Related papers: The Representation Dimension of a Selfinjective Al…
We prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three, and give an explicit construction of the Auslander generator of its module category.
Let $\Lambda$ be an artin algebra with representation dimension equal to three and $M$ an Auslander generator of $\Lambda$. We show how, under certain assumptions, we can mutate $M$ to get a new Auslander generator whose endomorphism ring…
The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the…
We are going to show that the representation dimension of a cluster-concealed algebra B is 3. We compute its representation dimension by showing an explicit Auslander generator for the cluster-tilted algebra.
For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of…
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…
In this paper, we initiate the study of higher-dimensional Auslander-Reiten theory of self-injective algebras. We give a systematic construction of (weakly) $d$-representation-finite self-injective algebras as orbit algebras of the…
Let $A$ be a selfinjective algebra over an algebraically closed field. We study the stable dimension of $A$, which is the dimension of the stable module category of $A$ in the sense of Rouquier. Then we prove that $A$ is…
Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $T_2(A)=(\begin{array}{cc}A&0 A&A\end{array})$ be the triangular matrix algebra and $A^{(1)}=(\begin{array}{cc}A&0 DA&A\end{array})$ be the…
We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least…
We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global…
We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a…
Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field and $A^{(m)}$ the $m$-replicated algebra of $A$. We prove that the representation dimension of $A^{(m)}$ is at most three, and that the dominant dimension…
The main goal of this paper is to study the class of algebras for which the global dimension of the endomorphism ring of the generator-cogenerator, given by the sum of the projective and injective modules, is equal to three. We will refer…
We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of the endomorphism algebra of M, then the representation dimension of the corresponding triangular matrix…
We show that every higher Auslander algebra $A_{n+1}^d$ of type $\mathbb{A}$ such that $\gcd(n,d)=1$ is derived equivalent to a certain replicated algebra $B=B_0^{(n+d)}$. Moreover ${\rm{gldim}} B = nd$ and $B$ admits an $nd$-cluster…
Let $\Lambda$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\Lambda$ is representation-finite if and only if the number of non-isomorphic indecomposable $\Lambda$-modules with projective dimension…
We prove that the representation dimension of finite dimensional selfinjective algebras over a field is invariant under socle equivalence and derive some consequences.
Let $\Lambda$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\mod\Lambda$, then for any indecomposable module $M \in \mod \Lambda$, we have that…
Let $A$ be a finite dimensional $G$-graded algebra with $G$ a finite group, and $A\# k[G]^{\ast}$ be the smash product of $A$ with the group $G$. Our results can be stated as follows: (1) If $A$ is a self-injective algebra and separably…