  We present new, unified proofs for the cell-like, $\mathbb{Z}/p$-, and
$\mathbb{Q}$-resolution theorems. Our arguments employ extensions that are much
simpler then those used by our predecessors. The techniques allow us to solve
problems involving cohomology groups by converting them into problems about
homology groups. We provide a coordinated general topological method for
constructing the maps needed to witness the resolution theorems simultaneously.
