Representation-tame algebras need not be homologically tame

TL;DR

This paper demonstrates that within representation-tame finite dimensional algebras, the big and little finitistic dimensions can differ significantly, even unboundedly, especially in special biserial algebras.

Contribution

It constructs explicit examples of special biserial algebras where the big and little finitistic dimensions differ arbitrarily, revealing new complexities in tame algebra representation theory.

Findings

The big finitistic dimension can be arbitrarily larger than the little.

Explicit constructions of special biserial algebras with unbounded dimension discrepancies.

Existence of infinite dimensional representations with finite projective dimension not approximable by finitely generated ones.

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.