122 research outputs found
Topological invariants of piecewise hereditary algebras
We investigate the Galois coverings of piecewise algebras and more
particularly their behaviour under derived equivalences. Under a technical
assumption which is satisfied if the algebra is derived equivalent to a
hereditary algebra, we prove that there exists a universal Galois covering
whose group of automorphisms is free and depends only on the derived category
of the algebra. As a corollary, we prove that the algebra is simply connected
if and only if its first Hochschild cohomology vanishes.Comment: The hypotheses of the main theorem were modified: The next now deals
mainly with piecewise hereditary algebras which are derived equivalent to a
hereditary algebra (instead of all piecewise hereditary algebras in the
previous version
On the Morita Reduced Versions of Skew Group Algebras of Path Algebras
Let R be the skew group algebra of a finite group acting on the path algebra
of a quiver. This article develops both theoretical and practical methods to do
computations in the Morita reduced algebra associated to R. Reiten and
Riedtmann proved that there exists an idempotent e of R such that the algebra
eRe is both Morita equivalent to R and isomorphic to the path algebra of some
quiver which was described by Demonet. This article gives explicit formulas for
the decomposition of any element of eRe as a linear combination of paths in the
quiver described by Demonet. This is done by expressing appropriate
compositions and pairings in a suitable monoidal category which takes into
account the representation theory of the finite group
Smash Products of Calabi-Yau Algebras by Hopf Algebras
Let H be a Hopf algebra and A be an H-module algebra. This article
investigates when the smash product A#H is (skew) Calabi-Yau, has Van den Bergh
duality or is Artin-Schelter regular or Gorenstein. In particular, if A and H
are skew Calabi-Yau, then so is A#H and its Nakayama automorphism is expressed
using the ones of A and H. This is based on a description of the inverse
dualising complex of A#H when A is a homologically smooth dg algebra and H is
homologically smooth and with invertible antipode. This description is also
used to explain the compatibility of standard constructions of Calabi-Yau dg
algebras with taking smash products.Comment: Minor corrections and changes to reflect the published articl
On the Morita Reduced Versions of Skew Group Algebras of Path Algebras
Let R be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita reduced algebra associated to R. Reiten and Riedtmann proved that there exists an idempotent e of R such that the algebra eRe is both Morita equivalent to R and isomorphic to the path algebra of some quiver which was described by Demonet. This article gives explicit formulas for the decomposition of any element of eRe as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category which takes into account the representation theory of the finite group
Degrees of irreducible morphisms and finite-representation type
We study the degree of irreducible morphisms in any Auslander-Reiten
component of a finite dimensional algebra over an algebraically closed field.
We give a characterization for an irreducible morphism to have finite left (or
right) degree. This is used to prove our main theorem: An algebra is of finite
representation type if and only if for every indecomposable projective the
inclusion of the radical in the projective has finite right degree, which is
equivalent to require that for every indecomposable injective the epimorphism
from the injective to its quotient by its socle has finite left degree. We also
apply the techniques that we develop: We study when the non-zero composite of a
path of irreducible morphisms between indecomposable modules lies in the
-th power of the radical; and we study the same problem for sums of such
paths when they are sectional, thus proving a generalisation of a pioneer
result of Igusa and Todorov on the composite of a sectional path.Comment: 20 page
Covering techniques in Auslander-Reiten theory
Given a finite dimensional algebra over a perfect field the text introduces
covering functors over the mesh category of any modulated Auslander-Reiten
component of the algebra. This is applied to study the composition of
irreducible morphisms between indecomposable modules in relation with the
powers of the radical of the module category.Comment: Minor modifications. Final version to appear in the Journal of Pure
and Applied Algebr
The strong global dimension of piecewise hereditary algebras
Let T be a tilting object in a triangulated category equivalent to the
bounded derived category of a hereditary abelian category with finite
dimensional homomorphism spaces and split idempotents. This text investigates
the strong global dimension, in the sense of Ringel, of the endomorphism
algebra of T. This invariant is expressed using the infimum of the lengths of
the sequences of tilting objects successively related by tilting mutations and
where the last term is T and the endomorphism algebra of the first term is
quasi-tilted. It is also expressed in terms of the hereditary abelian
generating subcategories of the triangulated category.Comment: Final published version. After refereeing, historical considerations
were added and the length of the article was reduced: Introduction and
Section 1 were reformulated; Subsection 2.1 was moved to Section 1 (with an
abridged proof); Subsection 3.2 was reformulated (with an abridged proof);
The proof in A.5 was rewritten (now shorter); And minor rewording was
processed throughout the articl
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