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Representations of finite partially ordered sets over commutative Artinian uniserial rings. (English) Zbl 1106.16016
Let \((S,\leq)\) be a finite partially ordered set (poset) with unique maximal element \(\infty\) and \(R\) a commutative ring. One can define the category \(\text{rep}_{\text{fg}}(S,R)\) of representations of \((S,\leq)\) over \(R\) with objects \(U=(U_s:s\in S)\), where each \(U_s\) is a finitely generated \(R\)-module and \(U_s\subseteq U_t\) if \(s\leq t\) in \(S\). A morphism from \(U\) to \(V\) is an \(R\)-module homomorphism \(f\colon U_\infty\to V_\infty\) with \(f(U_s)\subseteq V_s\) for each \(s\in S\).
The authors study the category \(\text{rep}_{\text{fg}}(S,R)\) for the case that \(R\) is a commutative Artinian uniserial ring and \(S\) is a finite poset with unique maximal element. By a series of functorial reductions and combinatorial analysis, the representation type of the category \(\text{rep}_{\text{fg}}(S,R)\) is characterized in terms of \(S\) and the index of nilpotency of the Jacobson radical \(J(R)\) of \(R\). As an application, the authors obtain a complete characterization of the representation type of a category of representations arising from pairs of finite rank completely decomposable Abelian groups.

MSC:
16G20 Representations of quivers and partially ordered sets
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
20K15 Torsion-free groups, finite rank
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