Moment angle complexes and big Cohen–Macaulayness

Let Z K C m be the moment angle complex associated to a simplicial complex K on Œm, together with the natural action of the torus T D U.1/ m .Let G T be a (possibly disconnected) closed subgroup and R WD T=G.Let ZŒK be the Stanley-Reisner ring of K and consider ZŒR WD H .BRI Z/ as a subring of ZŒT WD H .BTI Z/.We prove that H G .Z K I Z/ is isomorphic to Tor ZŒR .ZŒK; Z/ as a graded module over ZŒT .Based on this, we characterize the surjectivity ofin terms of the vanishing of Tor ZŒR 1 .ZŒK; Z/ and discuss its relation to the freeness and the torsion-freeness of ZŒK over ZŒR .For various toric orbifolds X , by which we mean quasitoric orbifolds or toric Deligne-Mumford stacks, the cohomology of X can be identified with H G .Z K / with appropriate K and G and the above results mean that H .X I Z/ Š Tor ZŒR .ZŒK; Z/ and that H odd .X I Z/ D 0 if and only if H .X I Z/ is the quotient H R .X I Z/.