Representation-Tame Algebras Need not be Homologically Tame

Abstract

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 − fin dim fail to be bounded as traces the special biserial algebras. More precisely: For every positive integer r, we exhibit a special biserial algebra with the property that fin dim = r + 1, while Fin dim = 2r + 1.