Wild algebras have one-point extensions of representation dimension at least four

We show that any wild algebra has a one-point extension of representation dimension at least four, and more generally that it has an n-point extension of representation dimension at least n+3. We give two explicit constructions, and obtain new examples of small algebras of representation dimension four.     

Graded Calabi Yau algebras of dimension 3

In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.     

Representation dimension and tilting

Let Λ be a finite dimensional k-algebra over an algebraically closed field k and let _ΛT be a splitting tilting module of projective dimension at most 1. Let Γ=End_ΛT. If the representation dimension of Λ is at most 3 then the main result asserts that the representation dimension of Γ does not exceed that of Λ.     

On module variety dimension for coverings and degenerations of algebras

Given a pair of G-covering functors F¹:R→R₁ and F⁰:R→R₀ such that F⁰ is a Galois covering, the inequality for all z,t, of the dimensions of the first kind module sets under some assumptions is proved (Theorem 2.2). The result is applied to show the equality of the module variety dimensions for some special degenerations of algebras. Certain consequences for preserving wild and tame representation types by G-covering functors are also presented (Theorems 2.4 and 3.1).     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

Tensor products of n-complete algebras

If A and B are n- and m-representation finite k-algebras, then their tensor product $\mathrm{\Lambda }=A{?}_{k}B$ is not in general $(n+m)$-representation finite. However, we prove that if A and B are acyclic and satisfy the weaker assumption of n- and m-completeness, then Λ is $(n+m)$-complete. This mirrors the fact that taking higher Auslander algebra does not preserve d-representation finiteness in general, but it does preserve d-completeness. As a corollary, we get the necessary condition for Λ to be $(n+m)$-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi–Yau property.     

Permanents,Doty coalgebras and dominant dimension of Schur algebras

A (hidden) multiplication on _AZ(n,r)$A_{\mathbb{Z}}(n,r)$, the Z$\mathbb{Z}$-dual of the integral Schur algebra _SZ(n,r)$S_{\mathbb{Z}}(n,r)$ is explicitly constructed, possibly without a unit. The image of the multiplication map is shown to be spanned by bipermanents. Let k   be any field of characteristic p>0$p>0$. The image of the induced multiplication on _Ak(n,r)=_AZ(n,r)_⊗Zk$A_k(n,r)=A_{\mathbb{Z}}(n,r){\otimes }_{\mathbb{Z}}k$ turns out to coincide with the Doty coalgebra _Dn,r,p$D_{n,r,p}$ of truncated symmetric powers. Combined with a new straightening formula for bipermanents, it is proved that such a multiplication induces an isomorphism _Ak(n,r)_⊗Sk(n,r)Ak(n,r)≅_Ak(n,r)$A_k(n,r){\otimes }_{S_k(n,r)}{A}_k(n,r)\cong A_k(n,r)$ as _Sk(n,r)$S_k(n,r)$-bimodules if and only if r≤n(p−1)$r\le n(p-1)$, if and only if _Dn,r,p=_Ak(n,r)$D_{n,r,p}=A_k(n,r)$. As a result, _Sk(n,r)$S_k(n,r)$ is a gendo-symmetric algebra, and its dominant dimension is at least two and admits a combinatorial characterization as long as r≤n(p−1)$r\le n(p-1)$.     

A homological characterization of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite. Dedicated to Otto Kerner on the occasion of his 65th birthday. The second author is supported by a grant from NSA. These results were obtained during a visit of the first author at Syracuse University. He would like to thank the second author for the hospitality during his stay.     

Exact categories,big Cohen-Macaulay modules and finite representation type

One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterisation of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce “big objects” and prove an Auslander-type “splitting-big-objects” theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Some characterisations of supported algebras

We give several equivalent characterisations of left (and hence, by duality, also of right) supported algebras. These characterisations are in terms of properties of the left and the right parts of the module category, or in terms of the classes L₀ and R₀ which consist respectively of the predecessors of the projective modules, and of the successors of the injective modules.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.     

Vanishing of self-extensions over symmetric algebras

We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type Z_A∞$\mathbb{Z}{A}_{\infty }$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander–Reiten Conjecture and the Extension Conjecture.     

Selfinjective algebras without short cycles of indecomposable modules

We describe the structure of finite dimensional selfinjective algebras over an arbitrary field without short cycles of indecomposable modules.