On Dranishnikovʹs cell-like resolution

We prove the following theorem: m 1. For a compactum X with c-dimZ/pX⩽n and c-dimZ(q)X⩽n for some distinct prime numbers p,q, and c-dimZX⩽n+1, where n>1, there exists an (n+1)-dimensional compactum Z with c-dimZ/pZ⩽n, c-dimZ(q)Z⩽n and a cell-like map f :Z→X. er, giving the following theorem, we note that Theorem 1 cannot be true in the case of n=1. m 2. For every pair p,q of distinct prime numbers there exists an infinite-dimensional compactum X such that c-dimZ/pX=1, c-dimZ(q)X=1 and c-dimZX=2.