Moment angle complexes and big Cohen–Macaulayness

Let $Z_{K} \subset ℂ^{m}$ 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 $G \subset T$ be a (possibly disconnected) closed subgroup and $R : = T ∕ G$ . Let $ℤ [K]$ be the Stanley–Reisner ring of $K$ and consider $ℤ [R^{*}] : = H^{*} (B R; ℤ)$ as a subring of $ℤ [T^{*}] : = H^{*} (B T; ℤ)$ . 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 $H_{T}^{*} (Z_{K}; ℤ) \to H_{G}^{*} (Z_{K}; ℤ)$ (ie $H_{G}^{odd} (Z_{K}; ℤ) = 0$ ) in terms of the vanishing of ${Tor}_{1}^{ℤ [R^{*}]} (ℤ [K], ℤ)$ and discuss its relation to the freeness and the torsion-freeness of $ℤ [K]$ over $ℤ [R^{*}]$ . For various toric orbifolds $X$ , 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 $H^{*} (X; ℤ) ≅ {Tor}_{ℤ [R^{*}]}^{*} (ℤ [K], ℤ)$ and that $H^{odd} (X; ℤ) = 0$ if and only if $H^{*} (X; ℤ)$ is the quotient $H_{R}^{*} (X; ℤ)$ .

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Keywords

orbifold, integral cohomology, equivariant cohomology, torus actions, toric orbifolds, Cohen–Macaulay, toric variety

Mathematical Subject Classification 2010

Primary: 55N91

Secondary: 57R18, 53D20, 14M25

References

Publication

Received: 5 August 2012

Revised: 10 March 2013

Accepted: 28 May 2013

Preview posted: 5 December 2014

Published: 9 January 2014

Authors

Shisen Luo
Department of Mathematics Cornell University 310 Malott Hall Ithaca, NY 14853 USA

Tomoo Matsumura
Department of Mathematical Sciences KAIST 291 Daehak-ro Yuseong-gu Daejeon Daejeon 305-701 South Korea

W Frank Moore
Department of Mathematics Wake Forest University PO Box 7388 127 Manchester Hall Winston-Salem, NC 27109 USA

Volume 14, issue 1 (2014)

Shisen Luo, Tomoo Matsumura and W Frank Moore

Abstract