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...of the complement in An k of the arrangement defined by a monomial ideal I can be obtained from the (algebraic) graded Betti numbers of I∨ . This fact was already proved using a different approach in =-=[8]-=-. 3. Extension problems. If M is a graded R-module and α ∈ Zn , as usual we denote by M(α) the graded R-module whose underlying R-module structure is the same as that of M and where the grading is giv...

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...r the cup products were understood by Gitler and López de Medrano [GL13]. We shall follow their approach to make a comparison of the two rings H∗((D1, S0)K) and H∗((D2, S1)K) in Example 5.4 (compare =-=[GPW01]-=-, [dL00]). The paper is organized as follows. Section 2 is devoted to the characterization of a real moment-angle manifold; as a corollary, the characterization of a momentangle manifold is given in S...

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...e Pv for the induced poset on {w ∈ P : v w}, and P≻v for the induced poset on {w ∈ P : v w}. Write U = An+1k rZ. The following was, perhaps, first observed by V. Gasharov, I. Peeva and V. Welker =-=[GPW02]-=-. Lemma 2.21. The map µ defines a duality between LA and LI∨ . Sketch. (See, also, [GPW02, Proof of Theorem 3.1].) The coatoms of LA correspond to the irreducible components of Z, which correspond, un...