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Quotient algebras of Stanley-Reisner rings and local cohomology. (English) Zbl 0761.55015
If a finite simplicial complex $$\Delta$$ and a field $$k$$ are given then construct the $$k$$-algebra $$k[\Delta]$$ generated by the vertices of $$\Delta$$ and subject to the relations determined by the requirement that a monomial should be zero if the set of generators appearing in it does not form a simplex of $$\Delta$$; this ring $$k[\Delta]$$ is called the “Stanley-Reiser ring of $$\Delta$$ over $$k$$” and it can be considered as a module over the corresponding polynomial ring. The simplicial complex $$\Delta$$ is called “Cohen-Macaulay” (resp. “Buchsbaum”) if the module $$k[\Delta]$$ is Cohen-Macaulay (resp. Buchsbaum) (see the author [Nagoya Math. J. 107, 91-113 (1987; Zbl 0601.13011)] and R. Stanley [Combinatorics and commutative algebra (1983; Zbl 0537.13009)]. The author’s main result – his “foggy rank selection theorem” – says that these properties are preserved by the transition to so-called quasi- hereditary subcomplexes. It is applied to obtain general versions of the rank selection theorems of J. Munkres [Mich. Math. J. 31, 113-128 (1984; Zbl 0585.57014)] and R. Stanley [Trans. Am. Math. Soc. 249, 139-157 (1979; Zbl 0411.05012)].

MSC:
 55U05 Abstract complexes in algebraic topology 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:
 [1] Hibi, T., Union and glueing of a family of Cohen-Macaulay partially ordered sets, Nagoya math. J., 107, 91-119, (1987) · Zbl 0601.13011 [2] Munkres, J., Topological results in combinatorics, Michigan math. J., 31, 113-128, (1984) · Zbl 0585.57014 [3] Reisner, G., Cohen-Macaulay quotients of polynomial rings, Adv. in math., 21, 30-49, (1976) · Zbl 0345.13017 [4] \scD. Smith, On the Cohen-Macaulay property in commutative algebra and simplicial topology, to appear. · Zbl 0686.13008 [5] Stanley, R., The upper bound conjecture and Cohen-Macaulay rings, Stud. appl. math., 54, 135-142, (1975) · Zbl 0308.52009 [6] Stanley, R., Balanced Cohen-Macaulay complexes, Trans. amer. math. soc., 249, 139-157, (1979) · Zbl 0411.05012 [7] Stanley, R., Combinatorics and commutative algebra, (1983), Birkhäuser Boston/Basel/Stuttgart · Zbl 0537.13009 [8] Stückrad, J.; Vogel, W., Buchsbaum rings and applications, (1986), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0606.13018
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