Patrick Le Meur

The universal cover of an algebra without double bypass

Let A be a basic and connected finite dimensional algebra over a field k of characteristic zero. We show that if the quiver of A has no double bypass then the fundamental group (as defined in [R. Martinez-Villa, J.A. de la Pena, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983) 277–292]) of any presentation of A by quiver and relations is the quotient of the fundamental group of a privileged presentation of A. Then we show that the Galois covering of A associated with this privileged presentation satisfies a universal property with respect to the connected Galois coverings of A in a similar fashion to the universal cover of a topological space.

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

Topological invariants of piecewise hereditary algebras

n

Coverings of laura algebras: The standard case

irreducible morphisms between indecomposable modules lies in the

Special biserial algebras with no outer derivations

n+1

Galois Coverings of Weakly Shod Algebras

-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.

On Galois coverings and tilting modules

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.

The strong global dimension of piecewise hereditary algebras

Abstract In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, has a standard connecting component), then it has Galois coverings associated to the coverings of the connecting component. As a consequence, the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings.

Covering techniques in Auslander–Reiten theory

Let

On Maximal Diagonalizable Lie Subalgebras of the First Hochschild Cohomology

A