Geometry of modules over tame quasi-tilted algebras
Graded modules and their completions
Higher-dimensional Auslander-Reiten sequences
Zhou and Zhu have shown that if is an -angulated category and is a cluster tilting subcategory of , then the quotient category is an -abelian category. We show that if has Auslander-Reiten -angles, then has Auslander-Reiten -exact sequences.
Hochschild cohomology of generalized multicoil algebras
We determine the Hochschild cohomology of all finite-dimensional generalized multicoil algebras over an algebraically closed field, which are the algebras for which the Auslander-Reiten quiver admits a separating family of almost cyclic coherent components. In particular, the analytically rigid generalized multicoil algebras are described.
Homological domino effects and the first Finitistic Dimension Conjecture.
Immersions of module varieties
We show that a homomorphism of algebras is a categorical epimorphism if and only if all induced morphisms of the associated module varieties are immersions. This enables us to classify all minimal singularities in the subvarieties of modules from homogeneous standard tubes.
Indecomposable modules in coils
We describe the structure of all indecomposable modules in standard coils of the Auslander-Reiten quivers of finite-dimensional algebras over an algebraically closed field. We prove that the supports of such modules are obtained from algebras with sincere standard stable tubes by adding braids of two linear quivers. As an application we obtain a complete classification of non-directing indecomposable modules over all strongly simply connected algebras of polynomial growth.
Indecomposable Modules over Multicoil Algebras.
Indecomposable representations for extended Dynkin quivers of type 𝔼̃₈
We discuss the problem of classification of indecomposable representations for extended Dynkin quivers of type 𝔼̃₈, with a fixed orientation. We describe a method for an explicit determination of all indecomposable preprojective and preinjective representations for those quivers over an arbitrary field and for all indecomposable representations in case the field is algebraically closed. This method uses tilting theory and results about indecomposable modules for a canonical algebra of type (5,3,2)...
Indecomposable representations of orders
Infinitesimal unipotent group schemes of complexity 1
We classify the uniserial infinitesimal unipotent commutative groups of finite representation type over algebraically closed fields. As an application we provide detailed information on the structure of those infinitesimal groups whose distribution algebras have a representation-finite principal block.
Irreducible morphisms, the Gabriel-valued quiver and colocalizations for coalgebras.
Iterated coil enlargements of algebras
Let Λ be a finite-dimensional, basic and connected algebra over an algebraically closed field, and mod Λ be the category of finitely generated right Λ-modules. We say that Λ has acceptable projectives if the indecomposable projective Λ-modules lie either in a preprojective component without injective modules or in a standard coil, and the standard coils containing projectives are ordered. We prove that for such an algebra Λ the following conditions are equivalent: (a) Λ is tame, (b) the Tits form...
Iterated tilted and tilted stably hereditary algebras
We prove that a stably hereditary bound quiver algebra A = KQ/I is iterated tilted if and only if (Q,I) satisfies the clock condition, and that in this case it is of type~Q. Furthermore, A is tilted if and only if (Q,I) does not contain any double-zero.
Jordan types for indecomposable modules of finite group schemes
In this article we study the interplay between algebro-geometric notions related to -points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that -points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on -points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial modules.
Ladder functors with an application to representation-finite Artinian rings.
Laura algebras and quasi-directed components
Using a notion of distance between indecomposable modules we deduce new characterizations of laura algebras and quasi-directed Auslander-Reiten components. Afterwards, we investigate the infinite radical of Artin algebras and show that there exist infinitely many non-directing modules between two indecomposable modules X and Y if . We draw as inference that a convex component is quasi-directed if and only if it is almost directed.
Left sections and the left part of an artin algebra
We define a notion of left section in an Auslander-Reiten component, by weakening one of the axioms for sections. We derive a generalisation of the Liu-Skowroński criterion for tilted algebras, then apply our results to describe the Auslander-Reiten components lying in the left part of an artin algebra.
Left-right projective bimodules and stable equivalences of Morita type
We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for self-injective algebras are also given.
Matrix factorizations for domestic triangle singularities
Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities of domestic type, that is, we assume that (a,b,c) are integers at least two satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a,b,c). Equivalently, in a representation-theoretic context,...