On the existence of modules which are neither preprojectives nor preinjectives (Q1335077)

Let \(A\) be a finite dimensional algebra over a field, \(\text{mod }A\) the category of finitely generated left \(A\)-modules, and \(\text{ind }A\) a full subcategory of \(\text{mod }A\) of chosen representatives of isoclasses of indecomposable \(A\)-modules. [In J. Algebra 66, 61-122 (1980; Zbl 0477.16013)] \textit{M. Auslander} and \textit{S. O. Smalø} have shown that \(\text{ind }A\) admits a unique partition \({\mathcal P}_ 0, {\mathcal P}_ 1,\dots,{\mathcal P}_ \infty\) of \(\text{ind }A\), called preprojective partition, such that \(\text{ind }A = \bigcup_{i \leq \infty} {\mathcal P}_ i\), \({\mathcal P}_ i \cap {\mathcal P}_ j = \emptyset\) if \(i \neq j\), and for each \(i < \infty\), \({\mathcal P}_ i\) is a finite minimal cover of \(\bigcup_{j \geq i} {\mathcal P}_ j\). The modules in \(\bigcup_{i < \infty} {\mathcal P}_ i\) are called preprojective. Dually, one defines a preinjective partition of \(\text{ind }A\) and preinjective modules. More than 16 years ago C. M. Ringel raised the question whether \(A\) is of finite representation type if any indecomposable module from \(\text{ind }A\) is preprojective or preinjective. It is shown in the paper that this is the case for finite dimensional algebras over infinite perfect fields. In the proof the validity of the second Brauer-Thrall conjecture for such algebras is used. Recently, the above problem has been solved in the affirmative for arbitrary (even Artin) algebras by the reviewer and \textit{S. O. Smalø} [in Arch. Math. 64, 8-10 (1995; Zbl 0813.16007)].