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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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References:
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