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Type and conductor of simplicial affine semigroups. (English) Zbl 1475.13048

Let \(S\), \(\mathbb{K}\) and \(\mathbb{K}[S]\) be a simplicial affine semigroup, a field and the affine semigroup ring, respectively. In this paper, the auhtors characterize the property of being Cohen-Macaulay and Buchsbaum using the Apery set of \(S\) for the ring \(\mathbb{K}[S]\).
They also prove that if \(\mathbb{K}[S]\) is Cohen-Macaulay then the type of \(S\) is bounded and they give this boundary. Morever, it is shown that if this ring is not Cohen-Macaulay, then the type of \(S\) is unbounded.
If we define the normalization of \(S\) as \(\bar{S}=\{a\in\mbox{group}(S):na\in S\mbox{ for some }n\in\mathbb{N}\}\), we say that \(S\) is normal if \(S=\bar{S}\). Characterization of normal simplicial affine semigroups can be found in this paper using its Apery set.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
20M25 Semigroup rings, multiplicative semigroups of rings
05E40 Combinatorial aspects of commutative algebra
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