Moment-angle complexes from simplicial posets

الأصل · EN

We extend the construction of moment-angle complexes to simplicial posets by associating a certain Tᵐ-space Zₛ to an arbitrary simplicial poset S on m vertices. Face rings Z[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen--Macaulay rings. Our primary motivation is to study the face rings Z[S] by topological methods. The space Zₛ has many important topological properties of the original moment-angle complex Zₖ associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Zₛ is isomorphic to the Tor-algebra of the face ring Z[S]. This leads directly to a generalisation of Hochster's theorem, expressing the algebraic Betti numbers of the ring Z[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Zₛ from below by proving the toral rank conjecture for the moment-angle complexes Zₛ.