Methods of commutative and homological algebra yield information on the Stanley-Reisner ring k[K] of a simplicial complex K. Consider the following problem: describe topological properties of simplicial complexes with given properties of the ring k[K]. It is known that for a simplicial complex K = ∂P * , where P * is a polytope dual to the simple polytope P of dimension n, the depth of depth k[K] equals n. A recent construction allows us to associate a simplicial complex K P to any convex polytope P. As a consequence, one wants to study the properties of the rings k[K P ]. In this paper, we report on the obtained results for both of these problems. In particular, we characterize the depth of k[K] in terms of the topology of links in the complex K and prove that depth k[K P ] = n for all convex polytopes P of dimension n. We obtain a number of relations between bigraded betti numbers of the complexes K P. We also establish connections between the above questions and the notion of a k-Cohen-Macaulay complex, which leads to a new filtration on the set of simplicial complexes.
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