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 Mathematics , 2010, 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$and the cohomology of the generalized moment-angle complexes.  Mathematics , 2015, Abstract: Associated to every finite simplicial complex$K$, there is a moment-angle complex$\mathcal {Z}_{K}$;$\mathcal {Z}_{K}$is a compact manifold if and only if$|K|$is a generalized homology sphere. The main goal of this article is to study the cohomology rings of moment-angle complexes associated to some special simplicial complexes. First, we give the cohomological transformation formulae of$\mathcal {Z}_{K}$induced by some combinatorial operations on the base space$K$, such as the connected sum operation on Gorenstein* complexes and the stellar subdivisions on simplicial spheres. Second, we prove the indecomposable property of$\mathcal {Z}_{K}$(i.e.$\mathcal {Z}_{K}$is a prime manifold) when$K$is a flag$2$-sphere by proving the indecomposable property of their cohomology rings. Then we use these results to solve the cohomological rigidity problem for some moment-angle manifolds and the$B$-rigidity problem for some simplicial spheres.  Yury Ustinovsky Mathematics , 2009, Abstract: We consider an operation K \to L(K) on the set of simplicial complexes, which we call the "doubling operation". This combinatorial operation has been recently brought into toric topology by the work of Bahri, Bendersky, Cohen and Gitler on generalised moment-angle complexes (also known as K-powers). The crucial property of the doubling operation is that the moment-angle complex Z_K can be identified with the real moment-angle complex RZ_L(K) for the double L(K). As an application we prove the toral rank conjecture for Z_K by estimating the lower bound of the cohomology rank (with rational coefficients) of real moment-angle complexes RZ_K$. This paper extends the results of our previous work, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.