CONICET | Buscador de Institutos y Recursos Humanos

The class of special biserial algebras is an important class of tame algebras among the family of associative algebras of finite dimension over an algebraically closed eld. There are several families of examples: blocks of group algebras with cyclic or dihedral defect group, or algebras with top and socle without multiplicity. They have been rst studied by Gelfand and Ponomarev in order to describe their representation theory. These algebras are tame and their modules can be of non-polynomial growth.Butler and Ringel studied their Auslander-Reiten quiver.A k-algebra is said to be special biserial if it is Morita equivalent to a k-algebra kQ/I with (Q;I ) special biserial. In these cases, kQ/I is finite dimensional if and only if Q has a finite number of vertices.There exist several examples of special biserial algebras whose Hochschild cohomology is known, but the problem of computing these invariants is dicult. The particular case of monomial algebras is easier to handle, since Bardzell gave a resolution of a monomial algebra as bimodule over itself which is well adapted to computations.Using Bergman's Diamond Lemma and Grobner bases we construct inductively from Bardzell's resolution for monomial algebras a resolution for more general algebras and use it for several families of special biserial algebras, with special interest on Brauer tree algebras.We also present several applications of this resolution to algebras of dierent kind.