…”

Section: An Example: Groups With Dihedral Sylow 2-subgroupsmentioning

confidence: 99%

“…We use functorial methods to establish these properties. This may seem complicated, but the generality of the methods avoids the question of extending the string and band module description of [7] from dihedral 2-groups to groups with dihedral Sylow 2-subgroups. The approach depends on the shape of the quiver, and seems to have potential applications in other situations with a quiver of this shape.…”

Section: An Example: Groups With Dihedral Sylow 2-subgroupsmentioning

confidence: 99%

The Graded Center of a Triangulated Category

Carlson¹,

Webb²

2016

J. Aust. Math. Soc.

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Abstract. With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor We show that such natural transformations which have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities.Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

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“…De nition 7.1. BS,112] (1) Let W be a word (not necessarily irreducible) as above. De ne the associated string module M(W) to be the pq -module with generators u 1 ; : : : ; u n and relations a i 1 u 1 = 0; b j k u k = a i k+1 u k+1 (k = 1; : : : ; n ?…”

Section: Classification Of Indecomposablesmentioning

confidence: 99%

“…Theorem 7.3. ( BS,p. 112]) Artinian string and band modules account for all nitely generated indecomposable pq {modules.…”

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confidence: 99%

Finitely Generated Modules over Pullback Rings

Arnold

Laubenbacher

1996

Journal of Algebra

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The purpose of this paper is to outline a new approach to the classification of finitely generated indecomposable modules over certain kinds of pullback rings. If R is the pullback of two hereditary noetherian homogeneously serial rings, finitely generated over their centers, over a common semi-simple artinian ring, then this classification can be divided into the classification of indecomposable artinian modules and those modules over the coordinate rings with no non-trivial artinian submodules. The classification of the artinian modules can be reduced to the case of a finite dimensional algebra over a semi-simple ring. This approach is then used as the key step in the case where the coordinate rings are hereditary noetherian serial rings over a common quotient which is a matrix ring over a field, resulting in a complete module classification.

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The construction of almost split sequences, III: Modules over two classes of tame local algebras

Cited by 15 publications

References 7 publications

On Modules of Finite Complexity Over Selfinjective Artin Algebras

On Modules of Finite Complexity Over Selfinjective Artin Algebras

The Graded Center of a Triangulated Category

Finitely Generated Modules over Pullback Rings

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