  We prove that for every compactum X and every integer $n \geq 2$ there are a
compactum Z of $\dim \leq n$ and a surjective $UV^{n-1}$-map $r: Z \lo X$
having the property that: for every finitely generated 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, or equivalently: for every simply connected
CW-complex K with finitely generated homotopy groups such that $\edim X \leq K$
we have $\edim Z \leq K$ 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.)
