Volume 14, issue 1 (2014)

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Moment angle complexes and big Cohen–Macaulayness

Shisen Luo, Tomoo Matsumura and W Frank Moore

Algebraic & Geometric Topology 14 (2014) 379–406
Abstract

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

Keywords
orbifold, integral cohomology, equivariant cohomology, torus actions, toric orbifolds, Cohen–Macaulay, toric variety
Mathematical Subject Classification 2010
Primary: 55N91
Secondary: 57R18, 53D20, 14M25