Moment-angle complexes from simplicial posets

TL;DR

This paper generalizes moment-angle complexes to simplicial posets, linking their topological properties with algebraic face rings, and extends key theorems like Hochster's and the toral rank conjecture to this broader setting.

Contribution

It introduces a new construction of moment-angle complexes from simplicial posets and establishes their topological and algebraic properties, including a generalized Hochster's theorem.

Findings

Integral cohomology algebra is isomorphic to the Tor-algebra of the face ring.

Generalization of Hochster's theorem for simplicial posets.

Proof of the toral rank conjecture for these complexes.

Abstract

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T^m-space Z_S 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 Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z_S 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 Z_S from below by proving the toral rank conjecture for the moment-angle complexes Z_S.