Representation-Tame Algebras Need not be Homologically Tame

We show that even within the class of special biserial algebras, one of the most thoroughly studied classes of representation-tame finite dimensional algebras, the (left) big finitistic dimension may be strictly larger than the little. In fact, we find that the discrepancies Fin dim Λ – fin dim Λ fail to be bounded as Λ traces the special biserial algebras. More precisely: For every positive integer r, we exhibit a special biserial algebra Λ with the property that fin dim Λ = r+1, while Fin dim Λ=2r+1.