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Quite free complicated abelian groups, pcf and black boxes

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Abstract

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even \({\aleph _{{\omega _1} \cdot n}}\)-free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every \({\aleph _{{\omega _1} \cdot \omega }}\)-free Abelian group has non-trivial homomorphisms to ℤ.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Saharon Shelah

  2. Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

Authors
  1. Saharon Shelah
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Correspondence to Saharon Shelah.

Additional information

The author thanks the Israel Science Foundation for support of this paper, Grant No. 1053/11. Publication 1028.

The author thanks Alice Leonhardt for the beautiful typing. The reader should note that the version in my website is usually more updated that the one in the mathematical archive.

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Shelah, S. Quite free complicated abelian groups, pcf and black boxes. Isr. J. Math. 240, 1–64 (2020). https://doi.org/10.1007/s11856-020-2051-7

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  • DOI: https://doi.org/10.1007/s11856-020-2051-7

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