  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.
