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Characterizations of Buchsbaum complexes. (English) Zbl 0671.13014
Let K be a field, \(\Delta\) a simplicial complex with vertex set \(V\subset \{x_ 1,...,x_ n\}\), K[\(\Delta\) ] the associated Stanley-Reisner ring, \(A=k[x_ 1,...,x_ n]\) and \({\mathfrak m}_ j=(x^ j_ 1,...,x^ j_ n)\). The author computes the modules \(Ext^ i_ A(A/{\mathfrak m}_ j,K[\Delta])\) 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 \(Ext^ i_ A(A/{\mathfrak m},K[\Delta])\) and \(Ext^ i_ A(A/{\mathfrak m}_ 2,K[\Delta])\) have the same length.
Reviewer: W.Bruns

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
55U10 Simplicial sets and complexes in algebraic topology
Full Text: DOI EuDML
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