Finitistic Dimensions and Piecewise Hereditary Property of Skew Group Algebras

Abstract

Abstract. Let Λ be a finite dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Under the assumption that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S G. If the action of S on E is free, we show that the skew group algebra ΛG and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra Λ S is a direct summand of the Λ S -bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for ΛG to be piecewise hereditary. IntroductionThroughout this note let Λ be a finite dimensional k-algebra, where k is an algebraically closed field with characteristic p 0, and let G be a finite group whose elements act on Λ as algebra automorphisms. The skew group algebra ΛG is the vector space Λ ⊗ k kG equipped with a bilinear product · determined by the following rule:is the image of µ under the action of g. We write λg rather than λ ⊗ g to simplify the notation. 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. This problem is considered in [9], where under the hypothesis that Λ has a complete set E of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S G, we show that Λ and ΛG share certain properties such as finite global dimension, finite representation type, etc., if and only if the action of S on E is free. Clearly, this answer generalizes results in [13] since if |G| is invertible in k, the only Sylow p-subgroup of G is the trivial group.In this note we continue to study representations and homological properties of modular 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:2010 Mathematics Subject Classification. 16G10, 16E10. Key words and phrases. skew group algebras, finitistic dimension, piecewise hereditary, strong global dimension.The author would like to thank the referee for carefully reading and checking this paper, and for his/her valuable comments.

Homological Dimensions of Crossed Products

“…A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S G, we show that A and A σ α G share the same homological dimensions under extra assumptions, extending the main results in [15,16]. …”

“…We then focus on a special case that A is a semiprimary left Noetherian algebra over an algebraically closed field k, and suppose that there are a Sylow p-subgroup S G and a complete set E = {e i } i∈ [n] of primitive orthogonal idempotents in A closed under the action of S. Denote by C(A) and A S the center of A and the fixed algebra respectively. The following conclusion gives us a feasible criterion for the global dimension of A σ α G to be finite, generalizing a main result in [15,16] for skew group algebras. Theorem 1.3.…”

“…This makes the conjecture posted in [16] by the author trivial, which asks whether fin. dim A σ G < ∞ whenever fin.…”

“…For instance, their homological dimensions are studied by Yi and Aljadeff in [1,2,24], and several criteria for the global dimension to be finite are described; Reiten and Riedtmann show that A and its skew group algebra A σ G share many properties for finite dimensional algebras A and finite groups G when the order of G is invertible in A ( [23]); in [15,16], the author proves that these properties are still shared by A and A σ G for arbitrary finite groups G under a much weaker assumption.…”

“…Based on techniques and known results introduced in [1,2,15,16], we attack this problem from the viewpoint of representation theory. As the first step, we lift the classical induction and…”