An Independence Result on Cotorsion Theories over Valuation Domains

It is shown that, over suitable valuation domains R with field of quotients Q, the cotorsion theory K generated by K = Q/R coincides with the cotorsion theory ∂ cogenerated by the Fuchsʹ divisible module ∂, provided that Gödelʹs Axiom of Constructibility V = L is assumed. On the other hand, assuming Martinʹs Axiom and the negation of the Continuum Hypothesis, it is proved that the cotorsion theory K is strictly smaller than ∂ by exhibiting a strongly ( 1 − K)-free divisible module M of projective dimension 2 such that Ext1R(M, K) = 0. Applications to Whitehead modules are derived.