arXiv subjects
Publications and source records attributed to Paul C. Eklof.
We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns out to be equivalent to the question of when the tensor product of two Whitehead groups is Whitehead. We investigate what happens in different models of set theory.
We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality aleph_1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that Z is a p.i.d. with infinitely many primes is essential for this result.
Let R be a Dedekind domain. Enochs' solution of the Flat Cover Conjecture was extended as follows: (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set. We show that (*) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP^+ implies that C is not cogenerated by a set whenever C is a cotorsion pair generated by a set which contains a non-cotorsion module.
We consider what Ext(A,Z) can be when A is torsion-free and Hom(A,Z)=0. We thereby give an answer to a question of Golasinski and Goncalves which asks for the divisible Abelian groups which can be the type of a co-Moore space.
It is proved undecidable in ZFC + GCH whether every Z-module has a^{perp} {Z}-precover.
We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.'s of size >= aleph_{2} have non-free Whitehead modules even though they are not complete discrete valuation rings.
We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal kappa(omega).
We prove that it is consistent that there is a non-reflexive Whitehead group, in fact one whose dual group is free. We also prove that it is consistent that there is a group A such that Ext(A,Z) is torsion and Hom(A,Z)=0. As an application we show the consistency of the existence of new co-Moore spaces.
We show that it is consistent that there is a strongly aleph_1-free aleph_1-coseparable group of cardinality aleph_1 which is not aleph_1-separable.
We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph_1-separable abelian groups of cardinality aleph_1. In fact, there is an aleph_1-separable abelian group M such that M is isomorphic to M oplus M oplus M but not to M oplus M .
Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (X_i)_{i in I} of subsets of L with the property that bigcap_{i in I} A_i not= 0 whenever A_i in X_i. Finally, we show that, for domains with Krull dimension (and countably generated extensions thereof), these lattice-theoretic conditions are equivalent to productive boundedness.
We study the classification of omega_1-separable groups using Ehrenfeucht-Fraisse games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming diamond.
As a consequence of identifying the principle described in the title, we prove that for any uncountable cardinal lambda, if there is a lambda-free Whitehead group of cardinality lambda which is not free, then there are many ``nice'' Whitehead groups of cardinality lambda which are not free.
We consider the question of which valuation domains (of cardinality aleph_1) have non-standard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ZFC + GCH.
It is proved consistent with either CH or the negation of CH that there is an aleph_1-separable group of cardinality aleph_1 which does not have a coherent system of projections. It had previously been shown that it is consistent with not CH that every aleph_1-separable group of cardinality aleph_1 does have a coherent system of projections.
A new construction is given of non-standard uniserial modules over certain valuation domains; the construction resembles that of a special Aronszajn tree in set theory. A consequence is the proof of a sufficient condition for the existence of non-standard uniserial modules; this is a theorem of ZFC which complements an earlier independence result.
The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, aleph_1-coseparable) group of cardinality aleph_1 if and only if there is a ladder system on a stationary subset of omega_1 which satisfies 2-uniformization (respectively, omega-uniformization). These techniques allowed also the proof of various independence and consistency results about Whitehead groups, for example that it is consistent that there is a non-free Whitehead group of cardinality aleph_1 but no non-free aleph_1-coseparable group. However, some natural questions remained open, among them the following two: (i) Is it consistent that the class of W-groups of cardinality aleph_1 is exactly the class of strongly aleph_1-free groups of cardinality aleph_1 ? (ii) If every strongly aleph_1-free group of cardinality aleph_1 is a W-group, are they also all aleph_1-coseparable? In this paper we use the techniques of uniformization to answer the first question in the negative and give a partial affirmative answer to the second question.