The homology of simplicial complement and the cohomology of the moment-angle complexes

Abstract: A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,\sigma}(\Lambda^{,}[P], d)$ over a principle ideal domain k and proved that $H_{,}(\Lambda[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{,}^{k[x]}(k(K), k)$, $link_{K}\sigma$, $star_{K}\sigma$ and the cohomology of the generalized moment-angle complexes.