zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] D. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, CMS Books in Mathematics, Springer, New York, 2000. · Zbl 0959.16011
[2] D. Arnold, D. Simson, Representations of finite posets over discrete valuation rings, 2004, preprint. · Zbl 1142.16003
[3] Arnold, D.; Simson, D., Endo-wild representation type and generic representations of finite posets, Pacific J. math., 219, 101-126, (2005) · Zbl 1108.16010
[4] Atiyah, M.F.; MacDonald, I.G., Introduction to commutative algebra, (1969), Addison-Wesley Reading, MA · Zbl 0175.03601
[5] M. Auslander, I. Reiten, S.O. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. · Zbl 0834.16001
[6] Auslander, M.; Smalø, S.O., Almost split sequences in subcategories, J. algebra, 69, 426-454, (1981) · Zbl 0457.16017
[7] Butler, M.C.R., Torsion-free modules and diagrams of vector spaces, Proc. London math. soc., 18, 635-652, (1968) · Zbl 0179.32603
[8] Dugas, M.; Rangaswamy, K.M., Completely decomposable abelian groups with a distinguished subgroup, Rocky mountain J. math., 32, 1383-1395, (2002) · Zbl 1035.20042
[9] P. Gabriel, A.V. Roiter, Representations of Finite Dimensional Algebras, Algebra VIII, Encyclopaedia of Mathematical Science, vol. 73, Springer, Berlin, 1992. · Zbl 0839.16001
[10] Kasjan, S.; Simson, D., Fully wild prinjective type of posets and their quadratic forms, J. algebra, 172, 506-529, (1995) · Zbl 0831.16010
[11] Kleiner, M.M., Partially ordered sets of finite type, Zap. nauchn. sem. leningrad. otdel. mat. inst. steklova (LOMI), 28, 32-41, (1972) · Zbl 0345.06001
[12] Lang, S., Algebra, (1993), Addison-Wesley Reading, MA · Zbl 0848.13001
[13] Nazarova, L.A., Partially ordered sets of infinite type, Izv. akad. nauk SSSR, 39, 963-991, (1975), (in Russian) · Zbl 0362.06002
[14] L.A. Nazarova, A.G. Zavadskij, Partially ordered sets of tame type, in: Matrix Problems, Acad. Nauk Ukr. S.S.R., Inst. Mat., Kiev, 1977, pp. 122-143 (in Russian). · Zbl 0461.16024
[15] de la Peña, J.A.; Simson, D., Prinjective modules, reflection functors, quadratic forms and auslander – reiten sequences, Trans. amer. math. soc., 329, 733-753, (1992) · Zbl 0789.16010
[16] Plahotnik, V.V., Representations of partially ordered sets over commutative rings, Izv. akad. nauk SSSR, 40, 527-543, (1976), (in Russian)
[17] Richman, F., Isomorphism of butler groups at a prime \(p\), abelian group theory, Contemp. math., 171, 333-337, (1995) · Zbl 0857.20035
[18] Richman, F.; Walker, E., Subgroups of \(p^5\) bounded groups, (), 55-74 · Zbl 0953.20045
[19] C. Ringel, M. Schmidmeier, Submodule categories of wild representation type, J. Pure Appl. Algebra, in press, doi: 10.1016/j.jpaa.2005.07.002. · Zbl 1147.16019
[20] C. Ringel, M. Schmidmeier, The Auslander-Reiten translation in submodule categories, 2005, preprint. · Zbl 1154.16011
[21] M. Schmidmeier, Bounded submodules of modules, J. Pure Appl. Algebra, in press, doi: 10.1016/j.jpaa.2005.02.003. · Zbl 1128.16012
[22] Simson, D., Linear representations of partially ordered sets and vector space categories, (1992), Gordon and Breach Switzerland, Australia · Zbl 0818.16009
[23] D. Simson, Socle projective representations of partially ordered sets and Tits quadratic forms with applications to lattices over orders, Abelian Groups and Modules, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1996, pp. 73-112. · Zbl 0869.16008
[24] Simson, D., Chain categories of modules and subprojective representations of posets over uniserial algebras, Rocky mountain J. math., 32, 1627-1650, (2002) · Zbl 1048.16006
[25] D. Simson, Representation type of the category of subprojective representations of a finite poset over \(K [t] /(t^m)\), 2003, preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.