Algebraic & Geometric Topology

Moment angle complexes and big Cohen–Macaulayness

Shisen Luo, Tomoo Matsumura, and W Frank Moore

Full-text: Open access

Abstract

Let ZKm 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 GT 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;).

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 379-406.

Dates
Received: 5 August 2012
Revised: 10 March 2013
Accepted: 28 May 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715805

Digital Object Identifier
doi:10.2140/agt.2014.14.379

Mathematical Reviews number (MathSciNet)
MR3158763

Zentralblatt MATH identifier
1288.55003

Subjects
Primary: 55N91: Equivariant homology and cohomology [See also 19L47]
Secondary: 57R18: Topology and geometry of orbifolds 53D20: Momentum maps; symplectic reduction 14M25: Toric varieties, Newton polyhedra [See also 52B20]

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

Citation

Luo, Shisen; Matsumura, Tomoo; Moore, W Frank. Moment angle complexes and big Cohen–Macaulayness. Algebr. Geom. Topol. 14 (2014), no. 1, 379--406. doi:10.2140/agt.2014.14.379. https://projecteuclid.org/euclid.agt/1513715805


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