### Abstract

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal I_{Δ} in the polynomial ring A = k [x_{1},...,x_{n}], and its quotient k[Δ] = A/I_{Δ} known as the Stanley-Reisner ring. This note considers a simplicial complex Δ* which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim_{k}Tor_{i} ^{A}(k[Δ],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ*. As corollaries, we prove that I_{Δ} has a linear resolution as A-module if and only if Δ* is Cohen-Macaulay over k, and show how to compute the Betti numbers dim_{k}Tor_{i} ^{A}(k[Δ],k) in some cases where Δ* is well-behaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.

Original language | English (US) |
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Pages (from-to) | 265-275 |

Number of pages | 11 |

Journal | Journal of Pure and Applied Algebra |

Volume | 130 |

Issue number | 3 |

DOIs | |

State | Published - Sep 17 1998 |

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## Cite this

*Journal of Pure and Applied Algebra*,

*130*(3), 265-275. https://doi.org/10.1016/S0022-4049(97)00097-2