The index of a numerical semigroup ring (Q392139)

In this article, the author studies the index of semigroup rings. Precisely, Theorem 2.2 shows a concrete formula for the index of three generated complete intersection numerical semigroups. Moreover if the semigroup is Gorenstein, Corollary 3.3 shows that the index is computed with respect to the Frobenius number and the generators of the semigroup. Other applications of these results are given in Proposition 3.9 and Remarks 3.1 and 3.5.NEWLINENEWLINELet \(\mathrm{mult}(S)\), \(\mathrm{index}(S)\) and \(\mathrm{codim}(S)\) be respectively the multiplicity, the index and the codimension of a noetherian local ring \(S\). It is known that \(\mathrm{mult}(S)-\mathrm{index}(S)-\mathrm{codim}(S)+1\) is always strictly greater than zero.NEWLINENEWLINEThe main motivation for the studies in the article is to give a negative answer to the following question: When \(S\) is an complete intersection ring, is \(\mathrm{mult}(S)-\mathrm{index}(S)-\mathrm{codim}(S)+1\) bounded above by a constant?NEWLINENEWLINELet \(k\) be an infinite field and consider the semigroup ring \(R_n\) associated to the numerical semigroup generated by \(4n\), \((4n+1)(2n-1)\) and \((4n+1)(2n+1)\). If \(n\geq 2\) then, in Example 2.5, the author shows that \(R_n\) is a complete intersection ring and NEWLINE\[NEWLINE\mathrm{mult}(R_n)-\mathrm{index}(R_n)-\mathrm{codim}(R_n)+1=2n-3.NEWLINE\]NEWLINE This gives a negative answer to the question.