Cohomological dimension and acyclic resolutions

Let G be an Abelian group admitting a homomorphism α: ℤ→G such that the induced homomorphisms α⊗id: ℤ⊗G→G⊗G and α*: Hom(G,G)→Hom(ℤ,G) are isomorphisms. We show that for every simplicial complex L there exists an Edwards-Walsh resolution ω: EW_G(L,n)→ L . As applications of it we give several resolution theorems. In particular, we have Theorem. Let G be an arbitrary Abelian group. For every compactum X with c-dim_GX≤n there exists a G-acyclic map f: Z→X from a compactum Z with dimZ≤n+2 and c-dim_GZ≤n+1. Our methods determine other results as well. If the group G is cyclic, then one can obtain Z with dimZ≤n. In certain other cases, depending on G, we may resolve X in such a manner that dimZ≤n+1 and c-dim_GZ≤n.