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Affine semigroups and Cohen-Macaulay rings generated by monomials. (English) Zbl 0631.13020
Let \(S\) be an affine semigroup, i.e. a finitely generated submonoid of the additive monoid \({\mathbb{N}}^ n\), and \(k[S]\) the semigroup ring of \(S\) over a field \(k\). The authors’ main result is a criterion for \(k[S]\) to be Cohen-Macaulay in terms of certain numerical and topological properties of \(S\). Following S. Goto and K. Watanabe [Tokyo J. Math. 1, 237-261 (1978; Zbl 0406.13007)] they first introduce a suitable extension \(S'\) of \(S\). Goto and Watanabe claimed the coincidence of \(S'\) with \(S\) be necessary and sufficient for \(k[S]\) to be Cohen-Macaulay. The authors show that this is not true in general. Actually the condition is not sufficient as they demonstrate by a counterexample. On the other hand they prove that it is in fact necessary, using the Cousin complex of a commutative noetherian ring introduced by R. Y. Sharp [Math. Z. 112, 340-356 (1969; Zbl 0182.061)]. (The arguments of Goto and Watanabe do not work even for this implication.) The investigation of the local cohomology modules of \(k[S']\) leads to the above criterion and an analogous criterion for \(k[S]\) to be Gorenstein. From this Goto’s and Hochster’s results are derived, who treated the cases in which \(S\) is simplicial or normal, respectively. In the last part of their paper the authors show that the vanishing of local cohomology modules (and the Cohen-Macaulayness) of \(k[S]\) depends on the characteristic of \(k\).
Reviewer: U.Vetter

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13B02 Extension theory of commutative rings
20M25 Semigroup rings, multiplicative semigroups of rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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