A generalization of Whitehead’s problem and its independence

Abstract

For certain classes of Dedekind domains S we want to characterize S -modules U such that Ext ( U , M ) = 0 for some module S ⊆ M ⊆ Q . We shall call these modules M -Whitehead modules. On the one hand we will show that assuming ( V = L ) all M -Whitehead modules U are S 0 -free, i.e. U ⊗ S 0 is a free S 0 -module where S 0 is the nucleus of M . On the other hand if there is a ladder system on a stationary subset of ω 1 that satisfies 2 -uniformization, then there exists a non- S 0 -free M -Whitehead module. Conversely, we will show that in the special case of Abelian groups the existence of a non- S 0 -free R -Whitehead group–here R is a rational group–implies that there is a ladder system on a stationary subset of ω 1 that satisfies 2 -uniformization.