Tilting-connected symmetric algebras. (English) Zbl 1348.16010
Let \(A\) be a finite dimensional algebra over a field. The main result of the paper asserts that if \(A\) is representation-finite symmetric then the homotopy category \(K^b(\mathrm{proj\,}A)\) of bounded complexes of projective \(A\)-modules is tilting-connected, that is, the action of iterated irreducible tilting mutation on this category is transitive. In fact, a stronger assertion is proved: the category \(K^b(\mathrm{proj\,}A)\) is silting-discrete, provided \(A\) is a representation-finite symmetric algebra. The theorem is obtained within the frames of the theory of silting mutations for self-injective algebras, introduced by T. Aihara and O. Iyama [“Silting mutation in triangulated categories”, arXiv:1009.3370]. Another important result is a silting version of the classical Bongartz’s Lemma on partial tilting modules.
Reviewer: Stanisław Kasjan (Toruń)
MSC:
16G10 | Representations of associative Artinian rings |
16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
18E30 | Derived categories, triangulated categories (MSC2010) |
16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
Keywords:
finite dimensional algebras; representation-finite symmetric algebras; derived categories; tilting objects; silting objects; silting mutations; Bongartz-type lemmaReferences:
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