Representation-tame algebras need not be homologically tame

Abstract

We show that, also within the class of representation-tame finite dimensional algebras $\Lambda$, the big left finitistic dimension of $\Lambda$ may be strictly larger than the little. In fact, the discrepancies $Findim \Lambda - findim \Lambda$ need not even be bounded for special biserial algebras which constitute one of the (otherwise) most thoroughly understood classes of tame algebras. More precisely: For every positive integer $r$, we construct a special biserial algebra $\Lambda$ with the property that $findim \Lambda = r + 1$, while $Findim \Lambda = 2r + 1$. In particular, there are infinite dimensional representations of $\Lambda$ which have finite projective dimension, while not being direct limits of {\it finitely generated\/} representations of finite projective dimension.