Selfinjective Artin Algebras Research Articles

Self-injective algebras under derived equivalences

The Nakayama permutations of two derived equivalent, self-injective Artin algebras are conjugate. A different but elementary approach is given to showing that the weak symmetry and self-injectivity of finite-dimensional algebras over an arbitrary field are preserved under derived equivalences.

On the monomorphism category of n-cluster tilting subcategories

Let $${\cal M}$$ be an n-cluster tilting subcategory of mod-Λ, where Λ is an Artin algebra. Let $${\cal S}({\cal M})$$ denote the full subcategory of $${\cal S}(\Lambda )$$ , the submodule category of Λ, consisting of all the monomorphisms in $${\cal M}$$ . We construct two functors from $${\cal S}({\cal M})$$ to $$\bmod - \underline {\cal M} $$ , the category of finitely presented additive contravariant functors on the stable category of $${\cal M}$$ . We show that these functors are full, dense and objective and hence provide equivalences between the quotient categories of $${\cal S}({\cal M})$$ and $$\bmod - \underline {\cal M} $$ . We also compare these two functors and show that they differ by the n-th syzygy functor, provided $${\cal M}$$ is an nℤ-cluster tilting subcategory. These functors can be considered as higher versions of the two functors studied by Ringel and Zhang (2014) in the case $$\Lambda = k\left[ x \right]/\left\langle {{x^n}} \right\rangle $$ and generalized later by Eiríksson (2017) to self-injective Artin algebras. Several applications are provided.

A Frobenius category within the monomorphism category

Given a selfinjective artin algebra Λ, we consider the category Sinj(Λ) of all embeddings of a left Λ-module in a finitely generated injective left Λ-module. We show that Sinj(Λ) is a Frobenius category with Auslander-Reiten sequences such that the categories Λ-mod and Sinj(Λ) are stably equivalent and Sinj(Λ) has twice as many indecomposable injective objects as Λ-mod.

Self-injective artin algebras without short cycles in the component quiver

The project was supported by the Polish National Science Center grant awarded on the basis of the decision number DEC-2011/01/N/ST1/02064.

Derived equivalences for Φ-Auslander-Yoneda algebras

In this paper, we first define a new family of Yoneda algebras, called Φ \Phi -Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets Φ \Phi in N \mathbb N , which includes higher cohomologies indexed by Φ \Phi , and then present a general method to construct a family of new derived equivalences for these Φ \Phi -Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters Φ \Phi are rather abundant. Among applications of our method are the following results: (1) if A A is a self-injective Artin algebra, then, for any A A -module X X and for any admissible set Φ \Phi in N \mathbb N , the Φ \Phi -Auslander-Yoneda algebras of A ⊕ X A\oplus X and A ⊕ Ω A ( X ) A\oplus \Omega _A(X) are derived equivalent, where Ω \Omega is the Heller loop operator. (2) Suppose that A A and B B are representation-finite self-injective algebras with additive generators A X _AX and B Y _BY , respectively. If A A and B B are derived equivalent, then so are the Φ \Phi -Auslander-Yoneda algebras of X X and Y Y for any admissible set Φ \Phi . In particular, the Auslander algebras of A A and B B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between Φ \Phi -Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles.

On Auslander-Reiten components of algebras without external short paths

We describe the structure of semi-regular Auslander–Reiten components of artin algebras without external short paths in the module category. As an application, we give a complete description of self-injective artin algebras whose Auslander–Reiten quiver admits a regular acyclic component without external short paths.

On selfinjective artin algebras having generalized standard quasitubes

We give a complete description of the Morita equivalence classes of all connected selfinjective artin algebras for which the Auslander–Reiten quiver admits a family of quasitubes having common composition factors, closed under composition factors, and consisting of modules not lying on infinite short cycles.

