Algebras with small homological dimensions

We consider the algebras Λ which satisfy the property that for each indecomposable module X, either its projective dimension pdΛX is at most one or its injective dimension idΛX is at most one. This clearly generalizes the so-called quasitilted algebras introduced by Happel–Reiten–Smalø. We show that some of the niciest features for this latter class of algebras can be generalized to the case we are considering, in particular the existence of a trisection in its module category.