We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective $UV^{n-1}$-map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that $dim_G X ≤ k ≤ n$ we have $dim_G Z ≤ k$ and r is G-acyclic.