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