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A fundamental result in toric topology identifies the cohomology ring of themoment-angle complex $\mathcal{Z}_K$ associated to a simplicial complex $K$with the Koszul homology of the Stanley--Reisner ring of $K$. By studyingcohomology operations induced by the standard torus action on the moment-anglecomplex, we extend this to a topological interpretation of the minimal freeresolution of the Stanley-Reisner ring. The exterior algebra module structurein cohomology induced by the torus action recovers the linear part of theminimal free resolution, and we show that higher cohomology operations inducedby the action (in the sense of Goresky-Kottwitz-MacPherson) can be assembledinto an explicit differential on the resolution. Describing these operations interms 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 actionson moment-angle complexes. For these spaces, we obtain complete algebraic andcombinatorial characterisations of which subtori of the naturally acting torusact equivariantly formally.

Cohomology operations for moment-angle complexes and resolutions of Stanley-Reisner rings