On the deviation and the type of certain local Cohen-Macaulay rings and numerical semigroups
Let \(R\) be a Noetherian local ring and \(\hat{R}=P/I\), where \(\hat{R}\) is the completion of \(R\) and \(P\) is a regular local ring. Let \(\mu(I)\) be the minimal number of generators of \(I\), then the number \(d(R):=\mu(I)-(\dim(P)-\dim(R))\) is independent of the chosen presentation of \(\hat{R}\) and is called the deviation of \(R\). In this article under review, the authors study certain Cohen-Macaulay rings for which there are close relations between deviation, type and embedding dimension. This relations apply to numerical semigroups (or equivalently monomial curves) and lead also to some further cases, generalizing [\textit{E. Kunz}, Semigroup Forum 93, No. 1, 205--210 (2016; Zbl 1347.20069)] with ring-theoretic proofs, in which a question of \textit{H. S. Wilf} [Am. Math. Mon. 85, 562--565 (1978; Zbl 0387.10009)] has a positive answer.
scientific article; zbMATH DE number 6695610
scientific article