Homological algebra and set theory

Eklof, Paul C.

doi:10.1090/S0002-9947-1977-0453520-X

Homological algebra and set theory
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by Paul C. Eklof

Trans. Amer. Math. Soc. 227 (1977), 207-225

DOI: https://doi.org/10.1090/S0002-9947-1977-0453520-X

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Abstract:

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of ${\operatorname {Ext}}_\Lambda ^1$ for rings $\Lambda$ of global dimension 1. Using Martin’s Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming ${\text {V}} = {\text {L}}$) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.

References

Reinhold Baer, The subgroup of the elements of finite order of an abelian group, Ann. of Math. (2) 37 (1936), no. 4, 766–781. MR 1503308, DOI 10.2307/1968617
William Boos, Infinitary compactness without strong inaccessibility, J. Symbolic Logic 41 (1976), no. 1, 33–38. MR 409184, DOI 10.2307/2272942
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
M. A. Dickmann, Large infinitary languages, Studies in Logic and the Foundations of Mathematics, Vol. 83, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Model theory. MR 0539973

Set theory, an introduction to large cardinals

Paul C. Eklof, Infinitary equivalence of abelian groups, Fund. Math. 81 (1974), 305–314. MR 354349, DOI 10.4064/fm-81-4-305-314
Paul C. Eklof, On the existence of $\kappa$-free abelian groups, Proc. Amer. Math. Soc. 47 (1975), 65–72. MR 379694, DOI 10.1090/S0002-9939-1975-0379694-0
L. Fuchs, Abelian groups, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York-Oxford-London-Paris, 1960. MR 0111783
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
Kurt Gödel, The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, No. 3, Princeton University Press, Princeton, N. J., 1940. MR 0002514, DOI 10.1515/9781400881635

Abelian groups infinitarily equivalent to free ones

20

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
Phillip A. Griffith, Infinite abelian group theory, University of Chicago Press, Chicago, Ill.-London, 1970. MR 0289638
R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI 10.1016/0003-4843(72)90001-0
H. Jerome Keisler, Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers, Studies in Logic and the Foundations of Mathematics, Vol. 62, North-Holland Publishing Co., Amsterdam-London, 1971. MR 0344115
Kenneth Kunen, Indescribability and the continuum, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 199–203. MR 0282829

Applications of logic to group theory

The number of

free groups and the size of

Shelah’s solution of Whitehead’s problem

E. Sa̧siada, An application of Kulikov’s basic subgroups in the theory of Abelian mixed groups, Bull. Acad. Polon. Sci. Cl. III. 4 (1956), 411–413. MR 0081903
Saharon Shelah, Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 18 (1974), 243–256. MR 357114, DOI 10.1007/BF02757281
Saharon Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21 (1975), no. 4, 319–349. MR 389579, DOI 10.1007/BF02757993
R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin’s problem, Ann. of Math. (2) 94 (1971), 201–245. MR 294139, DOI 10.2307/1970860
Karl Stein, Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem, Math. Ann. 123 (1951), 201–222 (German). MR 43219, DOI 10.1007/BF02054949