On Dranishnikov's cell-like resolution

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