Moment Angle Complexes and Big Cohen-Macaulayness

TL;DR

This paper establishes a deep algebraic connection between the equivariant cohomology of moment angle complexes and Tor modules, providing criteria for cohomology surjectivity and applications to toric orbifolds.

Contribution

It proves an isomorphism between G-equivariant cohomology and Tor modules for moment angle complexes, linking topological and algebraic properties.

Findings

Equivariant cohomology is isomorphic to Tor modules over the Stanley-Reisner ring.

Surjectivity of cohomology maps corresponds to vanishing Tor_1.

Conditions identified for cohomology of toric orbifolds to be quotients of equivariant cohomology.

Abstract

Let Z_K be the moment angle complex associated to a simplicial complex K, with the canonical torus T-action. In this paper, we prove that, for any possibly disconnected subgroup G of T, G-equivariant cohomology of Z_K over the integer Z is isomophic to the Tor-module Tor_{H(BR;Z)}(Z[K],Z) as graded modules, where Z_[K] is the Stanley-Reisner ring of K. Based on this, we prove that the surjectivity of the natural map H_T(Z_K;Z) to H_G(Z_K;Z) is equivalent to the vanishing of Tor^{H(BR;Z)}_1(Z[K],Z). Since the integral cohomology of various toric orbifolds can be identified with H_G(Z_K;Z), we studied the conditions for the cohomology of a toric orbifold to be a quotient of its equivariant cohomology by linear terms.