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Squared cycles in monomial relations algebras

100%
Jue B.
Open Mathematics
|
2006
|
tom 4
|
nr 2
250-259
EN
Let $$\mathbb{K}$$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $$\Lambda = {{\mathbb{K}\Gamma } \mathord{\left/ {\vphantom {{\mathbb{K}\Gamma } I}} \right. \kern-\nulldelimiterspace} I}$$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra $$\mathbb{K}\Gamma $$ . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern in the syzygies of a specific factor module of the corresponding indecomposable projective module is found, allowing us to conclude that the square of any cycle must lie in the ideal I.
2
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Finiteness of the strong global dimension of radical square zero algebras

84%
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.
3
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The representation dimension of domestic weakly symmetric algebras

84%
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.
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