Algebra Seminar

Given a finite dimensional algebra A over a field, Auslander defined in 1971 the representation dimension of A to be the minimum of the global dimensions of the endomorphism algebras of generator-cogenerators of the A-module category. It was shown that an algebra is of representation-finite type if and only if the representation dimension is at most 2. However, it is still open whether the representation dimension of an algebra is finite or not. In this talk, we shall show that if there is a stable equivalence of Morita-type between two algebras A and B, then they have the same representation dimension. So representation dimension is an invariant of stable equivalence of Morita-type (but note that it is not an invariant of derived equivalences).