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Graded algebras with cyclotomic Hilbert series. (English) Zbl 1469.13035

In [E.-A. Ciolan et al., SIAM J. Discrete Math. 30, No. 2, 650–668 (2016; Zbl 1343.20060)], the following conjecture can be found: “A numerical semigroup is cyclotomic if and only if its semigroup ring is a complete intersection”. In this paper, the authors consider a more general situation with \(R\) being a positively graded algebra over a field \(K\) and reformulate this conjecture in terms of graded domains.
In this context, it is proven that, in the case of Koszul algebras, it is equivalent to be cyclotomic to be a complete intersection. The authors also prove that, under the assumption that \(R\) is standard graded and its \(h\)-polynomial is irreducible over \(\mathbb{Q}\), cyclotomic algebras coincide with complete intersections.

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13A02 Graded rings
16S37 Quadratic and Koszul algebras
20M14 Commutative semigroups
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Citations:

Zbl 1343.20060

Software:

Macaulay2; Normaliz
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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