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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

##### MSC:
 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
##### Keywords:
Stanley-Reisner ring; Ext; Buchsbaum ring
Full Text:
##### References:
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