On Bernstein-Gelfand-Gelfand equivalence and tilting theory
On Cohen-Macaulay modules over non-commutative surface singularities
We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.
On components of the Auslander-Reiten quiver of trivial extensions of 2-fundamental algebras which contain projective modules
Trivial extensions of a certain subclass of minimal 2-fundamental algebras are examined. For such algebras the characterization of components of the Auslander-Reiten quiver which contain indecomposable projective modules is given.
On corepresentations of equipped posets and their differentiation.
On cyclic vertices in valued translation quivers
Let x and y be two vertices lying on an oriented cycle in a connected valued translation quiver (Γ,τ,δ). We prove that, under certain conditions, x and y belong to the same cyclic component of (Γ,τ,δ) if and only if there is an oriented cycle in (Γ,τ,δ) passing through x and y.
On derived equivalence classification of gentle two-cycle algebras
We classify, up to derived (equivalently, tilting-cotilting) equivalence, all nondegenerate gentle two-cycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case.
On domestic algebras of semiregular type
We describe the structure of finite-dimensional algebras of domestic representation type over an algebraically closed field whose Auslander-Reiten quiver consists of generalized standard and semiregular components. Moreover, we prove that this class of algebras contains all special biserial algebras whose Auslander-Reiten quiver consists of semiregular components.
On faithful projective representations of finite abelian p-groups over a field of characteristic p
Let G be a noncyclic abelian p-group and K be an infinite field of finite characteristic p. For every 2-cocycle λ ∈ Z²(G,K*) such that the twisted group algebra is of infinite representation type, we find natural numbers d for which G has infinitely many faithful absolutely indecomposable λ-representations over K of dimension d.
On fuzzification of the notion of quantaloid
The paper considers a fuzzification of the notion of quantaloid of K. I. Rosenthal, which replaces enrichment in the category of -semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At...
On Generalized Uniserial Rings and Decompositions that Complement Direct Summands.
On hereditary artinian rings and the pure semisimplicity conjecture: rigid tilting modules and a weak conjecture
A weak form of the pure semisimplicity conjecture is introduced and characterized through properties of matrices over division rings. The step from this weak conjecture to the full pure semisimplicity conjecture would be covered by proving that there do not exist counterexamples to the conjecture in a particular class of rings, which is also studied.
On hereditary rings and the pure semisimplicity conjecture II: Sporadic potential counterexamples
It was shown in [Colloq. Math. 135 (2014), 227-262] that the pure semisimplicity conjecture (briefly, pssC) can be split into two parts: first, a weak pssC that can be seen as a purely linear algebra condition, related to an embedding of division rings and properties of matrices over those rings; the second part is the assertion that the class of left pure semisimple sporadic rings (ibid.) is empty. In the present article, we characterize the class of left pure semisimple sporadic rings having finitely...
On Hom-spaces of tame algebras
Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules.
On indecomposable projective representations of finite groups over fields of characteristic p > 0
Let G be a finite group, F a field of characteristic p with p||G|, and the twisted group algebra of the group G and the field F with a 2-cocycle λ ∈ Z²(G,F*). We give necessary and sufficient conditions for to be of finite representation type. We also introduce the concept of projective F-representation type for the group G (finite, infinite, mixed) and we exhibit finite groups of each type.
On indecomposable representations of quivers with zero-relations
On induced modules over strongly group-graded algebras.
On intecomposable Lattices over Twisted group rings
On large selforthogonal modules
We construct non faithful direct summands of tilting (resp. cotilting) modules large enough to inherit a functorial tilting (resp. cotilting) behaviour.
On locally bounded categories stably equivalent to the repetitive algebras of tubular algebras
On minimal non-tilted algebras
A minimal non-tilted triangular algebra such that any proper semiconvex subcategory is tilted is called a tilt-semicritical algebra. We study the tilt-semicritical algebras which are quasitilted or one-point extensions of tilted algebras of tame hereditary type. We establish inductive procedures to decide whether or not a given strongly simply connected algebra is tilted.