Volume 14, issue 1 (2014)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Moment angle complexes and big Cohen–Macaulayness

Shisen Luo, Tomoo Matsumura and W Frank Moore

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

Let ZK 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 T be a (possibly disconnected) closed subgroup and R := TG. Let [K] be the Stanley–Reisner ring of K and consider [R] := H(BR; ) as a subring of [T] := H(BT; ). We prove that HG(ZK; ) is isomorphic to Tor[R]([K], ) as a graded module over [T]. Based on this, we characterize the surjectivity of HT(ZK; ) HG(ZK; ) (ie HGodd(ZK; ) = 0) in terms of the vanishing of Tor1[R] ([K], ) and discuss its relation to the freeness and the torsion-freeness of [K] over [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 HG(ZK) with appropriate K and G and the above results mean that H(X; )Tor[R]([K], ) and that Hodd(X; ) = 0 if and only if H(X; ) is the quotient HR(X; ).

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