On the Cohen-Macaulay property in commutative algebra and simplicial topology.

*(English)*Zbl 0686.13008A ring R is called a “ring of sections” provided R is the section ring of a sheaf (\({\mathcal A},X)\) of commutative rings defined over a base space X which is a finite partially ordered set given the order topology. Regard X as a finite abstract complex, where a chain in X corresponds to a simplex. In specific instances of (\({\mathcal A},X)\), certain algebraic invariants of R are equivalent to certain topological invariants of X. (Author)

The author investigates the depth of factor rings of \(SR(F,\Sigma)\), the Stanley-Reisner ring of a complex \(\Sigma\) with coefficients in a field F. \(SR(F,\Sigma)\) is viewed as the ring of sections of a sheaf of polynomial rings over the partially ordered set of all simplices of \(\Sigma\). The complex \(\Sigma\) is defined to be Cohen-Macaulay (CM) provided the reduced singular cohomology of the link subcomplexes vanish except in maximal degree. The main theorem goes as follows: Let S be the polynomial ring \(S=F[X_ 0,...,X_ n]\), put \(\alpha =n-pd_ SSR(F,\Sigma)\), then the skeleton \(\Sigma^{\alpha}\) is maximal with respect to the property of being CM.

The author investigates the depth of factor rings of \(SR(F,\Sigma)\), the Stanley-Reisner ring of a complex \(\Sigma\) with coefficients in a field F. \(SR(F,\Sigma)\) is viewed as the ring of sections of a sheaf of polynomial rings over the partially ordered set of all simplices of \(\Sigma\). The complex \(\Sigma\) is defined to be Cohen-Macaulay (CM) provided the reduced singular cohomology of the link subcomplexes vanish except in maximal degree. The main theorem goes as follows: Let S be the polynomial ring \(S=F[X_ 0,...,X_ n]\), put \(\alpha =n-pd_ SSR(F,\Sigma)\), then the skeleton \(\Sigma^{\alpha}\) is maximal with respect to the property of being CM.

Reviewer: Y.Felix

##### MSC:

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

55U05 | Abstract complexes in algebraic topology |

13D25 | Complexes (MSC2000) |

55M99 | Classical topics in algebraic topology |

57Q99 | PL-topology |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |