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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
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References:
[1] S.Goto and K.-i.Watanabe, On graded rings, II, Tokyo J. Math. 1 (1978), 237-261 · Zbl 0406.13007 · doi:10.3836/tjm/1270216496
[2] M.Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring Theory II, Proc. of the second Oklahoma Conf. (B.R.McDonald and R.Morris, ed.), Lect. Notes in Pure and Appl. Math., No.26, Dekker, New York, 1977, 171-223 · Zbl 0351.13009
[3] M.Hochster and J.L.Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115-175 · Zbl 0289.14010 · doi:10.1016/0001-8708(74)90067-X
[4] J.Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1984), 113-128 · Zbl 0585.57014 · doi:10.1307/mmj/1029002969
[5] G.Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), 30-49 · Zbl 0345.13017 · doi:10.1016/0001-8708(76)90114-6
[6] P.Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 178 (1981), 125-142 · Zbl 0472.13012 · doi:10.1007/BF01218376
[7] R.Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), 135-142 · Zbl 0308.52009
[8] R.Stanley, Cohen-Macaulay Complexes, in Higher Combinatorics (M.Aigner, ed.), Reidel, Dordrecht and Boston, 1977, pp. 51-62
[9] R.Stanley, ?Combinatorics and Commutative Algebra?, Progress in Math., Vol.41, Birkhäuser, Boston/ Basel/ Stuttgart, 1983 · Zbl 0537.13009
[10] J.Stückrad and W.Vogel, Toward a theory of Buchsbaum Singularities, Amer. J. Math. 100, (1978), 727-746 · Zbl 0429.14001 · doi:10.2307/2373908
[11] J.Stückrad and W.Vogel, ?Buchsbaum Rings and Applications?, Springer-Verlag Berlin Heidelberg New York London Tokyo, 1986
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