We describe certain operations on resolutions in abelian categories, and apply them to calculate part of a reverse Adams spectral sequence, going "from homotopy to homology", for the space
K
(
Z
/
2
,
n
)
{\mathbf {K}}(\mathbb {Z}/2,n)
. This calculation is then used to deduce that there is no space whose homotopy groups are the reduction
mod
2
\bmod \; 2
of
π
∗
S
r
{\pi _\ast }{{\mathbf {S}}^r}
. As another application of the operations we give a short proof of T. Y. Lin’s theorem on the infinite projective dimension of all nonfree
π
\pi
-modules.