an:04097527
Zbl 0671.13014
Miyazaki, Mitsuhiro
Characterizations of Buchsbaum complexes
EN
Manuscr. Math. 63, No. 2, 245-254 (1989).
00170749
1989
j
13H10 13D03 55U10
Stanley-Reisner ring; Ext; Buchsbaum ring
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 St??ckrad-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.
W.Bruns