Lattices over subhereditary orders and socle-projective modules
A classical order \(\Lambda\) is called subhereditary if there is a hereditary overorder \(\Gamma\) with Rad \(\Gamma\subset \Lambda\). The \(\Lambda\)-lattices are then closely related to the socle-projective modules over the algebra \[ {\mathcal D}=\begin{pmatrix} \Gamma /Rad \Gamma & \Gamma /Rad \Gamma \\ 0& \Lambda /Rad \Gamma \end{pmatrix}. \] In the case of a generalized Bäckström order \(\Lambda\), \({\mathcal D}\) is a hereditary algebra. This case has been treated in previous work of the author [Mitt. Math. Semin. Gießen 159 (1983; Zbl 0513.16003)]. The theorems of equivalence concerning \(\Lambda\) and \({\mathcal D}\), existence of Auslander-Reiten sequences, and Brauer-Thrall 1 1/2 are presented in a very general context. In the classical situation where \(\Lambda\) is an order in a separable algebra over a local field, the Auslander-Reiten quiver \(A(\Lambda)\) of \(\Lambda\) is described. It is shown (Theorem IV) that indecomposable \(\Gamma\)-lattices have \(\Lambda\)- projective successors in A(\(\Lambda)\). The representation-finite subhereditary orders \(\Lambda\), for which \({\mathcal S}({\mathcal D})\) (the category of f.g. \({\mathcal D}\)-modules with projective socle) has preprojective components, are characterized in terms of A(\(\Lambda)\).
representation-finite subhereditary orders
scientific article; zbMATH DE number 4087853
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