Cotorsion theories and splitters

Göbel, Rüdiger; Shelah, Saharon

doi:10.1090/S0002-9947-00-02475-2

Cotorsion theories and splitters
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by Rüdiger Göbel and Saharon Shelah PDF

Trans. Amer. Math. Soc. 352 (2000), 5357-5379 Request permission

Abstract:

Let $R$ be a subring of the rationals. We want to investigate self splitting $R$-modules $G$ (that is $\operatorname {Ext}_R(G,G) = 0)$. Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism $R$-algebras with a free $R$-module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.

References

Thomas Becker, László Fuchs, and Saharon Shelah, Whitehead modules over domains, Forum Math. 1 (1989), no. 1, 53–68. MR 978975, DOI 10.1515/form.1989.1.53
A. L. S. Corner and Rüdiger Göbel, Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), no. 3, 447–479. MR 779399, DOI 10.1112/plms/s3-50.3.447
Yoshiomi Furuta, The genus field and genus number in algebraic number fields, Nagoya Math. J. 29 (1967), 281–285. MR 209260, DOI 10.1017/S0027763000024387
Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
Berthold Franzen and Rüdiger Göbel, Prescribing endomorphism algebras. The cotorsion-free case, Rend. Sem. Mat. Univ. Padova 80 (1988), 215–241 (1989). MR 988123
Walter Leighton and W. T. Scott, A general continued fraction expansion, Bull. Amer. Math. Soc. 45 (1939), 596–605. MR 41, DOI 10.1090/S0002-9904-1939-07046-8
Phillip Griffith, A solution to the splitting mixed group problem of Baer, Trans. Amer. Math. Soc. 139 (1969), 261–269. MR 238957, DOI 10.1090/S0002-9947-1969-0238957-1
Rüdiger Göbel, Abelian groups with small cotorsion images, J. Austral. Math. Soc. Ser. A 50 (1991), no. 2, 243–247. MR 1094921, DOI 10.1017/S1446788700032729
Rüdiger Göbel, New aspects for two classical theorems on torsion splitting, Comm. Algebra 15 (1987), no. 12, 2473–2495. MR 917750, DOI 10.1080/00927878708823548
Rüdiger Göbel and Rainer Prelle, Solution of two problems on cotorsion abelian groups, Arch. Math. (Basel) 31 (1978/79), no. 5, 423–431. MR 526605, DOI 10.1007/BF01226469
R. Göbel, S. Shelah, Almost free splitters, Colloq. Math. 81 (1999), 193–221.
R. Göbel, J. Trlifaj, Cotilting and a hierarchy of almost cotorsion groups, to appear in Proc. Amer. Math. Soc.
Jutta Hausen, Automorphismengesättigte Klassen abz̈ahlbarer abelschen Gruppen, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 147–181 (German). MR 0244376
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
Otto Kerner, Elementary stones, Comm. Algebra 22 (1994), no. 5, 1797–1806. MR 1264742, DOI 10.1080/00927879408824936
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
Claus Michael Ringel, The braid group action on the set of exceptional sequences of a hereditary Artin algebra, Abelian group theory and related topics (Oberwolfach, 1993) Contemp. Math., vol. 171, Amer. Math. Soc., Providence, RI, 1994, pp. 339–352. MR 1293154, DOI 10.1090/conm/171/01786
Claus Michael Ringel, Bricks in hereditary length categories, Results Math. 6 (1983), no. 1, 64–70. MR 714659, DOI 10.1007/BF03323325
P. Rothmaler, Purity in model theory, pp. 445 – 469 in Advances in Algebra and Model Theory, Series Algebra, Logic and Applications, Vol. 9, Gordon and Breach, Amsterdam 1997.
Helices and vector bundles, London Mathematical Society Lecture Note Series, vol. 148, Cambridge University Press, Cambridge, 1990. Seminaire Rudakov; Translated from the Russian by A. D. King, P. Kobak and A. Maciocia. MR 1074776
Luigi Salce, Cotorsion theories for abelian groups, Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977) Academic Press, London-New York, 1979, pp. 11–32. MR 565595
P. Schultz, Self-splitting groups, Preprint series of the University of Western Australia at Perth (1980).
Saharon Shelah, A combinatorial theorem and endomorphism rings of abelian groups. II, Abelian groups and modules (Udine, 1984) CISM Courses and Lect., vol. 287, Springer, Vienna, 1984, pp. 37–86. MR 789808, DOI 10.1007/978-3-7091-2814-5_{3}
Luise Unger, Schur modules over wild, finite-dimensional path algebras with three simple modules, J. Pure Appl. Algebra 64 (1990), no. 2, 205–222. MR 1055030, DOI 10.1016/0022-4049(90)90157-D
Takayoshi Wakamatsu, On modules with trivial self-extensions, J. Algebra 114 (1988), no. 1, 106–114. MR 931903, DOI 10.1016/0021-8693(88)90215-3
R. B. Warfield Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719. MR 242885, DOI 10.2140/pjm.1969.28.699
Martin Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), no. 2, 149–213. MR 739577, DOI 10.1016/0168-0072(84)90014-9