Algebras of Finite Global Dimension
We survey some results on finite dimensional algebras of finite global dimension and address some open problems.
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Abstract We survey some results on finite dimensional algebras of finite global dimension and address some open problems.
1 Introduction Let Λ be a finite dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. In this article we are mainly interested in algebras of finite global dimension, so each X ∈ mod Λ admits a finite projective resolution, or equivalently each simple Λ-module admits a finite projective resolution. It is well-known that the global dimension, gldim Λ, of Λ is the maximum of the lengths of these finitely many minimal projective resolutions of the simple Λ-modules. These notions go back to the pioneering book by Cartan and Eilenberg [11]. They were intensively studied in a famous series of ten papers in the Nagoya Journal written by various authors and published over the years 1955 to 1958, see [1, 2, 8, 17–21, 40, 41] for this series of articles. The global dimension being preserved under Morita equivalence, implies that we may assume without loss of generality that Λ is basic. As the field k is algebraically closed, Λ is given by a quiver with relations. We briefly recall the construction. We start by recalling the definition of an admissible ideal. Let Q be a finite quiver and let kQ be the path algebra over k. Recall that the set W = {paths in Q} forms a k-basis for kQ. Denote by Q≥t the two sided ideal of kQ generated by all the paths in Q of length t. A two sided ideal I in kQ is called admissible if there exists a natural number t ≥ 2 such that Q≥t ⊆ I ⊆ Q≥2 .
D. Happel Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany D. Zacharia (B) Department of Mathematics, Syracuse University, Syracuse, NY 13244-0001, USA e-mail: [email protected] A.B. Buan et al. (eds.), Algebras, Quivers and Representations, Abel Symposia 8, DOI 10.1007/978-3-642-39485-0_5, © Springer-Verlag Berlin Heidelberg 2013
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Then it is well-known (see [23]) that every basic finite dimensional k-algebra Λ satisfies Λ kQ/I for some finite quiver Q and admissible ideal I in kQ. Note that the quiver Q is uniquely determined by Λ. By abuse of language, a minimal generating set of the ideal I is called a set of relations for Λ. This set is always finite, but we may have different choices for the relations for Λ. In particular there is usually no canonical choice for the relations. Note also that due to our assumptions on the admissible ideal I , the quiver Q can be recovered from Λ = kQ/I . To be more precise, the vertices of Q correspond to the (isomorphism classes of) simple Λ-modules and the number of arrows from a simple Λ-module S to a simple Λmodule S coincides with dimk Ext1Λ (S, S ). We remind the reader that there is also a more ring theoretic version to find the quiver of Λ (see for example [5]). We will address two basic questions for algebras of finite global dimension. First we deal with the question on possible obstructions for the quiver of an algebra of finite global dim
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