Splitting kernels into small summands

Let λ be a regular cardinal. An epimorphism between abelian groups is λ -pure if it is projective with respect to abelian groups of size less than λ . We show that cotorsion groups A have λ -pure projective dimension greater than 1 for all uncountable λ ≤ | A / tA |, where tA denotes the torsion subgroup of A . For λ > | A / tA |, cotorsion groups A are λ -pure projective. This is related to a (hard) problem of Neeman in module theory about writing modules as factors of direct sums of small modules. We hope that this is a first step towards a counterexample of this problem.