Sh:981

• Göbel, R., Shelah, S., & Strüngmann, L. H. (2013). \aleph_n-free modules over complete discrete valuation domains with almost trivial dual. Glasg. Math. J., 55(2), 369–380. DOI: 10.1017/S0017089512000614 MR: 3040868
• Abstract:
  Let M^*Hom_R(M,R) be the dual module of M for any commutative ring R. In [GS] we applied a recent prediction principle from Shelah [Sh] to find \aleph_n-free R-modules M with trivial dual M^*=0 for each natural number n. This can be achieved for a large class of rings R including all countable, principle ideal domains which are not fields. (Recall that an R-module M is \kappa-free for some infinite cardinal \kappa if all its submodules generated by <\kappa elements are contained in a free R-submodule.) However, the result fails if R is uncountable, as can be seen from Kaplansky’s [Ka] well-known splitting theorems for modules over the ring J_p of p-adic integers. Nevertheless we want to extend the main result from [GS] to complete discrete valuation domains (DVDs) R, in particular to p-adic modules R and define a duality-test which circumvents Kaplansky’s counterexamples. We say that M has almost a trivial dual if there is no homomorphism from M onto a free R-module of countable (infinite) rank. In the first part of this paper we must strengthen and adjust the new combinatorial principle (called the \aleph_n-Black Box) and in the second part we will apply it to find arbitrarily large \aleph_n-free R-modules over complete DVD with almost trivial dual. A corresponding result for torsion modules is obtained as well. Also observe, that the existence of such modules can easily be established assuming GCH (see e.g. Eklof-Mekler [EM] on diamonds), so the problem rests on the fact that we want to stay in ordinary set theory.
• published version (12p)

@article{Sh:981,
 author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon and Str{\"u}ngmann, Lutz H.},
 title = {{$\aleph_n$-free modules over complete discrete valuation domains with almost trivial dual}},
 journal = {Glasg. Math. J.},
 fjournal = {Glasgow Mathematical Journal},
 volume = {55},
 number = {2},
 year = {2013},
 pages = {369--380},
 issn = {0017-0895},
 mrnumber = {3040868},
 mrclass = {13F30 (13C99 20K99)},
 doi = {10.1017/S0017089512000614}
}