Piecewise hereditary algebras

Happel, Dieter; Reiten, Idun; Smalø, Sverre O.

doi:10.1007/bf01195702

Cited by 42 publications

(45 citation statements)

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“…A finite dimensional k-algebra A is called piecewise hereditary if the derived category D b (A) is equivalent to the derived category D b (H) of a hereditary abelian category H as triangulated categories; see [5,6,7,8]. If A is piecewise hereditary, then gl.…”

Section: Strong Global Dimensions and Piecewise Hereditary Algebrasmentioning

confidence: 99%

Finitistic Dimensions and Piecewise Hereditary Property of Skew Group Algebras

2014

Glasgow Math. J.

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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.

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Section: Strong Global Dimensions and Piecewise Hereditary Algebrasmentioning

confidence: 99%

Finitistic Dimensions and Piecewise Hereditary Property of Skew Group Algebras

2014

Glasgow Math. J.

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“…Recall also the characterisation proved by Happel, Rickard and Schofield [15]: D b (mod A) is equivalent to bounded derived category of the module category of a finite-dimensional hereditary algebra if and only if there exists a sequence of algebras with first term such a hereditary algebra, last term A, and where each term is isomorphic to the opposite of the endomorphism algebra of a tilting module over the preceding term. If the sequence has ℓ + 2 terms, then gl.dim.…”

Section: Introductionmentioning

confidence: 89%

The strong global dimension of piecewise hereditary algebras

Alvares

Meur

Marcos³

2017

Journal of Algebra

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Abstract. Let A be a finite-dimensional piecewise hereditary algebra over an algebraically closed field. This text investigates the strong global dimension of A. This invariant is characterised in terms of the lengths of sequences of tilting mutations relating A to a hereditary abelian category, in terms of the generating hereditary abelian subcategories of the derived category of A, and in terms of the AuslanderReiten structure of that derived category.

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“…below. On the other hand, two hereditary finite-dimensional algebras are T -equivalent iff they are derived equivalent, by a result of Happel-Rickard-Schofield [34].…”

Section: Grothendieck Groups Thenmentioning

confidence: 99%

Derived categories and tilting

Keller¹

2007

Handbook of Tilting Theory

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Abstract. We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel's theorem which states that each tilting triple yields an equivalence between derived categories. We establish its link with Rickard's theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalences. Using t-structures we compare two abelian categories having equivalent derived categories. Finally, we briefly sketch a generalization of the tilting setup to differential graded algebras.

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Piecewise hereditary algebras

Cited by 42 publications

References 6 publications

Finitistic Dimensions and Piecewise Hereditary Property of Skew Group Algebras

Finitistic Dimensions and Piecewise Hereditary Property of Skew Group Algebras

The strong global dimension of piecewise hereditary algebras

Derived categories and tilting

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