Whitehead Modules over Large Principal Ideal Domains

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 ≥ א2 have nonfree Whitehead modules even though they are not complete discrete valuation rings. A module M is a Whitehead module if ExtR(M,R) = 0. The second author proved that the problem of whether every Whitehead Z-module is free is independent of ZFC + GCH (cf. [5], [6], [7]). This was extended in [1] to modules over principal ideal domains of cardinality at most א1. Here we consider the Whitehead problem for modules over principal ideal domains (p.i.d.’s) of cardinality > א1. If R is any p.i.d. which is not a complete discrete valuation ring, then an Rmodule of countable rank is Whitehead if and only if it is free (cf. [3]). On the other hand, if R is a complete discrete valuation ring, then it is cotorsion and hence every torsion-free R-module is a Whitehead module (cf. [2, XII.1.17]). It will be convenient to decree that a field is not a p.i.d. and to use the term “slender” to designate a p.i.d. which is not a complete discrete valuation ring, or equivalently, is not cotorsion (cf. [2, III.2.9]). We will say that a module is κgenerated if it is generated by a subset of size ≤ κ and that it is κ-free if every submodule generated by < κ elements is free. (Note that, by Pontryagin’s Criterion and induction on κ, every א1-free module which has rank ≤ κ is κ-generated.) An argument due to the second author (cf. [7] or [8]) shows that it is consistent with ZFC + GCH that for any p.i.d. R (of arbitrary size), there are Whitehead R-modules of rank ≥ |R| which are not free. If the p.i.d. R is slender and has cardinality at most א1, the Axiom of Constructibility (V = L) implies that every Whitehead R-module is free (cf. [1]). Our main result is that the story is different for p.i.d.’s of larger size. We will prove the following theorems in ZFC. Theorem 1. There is a slender p.i.d. R of cardinality 21 such that every א1-free א1-generated R-module is a Whitehead module. Hence there are non-free Whitehead R-modules which are א1-generated. Theorem 2. There is a p.i.d. R of cardinality א2 such that an א1-generated Rmodule is Whitehead only if it is free. Assuming V = L and using the existing theory (cf. [1]) one easily obtains the following: Date: February 1, 2008. First author partially supported by NSF DMS 98-03126. Second author supported by the German-Israeli Foundation for Scientific Research & Development. Publication 752.