Selfinjective Koszul Algebras

The study of Koszul algebras and their representations has accelerated significantly in the last few years. They have been used in Topology, Algebraic Geometry and Commutative Algebra and they are used more and more frequently in Representation Theory, see for instance [BGS],[GTM],[M],[MZ] and [R]. The aim of this paper is to present some of the results presented by the second author at the Luminy conference that will appear in [MZ], as well as some new facts about the shapes of the components of the Auslander-Reiten quivers of selfinjective Koszul algebras. Namely, we show that if we have a selfinjective Koszul algebra of Loewy length greater than three having a noetherian Koszul dual, then each component of its graded A-R quiver is of the form ZA∞ and we show that this need not be the case for the usual A-R quiver if we ignore the grading. By a Koszul algebra we mean a graded associative algebra over a field K, Λ = Λ0 ⊕ Λ1 ⊕ . . . satisfying the following conditions: (1) for each i, the Kdimension of Λi is finite, and for each i, j ≥ 0 we have ΛiΛj = Λi+j ; (2) Λ0 ' K × · · · × K, and (3) the Yoneda ext-algebra of Λ, E(Λ) = ⊕n≥0 ExtΛ(Λ0,Λ0) is generated as a graded K-algebra by the degree zero and one parts. For more general facts about Koszul algebras and modules with linear resolutions we refer to the wonderful survey article of Fröberg, as well as to [GM]. The ext algebra of a Koszul algebra Λ is also denoted Λ and is also called the Koszul dual, or the shriek algebra of Λ. Recall that a graded module has a linear (graded) projective resolution, if the module is generated in degree zero, and, there exists a graded resolution of the module such that the first projective module in the resolution is generated in degree zero, the second one in degree one, the third one in degree two, and so on. So in a way, these modules have projective resolutions whose terms have the generators distributed in the best possible places. Note that linear resolutions are always minimal. The modules having these types of resolutions, and their graded shifts are also called Koszul modules, and let KΛ be the subcategory of grΛ consisting of all the modules having linear resolutions. If Λ is a Koszul algebra, then so is E(Λ) and we have a contravariant functor E : modΛ → grE(Λ) that induces a duality between the subcategories KΛ and KE(Λ). The notion of Koszul modules was generalized in [GM] to what we call now weakly a Koszul module. Observe that the definition makes sense even in the nongraded case. If Λ is a Koszul algebra with graded radical J , then a module M is weakly Koszul, if there exists a projective resolution of M :· · · → Pn fn → Pn−1 → · · · → P0 f0 →M → 0 such that for each i, k ≥ 0 we have JPi∩Ker fi = J Ker fi. Note again, that such a resolution, if it exists, must be minimal. Throughout this paper we will always assume that we deal with Koszul algebras. The modules will always be finitely generated and graded, and unless specified, the homomorphisms will be degree zero homomorphisms.