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In [1] it is proved that one must take care trying to copy results from the case of modules to an arbitrary Grothendieck category in order to describe a hereditary torsion theory in terms of filters of a generator. By the other side, we usually have for a Grothendick category an infinite family of generators {Gi; i ∈ I} and, although each Gi has good properties the generator G = ⊕i ∈ I Gi is not easy to handle (for instance in categories like graded modules). In this paper the authors obtain a bijective...
In [2] an internal homology theory of crossed modules was defined (CCG-homology for short), which is very much related to the homology of the classifying spaces of crossed modules ([5]). The goal of this note is to construct a low-dimensional homology exact sequence corresponding to a central extension of crossed modules, which is quite similar to the one constructed in [3] for group homology.
Let be a triangulated category and be a cluster tilting subcategory of . Koenig and Zhu showed that the quotient category is Gorenstein of Gorenstein dimension at most one. But this is not always true when becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let be an extriangulated category with enough projectives and enough injectives, and a cluster...
We introduce the notion of Gorenstein star modules and obtain some properties and a characterization of them. We mainly give the relationship between -Gorenstein star modules and -Gorenstein tilting modules, see L. Yan, W. Li, B. Ouyang (2016), and a new characterization of -Gorenstein tilting modules.
We investigate gradings on tame blocks of group algebras whose defect groups are dihedral. For this subfamily of tame blocks we classify gradings up to graded Morita equivalence, we transfer gradings via derived equivalences, and we check the existence, positivity and tightness of gradings. We classify gradings by computing the group of outer automorphisms that fix the isomorphism classes of simple modules.
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