On Whitehead Precovers

It is proved undecidable in ZFC + GCH whether every Z-module has a {Z}-precover. Let F be a class of R-modules of the form C = {A : Ext(A,C) = 0 for all C ∈ C} for some class C. The first author and Jan Trlifaj proved [7] that a sufficient condition for every module M to have an F -precover is that there is a module B such that F = {B} (= {A : Ext(B,A) = 0}). In [8], generalizing a method used by Enochs [1] to prove the Flat Cover Conjecture, it is proved that this sufficient condition holds whenever C is a class of pure-injective modules; moreover, for R a Dedekind domain, the sufficient condition holds whenever C is a class of cotorsion modules. The following is also proved in [8]: Theorem 1. It is consistent with ZFC + GCH that for any hereditary ring R and any R-module N , there is an R-module B such that ({N}) = {B} and hence every R-module has a {N}-precover. This is a generalization of a result proved by the second author for the class W of all Whitehead groups ( = {Z}): Theorem 2. It is consistent with ZFC + GCH that W = {B} where B is any free abelian group. Proof. The second author proved that Gödel’s Axiom of Constructibility (V = L) implies that W is exactly the class of free groups. (See [11] or [4].) Under this hypothesis (which implies GCH), W is the class of all groups; if we take B to be any free group, then {B} is also the class of all groups. Our main result here is that the conclusions of Theorem 1 are not provable in ZFC + GCH for N = Z = R: Theorem 3. It is consistent with ZFC + GCH that Q does not have a W-precover. An immediate consequence is: Theorem 4. It is consistent with ZFC + GCH that there is no abelian group B such that W = {B}. Theorem 3 follows easily from 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 749.