The component quiver of a self-injective artin algebra

We prove that the component quiver ${\mit\Sigma }_A$ of a connected self-injective artin algebra $A$ of infinite representation type is fully cyclic, that is, every finite set of components of the Auslander–Reiten quiver ${\mit\Gamma }_A$ of $A$ lie

On Modules of Finite Complexity Over Selfinjective Artin Algebras

In this paper we study Auslander-Reiten sequences of modules with finite complexity over selfinjective artin algebras. In particular, we show that for all eventually Ω-perfect modules of finite complexity, the number of indecomposable non projective summands of the middle term of such sequences is bounded by 4. We also describe situations in which all non projective modules in a connected component of the Auslander-Reiten quiver are eventually Ω-perfect.

Irreducible morphisms in the bounded derived category

We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ , denoted D b ( Λ mod ) , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in C b ( Λ proj ) either f j is split monomorphism for all j ∈ Z or split epimorphism, for all j ∈ Z . Moreover, we prove that all the non-trivial components of the Auslander–Reiten quiver of C b ( Λ proj ) are of the form Z A ∞ .

Auslander-Reiten components containing modules with bounded Betti numbers

Let R R be a connected selfinjective Artin algebra, and M M an indecomposable nonprojective R R -module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of R R . We prove that the Auslander-Reiten sequence ending at M M has at most two indecomposable summands in the middle term. Furthermore we show that the component of the Auslander-Reiten quiver containing M M is either a stable tube or of type Z A ∞ \mathbb ZA_{\infty } . We use these results to study modules with eventually constant Betti numbers, and modules with eventually periodic Betti numbers.

Representation dimension and finitely generated cohomology

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.

Stable Equivalence of Selfinjective Artin Algebras of Dynkin Type

We prove new results on the stable equivalences of selfinjective Artin algebras of tilted Dynkin type, extending the main results of Skowronski and Yamagata to arbitrary tilted type.

Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.

On selfinjective artin algebras having nonperiodic generalized standard Auslander–Reiten components

On selfinjective artin algebras having nonperiodic generalized standard Auslander–Reiten components

ON DERIVED EQUIVALENT COHERENT RINGS

For a selfinjective artin algebra , the projectively stable category of finitely presented left -modules has the structure of a triangulated category[5] and the canonical functor is an equivalence ...

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB n andC n and obtain the following main results:

Modules without self-extensions and Nakayama's conjecture

Let A be a self-injective artin algebra and X a finitely generated module. In this paper, we are mainly concerned with the problem posed by Tachikawa [16]: I f Ext"(X, X) = 0 for all n >_ 1, is then X projective? This problem arose from Nakayama's conjecture [11] which states that a finite dimensional algebra of infinite dominant dimension is selfinjective. It should be noted that if the answer to the above problem is affirmative, then Nakayama's conjecture is true for finite dimensional QF-3 algebras R with minimal faithful ideal Re such that e R e is self-injective (see [16] for details). Note also that a finite dimensional algebra of dominant dimension > 1 is QF-3 [15]. Tachikawa [16] showed that the above problem has affirmative answer in case A is a group algebra of a p-group, and this result was recently generalized by Schulz [14] to the case of A being a group algebra of an arbitrary finite group. We will also give several partial answers to the above problem and thus to Nakayama's conjecture. Throughout this paper, all modules are finitely generated modules over artin algebras, and most modules are fight modules. Given an artin algebra A over the center C, we denote by D the duality H o m e ( , / ) , where I is the injective envelope of C/tad C over C, and by Jf~a the category of the finitely generated fight A-modules. For modules X, Y we denote by Horn(X, Y) the factor group of Horn(X, Y) modulo the subgroup of the homomorphisms which factor through projective modules, and for a module X we denote by f2"X the n-th syzygy module of X. Note that in case A is self-injective, induces a self-equivalence of the stable category [I0]. Moreover, in case A is symmetric, we have (22 = D Tr. We refer to [1] and [2] for Auslander-Reiten sequences, irreducible homomorphisms and so on, and to [13] and [8] for (stable) Auslander-Reiten quivers. In what follows, D Tr and Tr D are denoted by z and ~1 respectively, and the "AuslanderReiten sequence" and the "Auslander-Reiten quiver" are siflaply written by the "ARsequence" and the "AR-quiver" respectively.