Let $A$ be a finite-dimensional self-injective algebra over an algebraically closed field, $\mathcal{C}$ a stably quasi-serial component (i.e. its stable part is a tube) of rank $n$ of the Auslander-Reiten quiver of $A$, and $\mathcal{S}$ be a simple-minded system of the stable module category $\stmod{A}$. We show that the intersection $\mathcal{S}\cap\mathcal{C}$ is of size strictly less than $n$, and consists only of modules with quasi-length strictly less than $n$. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of $A$ cannot be in any simple-minded system.
Recently, we obtained in [7] a new characterization for an orthogonal system to be a simple-minded system in the stable module category of any representation-finite self-injective algebra. In this paper, we apply this result to give an explicit construction of simple-minded systems over self-injective Nakayama algebras.
Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.
Let $A$ be a representation-finite self-injective algebra over an algebraically closed field $k$. We give a new characterization for an orthogonal system in the stable module category $A$-$\stmod$ to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in $A$-$\stmod$ extends to a simple-minded system.
We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular…
Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings 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 simple-minded systems. We are particularly interested in…
Let $Q$ be an acyclic quiver and $w \geq 1$ be an integer. Let $\mathsf{C}_{-w} (\mathbf{k} Q)$ be the $(-w)$-cluster category of $\mathbf{k} Q$. We show that there is a bijection between simple-minded collections in $\mathsf{D}^b (\mathbf{k} Q)$ lying in a fundamental domain of $\mathsf{C}_{-w} (\mathbf{k} Q)$ and $w$-simple-minded systems in $\mathsf{C}_{-w} (\mathbf{k} Q)$. This generalises the same result of Iyama-Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf{H}$ of a bounded t-structure in a Hom-finite, Krull-Schmidt, $\mathbf{k}$-l…
Let A be a self-injective algebra over an algebraically closed field k. We show that if an A-module M of complexity one has an open orbit in the variety of d-dimensional A-modules, then M is periodic. As a corollary we see that any simple A-module of complexity one must be periodic. In the course of the proof, we also show that modules with open orbits are preserved by stable equivalences of Morita type between self-injective algebras.
Let $\mathcal{C}$ be an additive category equipped with an automorphism $Σ$. We show how to obtain $n$-angulations of $(\mathcal{C},Σ)$ using some particular periodic injective resolutions. We give necessary and sufficient conditions on $(\mathcal{C},Σ)$ admitting an $n$-angulation. Then we apply these characterizations to explain the standard construction of $n$-angulated categories and the $n$-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of $n$-angulated categories from quasi-periodic selfinjective algebras.
We 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 …
We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.
A triangulated category is said to be Calabi-Yau of dimension d if the dth power of its suspension is a Serre functor. We determine which stable categories of self-injective algebras A of finite representation type are Calabi-Yau and compute their Calabi-Yau dimensions. We achieve this by studying the minimal projective resolution of the stable Auslander algebra of A over its enveloping algebra, and use covering theory to reduce to (generalized) preprojective algebras of Dynkin graphs. We also describe how this problem can be approached by realizing the stable categories in question as orbit c…
Let $\mathbf{k}$ be an algebraically closed field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra and let $V$ be a $Λ$-module with stable endomorphism ring isomorphic to $\mathbf{k}$. If $Λ$ is self-injective, then $V$ has a universal deformation ring $R(Λ,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if $Λ$ is further a Frobenius $\mathbf{k}$-algebra, then $R(Λ,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $Λ_{m,N}$-modules whose c…
We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the Universal enveloping of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components with Euclidean tree class if p=2. Finally we deduce conditions for a smash product of a local basic algebra with a commutative semi-simple group algebra to have components with Euclidean t…
A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce …
A ring R satisfies the Generalized Auslander-Reiten Condition if any R-module M with no self-extensions in degrees higher than m must have projective dimension at most m. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type- a broad class of self-injective algebras where every module is periodic under the Nakayama automorphism.
Given a Frobenius category $\mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $\mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {\it "semi-derived Hall algebra"} $\mathcal{SDH(F, P(F))}.$ We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group $K_0(\mathcal{P(F)})$ of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category $\underline{\mathcal{F}}$. We…
For an $(n-1)$-Auslander algebra $Λ$ with global dimension $n$, we give some necessary conditions for $Λ$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $Λ$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $Λ$ with global dimension 2, we prove that $Λ$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $Λ$-module $M$, $M$ is injective is equivalent to that the reduced grade of $M$ is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence o…
