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Dualities for self-small groups

Authors: Simion Breaz and Phill Schultz
Journal: Proc. Amer. Math. Soc. 140 (2012), 69-82
MSC (2010): Primary 20K21, 20K30, 20K40
Published electronically: May 12, 2011
MathSciNet review: 2833518
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a family of dualities on some subcategories of the quasi-category $ \mathcal{S}$ of self-small groups of finite torsion-free rank which cover the class $ \mathcal{S}$. These dualities extend several of those in the literature. As an application, we show that a group $ A\in\mathcal{S}$ is determined up to quasi-isomorphism by the $ \mathbb{Q}$-algebras $ \{\mathbb{Q}\operatorname{Hom}(C,A): C\in\mathcal{S}\}$ and $ \{\mathbb{Q}\operatorname{Hom}(A,C): C\in\mathcal{S}\}$. We also generalize Butler's Theorem to self-small mixed groups of finite torsion-free rank.

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  • [AB09] U. Albrecht and S. Breaz: Quasi-isomorphisms and groups of quasi-homomorphisms, Journal of Algebra and its Applications, 8 (2009), 617-627. MR 2581988 (2011a:20136)
  • [AB10] U. Albrecht and S. Breaz: A note on mixed $ A$-reflexive groups, J. Algebra, 323 (2010), 509-516. MR 2564852 (2010k:20095)
  • [ABW09] U. Albrecht, S. Breaz, and W. Wickless: Self-small abelian groups, Bull. Aust. Math. Soc., 80, No. 2 (2009), 205-216. MR 2540354 (2010m:20083)
  • [AGW95] U. Albrecht, P. Goeters and W. Wickless: The flat dimension of Abelian groups as $ E$-modules, Rocky Mount. J. of Math., 25(2) (1995), 569-590. MR 1336551 (96f:20086)
  • [AM75] D. M. Arnold and C. E. Murley: Abelian groups, $ A$, such that $ \mathrm{Hom}(A,-)$ preserves direct sums of copies of $ A$, Pacific J. Math., 56 (1975), 7-21. MR 0376901 (51:13076)
  • [Ar82] D. M. Arnold: Finite Rank Torsion Free Abelian Groups and Rings, Lect. Notes in Math., 931, Springer-Verlag, (1982). MR 665251 (84d:20002)
  • [Ar721] D. M. Arnold: A duality for torsion-free modules of finite rank over a discrete valuation ring, Proc. London Math. Soc., Third Series, Vol. XXIV (1972), 204-216. MR 0292813 (45:1895)
  • [Ar722] D. M. Arnold: A duality for quotient divisible abelian groups of finite rank, Pac. J. Math., 42 (1972), 11-15. MR 0311799 (47:361)
  • [Aus82] M. Auslander: Representation theory of finite dimensional algebras, Contemp. Math., 13, Amer. Math. Soc. (1982), 27-39. MR 685936 (84b:16031)
  • [AW04] U. Albrecht and W. Wickless: Finitely generated and cogenerated QD groups. Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Appl. Math., 236, Dekker, New York, 13-26 (2004). MR 2050698 (2004m:20103)
  • [Bo89] K. Bongartz: A generalization of a theorem of M. Auslander, Bull. London Math. Soc., 21, 3 (1989), 255-256. MR 986367 (90b:16031)
  • [BP61] R. Beaumont and R. Pierce: Torsion-free rings, Ill. J. Math., 5 (1961), 61-98. MR 0148706 (26:6212)
  • [Br10] S. Breaz: Warfield dualities induced by self-small mixed groups, J. of Group Theory, 13 (2010), 391-409. MR 2653527
  • [Br04] S. Breaz: Quasi-decompositions for self-small mixed groups, Comm. Algebra 32, no. 4 (2004), 1373-1384. MR 2100362 (2005k:20131)
  • [But65] M.R.C. Butler: A class of torsion free abelian groups, Proc. London Math. Soc., 15 (1965), 680-698. MR 0218446 (36:1532)
  • [F70] L. Fuchs: Infinite Abelian Groups, Vol. I, Academic Press (1970). MR 0255673 (41:333)
  • [F73] L. Fuchs: Infinite Abelian Groups, Vol. II, Academic Press (1973). MR 0349869 (50:2362)
  • [Fo87] A. Fomin: Invariants and duality in some classes of torsion-free abelian groups of finite rank, Algebra and Logic, 26 (1987), 63-83. MR 950262 (89i:20079)
  • [Fo09] A. Fomin: Invariants for abelian groups and dual exact sequences, J. Algebra, 322 (2009), 2544-2565. MR 2553694 (2010h:20125)
  • [FoW95] A. Fomin and W. Wickless: Categories of mixed and torsion free Abelian groups, in Abelian Groups and modules, Kluwer Academic (1995), 185-192. MR 1378197 (97c:20083)
  • [FoW981] A. Fomin and W. Wickless: Quotient divisible abelian groups, Proc. A.M.S., 126 (1998), 45-52. MR 1443826 (98c:20100)
  • [FoW982] A. Fomin and W. Wickless: Self-small mixed abelian groups $ G$ with $ G/t(G)$ finite rank divisible, Comm. in Algebra, 26(11) (1998), 3563-3580. MR 1647118 (99j:20061)
  • [Ri90] F. Richman: The constructive theory of torsion-free abelian groups, Comm. Algebra, 18 (1990), 3913-3922. MR 1068629 (92d:20080)
  • [VW90] C. Vinsonhaler and W. J. Wickless: Dualities for torsion-free abelian groups of finite rank, J. Algebra, 128 (1990), 474-487. MR 1036403 (91b:20076)
  • [W94] W. Wickless: A functor from mixed groups to torsion free groups, Contemp. Math., 171, Amer. Math. Soc. (1994), 407-417. MR 1293158 (95k:20090)
  • [Wa64] E. Walker: Quotient categories and quasi-isomorphisms of Abelian groups, Proc. Colloq. Abelian Groups, Akadémiai Kiadó, Budapest (1963), 147-162. MR 0178069 (31:2327)
  • [Warf68] R. B. Warfield: Homomorphisms and duality, Math. Zeitschr., 107 (1968), 189-200. MR 0237642 (38:5923)

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Additional Information

Simion Breaz
Affiliation: Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, Str. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania

Phill Schultz
Affiliation: School of Mathematics and Statistics, The University of Western Australia, Nedlands, 6009, Australia

Keywords: Self-small abelian group, finite rank torsion-free group, quotient divisible group, quasi-homomorphism category
Received by editor(s): March 31, 2010
Received by editor(s) in revised form: November 8, 2010
Published electronically: May 12, 2011
Additional Notes: The first author is supported by the UEFISCSU-CNCSIS, grant ID489
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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