a b s t r a c tWe say that an algebra A is periodic if it has a periodic projective resolution as an (A, A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B → A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba's classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.
Abstract. A triangulated category (T , Σ) is said to be Calabi-Yau of dimension d if Σ d is a Serre functor. We determine which stable module categories of self-injective algebras Λ of finite type are Calabi-Yau and compute their Calabi-Yau dimensions, correcting errors in previous work. We first show that the Calabi-Yau property of mod-Λ can be detected in the minimal projective resolution of the stable Auslander algebra Γ of Λ, over its enveloping algebra. We then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the (generalized) preprojective algebras of Dynkin graphs. For representation-finite self-injective algebras of torsion order t = 1, we obtain a complete description of their stable Calabi-Yau properties, but only partial results for those algebras of torsion order t = 2. We also obtain some new information about the periods of the representation-finite self-injective algebras of torsion order t > 1. Finally, we describe how these questions can also be approached by realizing the stable categories of representation-finite selfinjective algebras as orbit categories of the bounded derived categories of hereditary algebras, and illustrate this technique with several explicit computations that our previous methods left unsettled.
In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. Using some of these characterizations, we prove that, if a quasi-duo ring R has stable range n, the same is true for any semisplit corner ring of R. This contrasts with earlier results of Vaserstein and Warfield, which showed that the stable range can increase unboundedly upon descent to (even) Peirce corner rings.
Abstract. Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu [14], we say that a family S ⊆ T of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of T . We construct torsion pairs in T associated to any subset X of a simple-minded system S, and use these to define left and right mutations of S relative to X . When T has a Serre functor ν and S and X are invariant under ν • [1], we show that these mutations are again simpleminded systems. We are particularly interested in the case where T = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in D b (mod-Λ) [15]. It follows that the mutation of the set of simple Λ-modules relative to X yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to X .
We investigate when an exact functor F ∼ = − ⊗ Λ M Γ : mod-Λ → mod-Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard's for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ , it is possible to replace Λ by a Morita equivalent k-algebra ∆ such that Γ is a subring of ∆ and the induction and restriction functors induce inverse stable equivalences.
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