On the Cohen-Macaulay property in commutative algebra and simplicial topology. (English) Zbl 0686.13008

Smith, Dean E.

Pac. J. Math. 141, No. 1, 165-196 (1990).

A 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.

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

Keywords:

section ring of a sheaf of commutative rings; ring of sections; finite partially ordered set; order topology; finite abstract complex; simplicial complex; Stanley-Reisner ring of a complex; sections of a sheaf of polynomial rings; Cohen-Macaulay