Cohomology of face rings, and torus actions

Abstract

In this survey article we present several new developments of `toric topology' concerning the cohomology of face rings (also known as Stanley-Reisner algebras). We prove that the integral cohomology algebra of the moment-angle complex Z_K (equivalently, of the complement U(K) of the coordinate subspace arrangement) determined by a simplicial complex K is isomorphic to the Tor-algebra of the face ring of K. Then we analyse Massey products and formality of this algebra by using a generalisation of Hochster's theorem. We also review several related combinatorial results and problems.