Inheritance of the Buchsbaum and Cohen-Macaulay properties in subcomplexes
Let \(\Sigma\) be a finite simplicial complex with underlying topological space \(|\Sigma |\). Let \(k [\Sigma]\) be the Stanley-Reisner ring of \(\Sigma\) over a field \(k\). It has been conjectured by M. Hochster that the property of \(k [\Sigma]\) to be Cohen-Macaulay is independent of the simplicial structure of \(\Sigma\), depending only on the space \(|\Sigma |\). Proving this conjecture, \textit{J. R. Munkres} [Michigan Math. J. 31, 113-128 (1984; Zbl 0585.57014)] defined two topological invariants \(\alpha\) and \(\gamma\) of \(\Sigma \). Namely, \(\gamma (\Sigma)\) is the smallest integer \(j\) for which at least one of the relative cohomological groups \(\widetilde H^j (|\Sigma |, |\Sigma |- p)\) for \(p \in |\Sigma |\) is not trivial. Further, \(\alpha (\Sigma)\) is the least integer \(j\) so that one of the groups \(\widetilde H^j (|\Sigma |)\), \(\widetilde H^j (|\Sigma |, |\Sigma |- p)\), \(p \in |\Sigma |\), does not vanish. It is clear that \(\dim \Sigma \geq \gamma (\Sigma) \geq \alpha (\Sigma) \geq - 1\). In the case \(\alpha (\Sigma) = \dim \Sigma\), the ring \(k [\Sigma]\) is to be Cohen-Macaulay and it is to be Buchsbaum if \(\gamma (\Sigma) = \dim \Sigma\). This paper is devoted to further investigations of the invariants \(\alpha (\Sigma) \) and \(\gamma (\Sigma)\). Here are partial statements of its main results (Theorems 2.3 and 2.5): Let \(T \subseteq V (\Sigma)\) be the subset of the vertex set of a complex \(\Sigma \) such that, for \(x,x' \in T\), \(x \neq x'\), \(\{x,x' \} \notin \Sigma\). Let \(\Sigma'\) be the subcomplex \(\Sigma' = \Sigma - \{\sigma \in \Sigma \mid \sigma\) contains some \(x \in T\}\). Then \(\alpha (\Sigma') \geq \alpha (\Sigma) - 1\) and \(\gamma (\Sigma') \geq \gamma (\Sigma) - 1\). Note that associating with a finite poset \(P\) its order complex \(\Sigma (X)\) [\textit{R. P. Stanley}, Combinatorics and commutative algebra (1983; Zbl 0537.13009)], one can speak about invariants \(\alpha\) and \(\gamma\) of posets. In the remainder part of the paper the author studies the behavior of \(\alpha (X)\) and \(\gamma (X)\) with respect to some operations of posets. For example, \(\gamma (X \times Y) = \gamma (X) + \gamma (Y)\), and the corresponding formula holds for \(\alpha (X \times Y)\), too.
scientific article; zbMATH DE number 814486