Moment angle complexes and big Cohen–Macaulayness

Abstract

Let ${\mathcal{Z}}_K\subset {ℂ}^m$ be the moment angle complex associated to a simplicial complex $K$ on $\left[m\right]$, together with the natural action of the torus $\mathsf{T}=U{\left(1\right)}^m$. Let $\mathsf{G}\subset \mathsf{T}$ be a (possibly disconnected) closed subgroup and $\mathsf{R}:=\mathsf{T}∕\mathsf{G}$. Let $\mathbb{Z}\left[K\right]$ be the Stanley–Reisner ring of $K$ and consider $\mathbb{Z}\left[{\mathsf{R}}^{\ast }\right]:=H^{\ast }\left(B\mathsf{R};\mathbb{Z}\right)$ as a subring of $\mathbb{Z}\left[{\mathsf{T}}^{\ast }\right]:=H^{\ast }\left(B\mathsf{T};\mathbb{Z}\right)$. We prove that $H_{\mathsf{G}}^{\ast }\left({\mathcal{Z}}_K;\mathbb{Z}\right)$ is isomorphic to ${Tor}_{\mathbb{Z}\left[{\mathsf{R}}^{\ast }\right]}^{\ast }\left(\mathbb{Z}\left[K\right],\mathbb{Z}\right)$ as a graded module over $\mathbb{Z}\left[{\mathsf{T}}^{\ast }\right]$. Based on this, we characterize the surjectivity of $H_{\mathsf{T}}^{\ast }\left({\mathcal{Z}}_K;\mathbb{Z}\right)\to H_{\mathsf{G}}^{\ast }\left({\mathcal{Z}}_K;\mathbb{Z}\right)$ (ie $H_{\mathsf{G}}^{odd}\left({\mathcal{Z}}_K;\mathbb{Z}\right)=0$) in terms of the vanishing of ${Tor}_1^{\mathbb{Z}\left[{\mathsf{R}}^{\ast }\right]}\left(\mathbb{Z}\left[K\right],\mathbb{Z}\right)$ and discuss its relation to the freeness and the torsion-freeness of $\mathbb{Z}\left[K\right]$ over $\mathbb{Z}\left[{\mathsf{R}}^{\ast }\right]$. For various toric orbifolds $\mathcal{X}$, by which we mean quasitoric orbifolds or toric Deligne–Mumford stacks, the cohomology of $\mathcal{X}$ can be identified with $H_{\mathsf{G}}^{\ast }\left({\mathcal{Z}}_K\right)$ with appropriate $K$ and $\mathsf{G}$ and the above results mean that $H^{\ast }\left(\mathcal{X};\mathbb{Z}\right)\cong {Tor}_{\mathbb{Z}\left[{\mathsf{R}}^{\ast }\right]}^{\ast }\left(\mathbb{Z}\left[K\right],\mathbb{Z}\right)$ and that $H^{odd}\left(\mathcal{X};\mathbb{Z}\right)=0$ if and only if $H^{\ast }\left(\mathcal{X};\mathbb{Z}\right)$ is the quotient $H_{\mathsf{R}}^{\ast }\left(\mathcal{X};\mathbb{Z}\right)$.