Cohen-Macaulay simplicial complexes
The following theorems are proved: 1. Let \(\Delta\) be a simplicial complex over K (a field), and R a finitely generated K-algebra. Let \(\{I_{\Delta_ i}\}\) be ideals in R indexed by the subcomplexes of \(\Delta\), satisfying \((i)\quad I_{\Delta_ 1\cup \Delta_ 2}=I_{\Delta_ 1}\cap I_{\Delta_ 2};\)\ \((ii)\quad I_{\Delta_ 1\cap \Delta_ 2}=I_{\Delta_ 1}+I_{\Delta_ 2};\)\ \((iii)\quad For\) some c and for each simplex \(\sigma\) in \(\Delta\), \(R/I_{\sigma}\) is C-M of dimension \(\dim (\sigma)+c\). - Then \(R/I_{\Delta}\) is C-M. (This is easily seen to imply one half of the characterization of C-M-complexes due to Reisner.) 2. Suppose that for every face \(\sigma\) of \(\Delta\), \(\tilde H^ i(link(\sigma))=0\), \(i<\dim (link(\sigma))\), and \(\tilde H^{\dim (link(\sigma))}(link(\sigma))\) is torsion free over \({\mathbb{Z}}\). If R is a ring with ideals satisfying (i), (ii) and (iii), then \(R/I_{\Delta}\) is C-M.
scientific article; zbMATH DE number 4071054