56 research outputs found
Tilted algebras and short chains of modules
We provide an affirmative answer for the question raised almost twenty years
ago concerning the characterization of tilted artin algebras by the existence
of a sincere finitely generated module which is not the middle of a short
chain
On Auslander-Reiten components of algebras without external short paths
We describe the structure of semi-regular Auslander-Reiten components of
artin algebras without external short paths in the module category. As an
application we give a complete description of self-injective artin algebras
whose Auslander-Reiten quiver admits a regular acyclic component without
external short paths
Tilting and cotilting modules over concealed canonical algebras
We study infinite dimensional tilting modules over a concealed canonical
algebra of domestic or tubular type. In the domestic case, such tilting modules
are constructed by using the technique of universal localization, and they can
be interpreted in terms of Gabriel localizations of the corresponding category
of quasi-coherent sheaves over a noncommutative curve of genus zero. In the
tubular case, we have to distinguish between tilting modules of rational and
irrational slope. For rational slope the situation is analogous to the domestic
case. In contrast, for any irrational slope, there is just one tilting module
of that slope up to equivalence. We also provide a dual description of infinite
dimensional cotilting modules and a classification result for the
indecomposable pure-injective modules.Comment: 25 page
Moduli stacks of Serre stable representations in tilting theory
We introduce a new moduli stack, called the Serre stable moduli stack, which
corresponds to studying families of point objects in an abelian category with a
Serre functor. This allows us in particular, to re-interpret the classical
derived equivalence between most concealed-canonical algebras and weighted
projective lines by showing they are induced by the universal sheaf on the
Serre stable moduli stack. We explain why the method works by showing that the
Serre stable moduli stack is the tautological moduli problem that allows one to
recover certain nice stacks such as weighted projective lines from their moduli
of sheaves. As a result, this new stack should be of interest in both
representation theory and algebraic geometry
Semi-invariants for concealed-canonical algebras
In the paper is we generalize known descriptions of rings of semi-invariants
for regular modules over Euclidean and canonical algebras to arbitrary
concealed-canonical algebras
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