Topological results in combinatorics.

*(English)*Zbl 0585.57014The paper studies the ring K(\(\Delta)\) associated with a finite simplicial complex \(\Delta\) and a field K. The main result is that the number \(\alpha (\Delta)=n-h(\Delta)\) (where \(n+1\) is the number of vertices of \(\Delta\) and h(\(\Delta)\) the homological dimension of K(\(\Delta)\)) can be described by the (reduced) global cohomology groups and the local cohomology groups of the topological space \(| \Delta |\) underlying \(\Delta\) ; more precisely, \(\alpha\) (\(\Delta)\) is shown to be the lowest dimension for which at least one of these cohomology groups does not vanish. As a consequence, \(\alpha\) (\(\Delta)\) is a topological invariant, i.e. depends only on the topological space \(| \Delta |\) and not on the triangulation \(\Delta\).

Since K(\(\Delta)\) is a Cohen-Macaulay ring if and only if \((n+1)- h(\Delta)=\dim \Delta +1\) \((=Krull\) dimension of K(\(\Delta)\)) it follows in particular that this property is a topological invariant.

There are applications for the case that \(\Delta\) is the complex of chains of a partially ordered set in which all maximal chains have the same length (as for instance the first barycentric subdivision of a triangulated manifold).

K(\(\Delta)\) is defined to be the quotient ring of the polynomial ring \(K[x_ 0,...,x_ n]\), with the vertices \(x_ 0,...,x_ n\) of \(\Delta\) as indeterminates, by the ideal which is generated by all square-free monomials whose corresponding vertices do not span a simplex of \(\Delta\). The topological invariance of the Cohen-Macaulay condition had been conjectured by M. Hochster [Proc. 2nd. Okla. Conf. 1975, 171-223 (1977; Zbl 0351.13009)] and almost proved by G. A. Reisner [Adv. Math. 21, 30-49 (1976; Zbl 0345.13017)] who translated this condition into a condition involving the links of simplices in \(\Delta\). Subsequently, R. P. Stanley [Proc. Conf. Berlin 1976, 51-62 (1977; Zbl 0376.55007)] indicated how to transform Reisner’s condition into a condition on the global and local cohomology groups of \(| \Delta |\), thus proving its topological invariance (loc. cit., Theorem 5). These indications are fully worked out in the present paper, independently and prior to the proof of the more general main result, the topological invariance of \(\alpha\) (\(\Delta)\), which was only conjectured by Stanley.

Now this result is also obtained by connecting \(\alpha\) (\(\Delta)\) to cohomology groups of \(| \Delta |\). The starting point is a formula of Hochster (loc. cit.) for the Betti numbers (of a minimal free resolution) of K(\(\Delta)\) in terms of the simplicial cohomology of full subcomplexes of \(\Delta\). Then techniques similar to the proof of Poincaré duality for triangulated manifolds are used to relate the vanishing of the local cohomology groups to information about these simplicial cohomology groups.

This elementary but involved argument can be replaced by a generalization of the Lefschetz duality theorem in terms of Čech cohomology with coefficients in the presheaf of local homology groups of \(| \Delta |\). This duality theorem was found by the author (in the context of the present paper, as it seems) as a consequence of the interpretation of C. McCrory [Trans. Am. Math. Soc. 250, 147-166 (1979; Zbl 0363.57014)] of Zeeman’s spectral sequence.

The special case of triangulation of the sphere or the ball was already known; in the case of a sphere, K(\(\Delta)\) is even a Gorenstein ring (Hochster [loc. cit.], Corollary 6.8)). In these situations, the author shows separately (at the beginning of the paper) how the results can be derived very easily from Hochster’s formula by using classical Alexander duality; this is much simpler than Hochster’s original argument or the author’s arguments for the general case.

Since K(\(\Delta)\) is a Cohen-Macaulay ring if and only if \((n+1)- h(\Delta)=\dim \Delta +1\) \((=Krull\) dimension of K(\(\Delta)\)) it follows in particular that this property is a topological invariant.

There are applications for the case that \(\Delta\) is the complex of chains of a partially ordered set in which all maximal chains have the same length (as for instance the first barycentric subdivision of a triangulated manifold).

K(\(\Delta)\) is defined to be the quotient ring of the polynomial ring \(K[x_ 0,...,x_ n]\), with the vertices \(x_ 0,...,x_ n\) of \(\Delta\) as indeterminates, by the ideal which is generated by all square-free monomials whose corresponding vertices do not span a simplex of \(\Delta\). The topological invariance of the Cohen-Macaulay condition had been conjectured by M. Hochster [Proc. 2nd. Okla. Conf. 1975, 171-223 (1977; Zbl 0351.13009)] and almost proved by G. A. Reisner [Adv. Math. 21, 30-49 (1976; Zbl 0345.13017)] who translated this condition into a condition involving the links of simplices in \(\Delta\). Subsequently, R. P. Stanley [Proc. Conf. Berlin 1976, 51-62 (1977; Zbl 0376.55007)] indicated how to transform Reisner’s condition into a condition on the global and local cohomology groups of \(| \Delta |\), thus proving its topological invariance (loc. cit., Theorem 5). These indications are fully worked out in the present paper, independently and prior to the proof of the more general main result, the topological invariance of \(\alpha\) (\(\Delta)\), which was only conjectured by Stanley.

Now this result is also obtained by connecting \(\alpha\) (\(\Delta)\) to cohomology groups of \(| \Delta |\). The starting point is a formula of Hochster (loc. cit.) for the Betti numbers (of a minimal free resolution) of K(\(\Delta)\) in terms of the simplicial cohomology of full subcomplexes of \(\Delta\). Then techniques similar to the proof of Poincaré duality for triangulated manifolds are used to relate the vanishing of the local cohomology groups to information about these simplicial cohomology groups.

This elementary but involved argument can be replaced by a generalization of the Lefschetz duality theorem in terms of Čech cohomology with coefficients in the presheaf of local homology groups of \(| \Delta |\). This duality theorem was found by the author (in the context of the present paper, as it seems) as a consequence of the interpretation of C. McCrory [Trans. Am. Math. Soc. 250, 147-166 (1979; Zbl 0363.57014)] of Zeeman’s spectral sequence.

The special case of triangulation of the sphere or the ball was already known; in the case of a sphere, K(\(\Delta)\) is even a Gorenstein ring (Hochster [loc. cit.], Corollary 6.8)). In these situations, the author shows separately (at the beginning of the paper) how the results can be derived very easily from Hochster’s formula by using classical Alexander duality; this is much simpler than Hochster’s original argument or the author’s arguments for the general case.

Reviewer: H.Hähl

##### MSC:

57Q99 | PL-topology |

13D05 | Homological dimension and commutative rings |

55M05 | Duality in algebraic topology |

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

55U05 | Abstract complexes in algebraic topology |

55N35 | Other homology theories in algebraic topology |

55N05 | Čech types |