Characterizations of Buchsbaum complexes
Let \(K\) be a field, \(\Delta\) a simplicial complex with vertex set \(V\subset \{x_ 1,\dots,x_ n\}\), K[\(\Delta\) ] the associated Stanley-Reisner ring, \(A=k[x_ 1,\dots,x_ n]\) and \({\mathfrak m}_ j=(x^ j_ 1,\dots,x^ j_ n)\). The author computes the modules \(\mathrm{Ext}^ i_ A(A/{\mathfrak m}_ j,K[\Delta])\) in terms of the reduced simplicial cohomology of certain subcomplexes of \(\Delta\). As a corollary he gets Hochster's fundamental theorem which relates the local cohomology of \(K[\Delta]\) with respect to \({\mathfrak m}\) and the reduced simplicial cohomology. After having recalled some criteria of Schenzel and Stückrad-Vogel for \(K[\Delta]\) to be a Buchsbaum ring, the author proves his characterization of the Buchsbaum property of Stanley-Reisner rings: \(K[\Delta]\) is Buchsbaum if and only if for all \(i<d\) the modules \(\mathrm{Ext}^ i_ A(A/{\mathfrak m},K[\Delta])\) and \(\mathrm{Ext}^ i_ A(A/{\mathfrak m}_ 2,K[\Delta])\) have the same length.
scientific article; zbMATH DE number 4097527