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