## Algebraic & Geometric Topology

### Moment angle complexes and big Cohen–Macaulayness

#### Abstract

Let $ZK⊂ℂm$ 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 $R:=T∕G$. Let $ℤ[K]$ be the Stanley–Reisner ring of $K$ and consider $ℤ[R∗]:=H∗(BR;ℤ)$ as a subring of $ℤ[T∗]:=H∗(BT;ℤ)$. We prove that $HG∗(ZK;ℤ)$ is isomorphic to $Torℤ[R∗]∗(ℤ[K],ℤ)$ as a graded module over $ℤ[T∗]$. Based on this, we characterize the surjectivity of $HT∗(ZK;ℤ)→HG∗(ZK;ℤ)$ (ie $HGodd(ZK;ℤ)=0$) in terms of the vanishing of $Tor1ℤ[R∗](ℤ[K],ℤ)$ 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 $HG∗(ZK)$ with appropriate $K$ and $G$ and the above results mean that $H∗(X;ℤ)≅Torℤ[R∗]∗(ℤ[K],ℤ)$ and that $Hodd(X;ℤ)=0$ if and only if $H∗(X;ℤ)$ is the quotient $HR∗(X;ℤ)$.

#### Article information

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

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

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

#### 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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