Not lim-acyclic sums of copies (Q1358081)

A projective system (= an inverse spectrum over a directed set) of abelian groups \(\{G_i\}_{i\in I}\) is called lim-acyclic if the values of the \(n\)-th derived functor \(\lim^n_{I^{\text{op}}}\{G_i\}\) of the (inverse) limit functor are equal to the zero abelian group for all natural \(n>0\). Let \(I\) be a directed set and \(I^{\text{op}}Ab\) be the category of projective systems (and natural transformations) of abelian groups over \(I\). A sum (in the category \(I^{\text{op}}Ab)\) of projective systems each equal to some \(\{G_i\}_{i\in I}\) is said to be the sum of copies of the projective system \(\{G_i\}_{i\in I}\). The main result is the following: Theorem 3.5. The following statements are equivalent: (A) The sum of \(E\) copies of an arbitrary lim-acyclic projective system of finitely generated abelian groups is lim-acyclic for every set \(E\). (B) If an abelian group \(G\) is such that \(\text{Ext}(G,Z)=0\) then it is free.