  A fundamental result in toric topology identifies the cohomology ring of the
moment-angle complex $\mathcal{Z}_K$ associated to a simplicial complex $K$
with the Koszul homology of the Stanley--Reisner ring of $K$. By studying
cohomology operations induced by the standard torus action on the moment-angle
complex, we extend this to a topological interpretation of the minimal free
resolution of the Stanley-Reisner ring. The exterior algebra module structure
in cohomology induced by the torus action recovers the linear part of the
minimal free resolution, and we show that higher cohomology operations induced
by the action (in the sense of Goresky-Kottwitz-MacPherson) can be assembled
into an explicit differential on the resolution. Describing these operations in
terms of Hochster's formula, we recover and extend a result due to Katth\"an.
We then apply all of this to study the equivariant formality of torus actions
on moment-angle complexes. For these spaces, we obtain complete algebraic and
combinatorial characterisations of which subtori of the naturally acting torus
act equivariantly formally.
