Abstract
Let a and d be two linearly independent vectors in \({\mathbb {N}}^2\), over the field of rational numbers. For a positive integer \(k \ge 2\), consider the sequence \(a, a+d, \ldots , a+kd\) such that the affine semigroup \(S_{a,d,k} = \langle a, a+d, \ldots , a+kd \rangle \) is minimally generated. We study the properties of affine semigroup ring \(K[S_{a,d,k}]\) associated to this semigroup. We prove that \(K[S_{a,d,k}]\) is always Cohen-Macaulay and it is Gorenstein if and only if \(k=2\). For \(k=2,3,4\), we explicitly compute the syzygies, the minimal graded free resolution and the Hilbert series of \(K[S_{a,d,k}].\) We also give a minimal generating set for the defining ideal of \(K[S_{a,d,k}]\) which is also a Gröbner basis. Consequently, we prove that \(K[S_{a,d,k}]\) is Koszul. Finally, we prove that the Castelnuovo–Mumford regularity of \(K[S_{a,d,k}]\) is 1 for any a, d, k.
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11 January 2023
A Correction to this paper has been published: https://doi.org/10.1007/s00233-022-10334-x
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The authors would like to thank the anonymous referee for valuable comments and suggestions.
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Communicated by Nathan Kaplan.
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Bhardwaj, O.P., Sengupta, I. On the algebraic invariants of certain affine semigroup rings. Semigroup Forum 106, 24–50 (2023). https://doi.org/10.1007/s00233-022-10332-z
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DOI: https://doi.org/10.1007/s00233-022-10332-z