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Finiteness of the strong global dimension of radical square zero algebras

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Kerner O., Skowroński A., Yamagata K., Zacharia D.
Open Mathematics
|
2004
|
tom 2
|
nr 1
103-111
EN
The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.
2
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The representation dimension of domestic weakly symmetric algebras

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Bocian R., Holm T., Skowroński A.
Open Mathematics
|
2004
|
tom 2
|
nr 1
67-75
EN
Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.
3
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Classification of discrete derived categories

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Bobiński G., Geiß C., Skowroński A.
Open Mathematics
|
2004
|
tom 2
|
nr 1
19-49
EN
The main aim of the paper is to classify the discrete derived categories of bounded complexes of modules over finite dimensional algebras.
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