Moment angle complexes and big Cohen-Macaulayness

Abstract

Let Z(K) subset of C-m be the moment angle complex associated to a simplicial complex K on [m], together with the natural action of the torus T = U(1)(m). Let G subset of T be a ( possibly disconnected) closed subgroup and R := T/G. Let Z[K] be the Stanley-Reisner ring of K and consider Z[R*] := H*(BR; Z) as a subring of Z[T*] := H*(BT; Z). We prove that H-G*(Z(K); Z) is isomorphic to Tor(Z[R*])*(Z[K]; Z) as a graded module over Z[T*]. Based on this, we characterize the surjectivity of H-T*(Z(K); Z) -> H-G*(Z(K); Z) (ie H-G(odd)(Z(K); Z) = 0) in terms of the vanishing of Tor(Z[R*])*(Z[K], Z) and discuss its relation to the freeness and the torsion-freeness of Z[K] over Z[R*]. For various toric orbifolds chi, by which we mean quasitoric orbifolds or toric Deligne-Mumford stacks, the cohomology of chi can be identified with H-G*(Z(K)) with appropriate K and G and the above results mean that H*(chi: Z) congruent to Tor(Z[R*])*(Z[K], Z) and that H-odd(chi; Z) = 0 if and only if H*(chi; Z) is the quotient H-R*(chi; Z).