On the problem of axiomatization of tame representation type (Q2773310)

Let \(\mathcal A\) be the class of all \(d\)-dimensional algebras (associative with 1) over an algebraically closed field of fixed characteristic and let \(\mathcal T\) denote the subclass of algebras of tame representation type. The author proves that \(\mathcal T\) is an axiomatisable subclass of \(\mathcal A\). (The language includes a sort for the base field as well as a sort for the algebra.) An argument from \textit{C. Geiss} [Arch. Math. 64, No. 1, 11-16 (1995; Zbl 0828.16013)] is one key ingredient.NEWLINENEWLINENEWLINEIt is also shown that this class is finitely axiomatisable (equivalently, the class of wild algebras is axiomatisable) if and only if for every algebraically closed field \(K\) of the given characteristic the set of (structure constants of) tame \(d\)-dimensional \(K\)-algebras is Zariski-open in the variety of (structure constants of) \(d\)-dimensional algebras over \(K\). Whether or not this is in fact the case remains open.