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