Skew group algebras of piecewise hereditary algebras are piecewise hereditary

Abstract

Communicated by I. Reiten MSC:16S35 18E30 a b s t r a c tWe show that the main results of Happel- Rickard-Schofield (1988) and Happel-ReitenSmalø (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if G is a finite group acting on a piecewise hereditary algebra A over an algebraically closed field whose characteristic does not divide the order of G, then the resulting skew group algebra A[G] is also piecewise hereditary.Let k be an algebraically closed field. For a finite dimensional k-algebra A, we denote by mod A the category of finite dimensional left A-modules, and by D b (A) the (triangulated) derived category of bounded complexes over mod A. Let H be a connected hereditary abelian k-category. Following [7] (compare [5,9]), we say that A is piecewise hereditary of type H if it is derived equivalent to H , that is D b (A) is triangle-equivalent to the derived category D b (H) of bounded complexes over H. Over the years, piecewise hereditary algebras have been widely investigated and proved to be related with many other topics, such as the simply connected algebras and the trivial extensions, the self-injective algebras of polynomial growth and the strong global dimension.Hereditary categories H having tilting objects are of special interest in representation theory of algebras. The endomorphism algebras End H T of tilting objects T in H, called quasitilted algebras, were introduced and studied in [8]. It is well-known that H and End H T are derived equivalent. When k is algebraically closed, it was shown by Happel [6] that H is either derived equivalent to a finite dimensional hereditary k-algebra H or derived equivalent to a category of coherent sheaves cohX on a weighted projective line X (in the sense of [3]).

Finitistic Dimensions and Piecewise Hereditary Property of Skew Group Algebras

“…In [4] Dionne, Lanzilotta, and Smith show that if Λ is a piecewise hereditary algebra, and |G| is invertible in k, then the skew group algebra ΛG is also piecewise hereditary. This result motivates us to characterize general piecewise hereditary skew group algebras.…”

“…Using the ideas and techniques described in [9], we show that Λ and ΛG share more common properties under the same hypothesis and condition. Explicitly, we have: In [4] it has been proved that if Λ is a piecewise hereditary algebra (defined in Section 3) and |G| is invertible in k, then ΛG is piecewise hereditary as well. The second part of the above theorem generalizes this result by using the homological characterization of piecewise hereditary algebras by Happel and Zacharia in [6] We introduce some notations and conventions here.…”

“…Correspondingly, the product can be written as λg · µh = λg(µ)gh. Denote the identity of Λ and the identity of G by 1 Λ and 1 G respectively.It has been observed that when |G|, the order of G, is invertible in k, ΛG and Λ share many common properties [4,11,13]. We wonder for arbitrary groups G, under what conditions this phenomen still happens.…”

Almost laura algebras

“…We are mainly motivated by the fact that skew group algebras generally retain most features from the algebras they arise, especially concerning homological properties. The study of the representation theory of skew group algebras was started in [30,32], and more recently pursued in [7,19,21]. We recall the relevant definitions and refer the reader to [7,12,32] for details.…”

On the Morita Reduced Versions of Skew Group Algebras of Path Algebras

“…Their article was followed-up by many research works illustrating the following principle: Artin algebras share many properties, whether of homological or of representation theoretic nature, with their associated skew group algebras. For instance, see [11] for interactions with Koszul duality; [4,10] for interactions with piecewise hereditary algebras; and [13,3] for interactions with preprojective algebras and McKay correspondence.…”