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Representations of finite posets over discrete valuation rings. (English) Zbl 1142.16003
The representations of finite posets over some particular rings play an important role, exhibited in D. M. Arnold’s book [Abelian groups and representations of finite partially ordered sets. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. 2. New York: Springer (2000; Zbl 0959.16011)], in the study of finite rank torsion-fre groups, in particular for finite rank Butler groups.
In the present paper representations of finite posets over discrete valuation rings (i.e. local principal ideal domains) are considered. It is proved in Theorem 1.1 that if \(S\) is a poset and \(R\) is a discrete valuation ring with \(k_R\) its residue field then there exists an additive functor \(\widetilde G:\mathbf{fpr}(S^*,R)\to\mathbf{fpr}(S^*,k_R[t])\) which is exact, dense and induces a 1-1 correspondence between isomorphism classes of indecomposable type. Here \(S^*\) is the poset obtained from \(S\) by adjoining a biggest element \(*\), and \(\mathbf{fpr}(S^*,R)\) is the category of filtered \(R\)-projective representations of \(S^*\). This functor is constructed using a functor \(G\) from the category of finitely generated free \(R\)-modules into the category of finitely generated free \(k_R[t]\)-modules defined on objects by \(G(M)=\bigoplus_{i\in\mathbb{N}}p^iM/p^{i+1}M\) and using some natural restrictions for the definition of \(G\) on homomorphisms. This functor induces in a natural way the functor \(\widetilde G\) (see Section 2).
Another important result of the paper is Theorem 1.2. In this theorem the authors give a complete characterization for the classification (as “of finite representation type” or as “wild representation type”) of some full subcategories of \(\mathbf{fpr}(S^*,R)\). These subcategories are important since \(\mathbf{fpr}(S^*,R)\) can be refined into a chain of these full subcategories.

MSC:
16G20 Representations of quivers and partially ordered sets
20K15 Torsion-free groups, finite rank
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16D90 Module categories in associative algebras
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