A fundamental result in toric topology identifies the cohomology
ring of the moment-angle complex ZK 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 [Invent. Math. 131
(1998), pp. 25–83]) 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 [Mathematics 7 (2019), no. 7, p. 605]. 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.