Moment angle complexes and big Cohen–Macaulayness

Luo, Shisen¹ ; Matsumura, Tomoo² ; Moore, W Frank³

¹Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA
²Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu Daejeon, Daejeon 305-701, South Korea
³Department of Mathematics, Wake Forest University, PO Box 7388, 127 Manchester Hall, Winston-Salem, NC 27109, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 379-406

Résumé

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).

Détail

DOI : 10.2140/agt.2014.14.379

Classification : 55N91, 57R18, 53D20, 14M25
Keywords: orbifold, integral cohomology, equivariant cohomology, torus actions, toric orbifolds, Cohen–Macaulay, toric variety

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