Dualities for self–small groups

Breaz, Simion; Schultz, Phill

doi:10.1090/S0002-9939-2011-10919-5

Dualities for self–small groups
HTML articles powered by AMS MathViewer

by Simion Breaz and Phill Schultz

Proc. Amer. Math. Soc. 140 (2012), 69-82

DOI: https://doi.org/10.1090/S0002-9939-2011-10919-5

Published electronically: May 12, 2011

PDF | Request permission

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.

References

Ulrich Albrecht and Simion Breaz, Quasi-isomorphisms and groups of quasi-homomorphisms, J. Algebra Appl. 8 (2009), no. 5, 617–627. MR 2581988, DOI 10.1142/S0219498809003539
Ulrich Albrecht and Simion Breaz, A note on mixed $A$-reflexive groups, J. Algebra 323 (2010), no. 2, 509–516. MR 2564852, DOI 10.1016/j.jalgebra.2009.10.005
Ulrich Albrecht, Simion Breaz, and William Wickless, Self-small abelian groups, Bull. Aust. Math. Soc. 80 (2009), no. 2, 205–216. MR 2540354, DOI 10.1017/S0004972709000185
Ulrich F. Albrecht, H. Pat Goeters, and William Wickless, The flat dimension of mixed abelian groups as $E$-modules, Rocky Mountain J. Math. 25 (1995), no. 2, 569–590. MR 1336551, DOI 10.1216/rmjm/1181072238
D. M. Arnold and C. E. Murley, Abelian groups, $A$, such that $H\textrm {om}(A,—)$ preserves direct sums of copies of $A$, Pacific J. Math. 56 (1975), no. 1, 7–20. MR 376901, DOI 10.2140/pjm.1975.56.7
David M. Arnold, Finite rank torsion free abelian groups and rings, Lecture Notes in Mathematics, vol. 931, Springer-Verlag, Berlin-New York, 1982. MR 665251, DOI 10.1007/BFb0094245
David M. Arnold, A duality for torsion-free modules of finite rank over a discrete valuation ring, Proc. London Math. Soc. (3) 24 (1972), 204–216. MR 292813, DOI 10.1112/plms/s3-24.2.204
David M. Arnold, A duality for quotient divisible abelian groups of finite rank, Pacific J. Math. 42 (1972), 11–15. MR 311799, DOI 10.2140/pjm.1972.42.11
M. Auslander, Representation theory of finite-dimensional algebras, Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981) Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 27–39. MR 685936
Ulrich Albrecht and Bill Wickless, Finitely generated and cogenerated QD groups, Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Appl. Math., vol. 236, Dekker, New York, 2004, pp. 13–26. MR 2050698
Klaus Bongartz, A generalization of a theorem of M. Auslander, Bull. London Math. Soc. 21 (1989), no. 3, 255–256. MR 986367, DOI 10.1112/blms/21.3.255
R. A. Beaumont and R. S. Pierce, Torsion-free rings, Illinois J. Math. 5 (1961), 61–98. MR 148706, DOI 10.1215/ijm/1255629647
Simion Breaz, Warfield dualities induced by self-small mixed groups, J. Group Theory 13 (2010), no. 3, 391–409. MR 2653527, DOI 10.1515/JGT.2009.057
Simion Breaz, Quasi-decompositions for self-small abelian groups, Comm. Algebra 32 (2004), no. 4, 1373–1384. MR 2100362, DOI 10.1081/AGB-120028786
M. C. R. Butler, A class of torsion-free abelian groups of finite rank, Proc. London Math. Soc. (3) 15 (1965), 680–698. MR 218446, DOI 10.1112/plms/s3-15.1.680
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
A. A. Fomin, Invariants and duality in some classes of torsion-free abelian groups of finite rank, Algebra i Logika 26 (1987), no. 1, 63–83, 126 (Russian). MR 950262
Alexander Fomin, Invariants for abelian groups and dual exact sequences, J. Algebra 322 (2009), no. 7, 2544–2565. MR 2553694, DOI 10.1016/j.jalgebra.2009.03.043
Alexander A. Fomin and William J. Wickless, Categories of mixed and torsion-free finite rank abelian groups, Abelian groups and modules (Padova, 1994) Math. Appl., vol. 343, Kluwer Acad. Publ., Dordrecht, 1995, pp. 185–192. MR 1378197
A. Fomin and W. Wickless, Quotient divisible abelian groups, Proc. Amer. Math. Soc. 126 (1998), no. 1, 45–52. MR 1443826, DOI 10.1090/S0002-9939-98-04230-0
A. Fomin and W. Wickless, Self-small mixed abelian groups $G$ with $G/T(G)$ finite rank divisible, Comm. Algebra 26 (1998), no. 11, 3563–3580. MR 1647118, DOI 10.1080/00927879808826360
Fred Richman, The constructive theory of torsion-free abelian groups, Comm. Algebra 18 (1990), no. 11, 3913–3922. MR 1068629, DOI 10.1080/00927879008824116
C. Vinsonhaler and W. Wickless, Dualities for torsion-free abelian groups of finite rank, J. Algebra 128 (1990), no. 2, 474–487. MR 1036403, DOI 10.1016/0021-8693(90)90035-M
W. J. Wickless, A functor from mixed groups to torsion-free groups, Abelian group theory and related topics (Oberwolfach, 1993) Contemp. Math., vol. 171, Amer. Math. Soc., Providence, RI, 1994, pp. 407–417. MR 1293158, DOI 10.1090/conm/171/01792
E. A. Walker, Quotient categories and quasi-isomorphisms of Abelian groups, Proc. Colloq. Abelian Groups (Tihany, 1963) Akadémiai Kiadó, Budapest, 1963, pp. 147–162. MR 0178069
R. B. Warfield Jr., Homomorphisms and duality for torsion-free groups, Math. Z. 107 (1968), 189–200. MR 237642, DOI 10.1007/BF01110257