Primary orders of finite representation type (Q1360883)

Over a complete discrete valuation domain \(R\) with quotient field \(K\), the representation-finite local orders \(\Lambda\) in a semisimple \(K\)-algebra \(A\) are classified in detail. Since \textit{Yu. A. Drozd} and \textit{V. V. Kirichenko}'s fundamental work [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 715-736 (1973; Zbl 0291.16005)], the powerful tool of Auslander-Reiten sequences has been developed. Using this concept, a new and up-to-date approach is given by the authors: Starting with a local order \(\Lambda\) which satisfies the necessary conditions for representation-finiteness, they calculate the Auslander-Reiten quiver. This yields a new sufficiency proof on the one hand, and a better insight into the relationship between the indecomposables on the other hand. As in the paper of Drozd and Kirichenko, the analysis of multipliers of the radical plays a decisive role. However, the somewhat intricate and theoretically unsatisfactory computations of Ext groups are completely avoided. As an interesting new result of this revision, the authors point out a minor inaccuracy in the formulation of Drozd-Kirichenko's theorem. Namely, there exist orders \(\Lambda\) in a \(2\times 2\) matrix algebra \(A\) over a skewfield \(D\), satisfying the necessary conditions of Drozd and Kirichenko, which nevertheless have left ideals \(I\) which cannot be generated by two (not even by three) elements. Such orders are constructed by means of twisted algebras over the residue class field with respect to some outer \(\theta\)-derivation of \(\Delta/\text{Rad }\Delta\), where \(\Delta\) denotes the unique maximal order in \(D\). In order to rectify Drozd and Kirichenko's criterion, the necessary conditions have to be completed by the property that \(\overline{\Lambda}(\text{Rad }\Lambda)\) is cyclic over \(\overline\Lambda\), the intersection of all maximal overorders of \(\Lambda\).