Universal acyclic resolutions for finitely generated coefficient groups

We prove that for every compactum X and every integer n⩾2 there are a compactum Z of dim⩽n and a surjective UVn−1-map r :Z→X having the property that: ery finitely generated Abelian group G and every integer k⩾2 such that dimGX⩽k⩽n we have dimGZ⩽k and r is G-acyclic, or equivalently: ery simply connected CW-complex K with finitely generated homotopy groups such that e-dimX⩽K we have e-dimZ⩽K and r is K-acyclic. (A space is K-acyclic if every map from the space to K is null-homotopic. A map is K-acyclic if every fiber is K-acyclic.)