Number of generators of derivation modules of some one-dimensional local domain (Q6594871)

The paper under review studies the set of generators of the modules of derivations of some one-dimensional local domains.\N\NThe author starts by providing an upper bound for \(\mu(\operatorname{Der}_k(R))\), the minimum number of generators of the derivation module of \(R\), when \(R\) is a Gorenstein ring of the form \(k[\![x_1(t),\dots,x_n(t)]\!]\), where \(x_i(t)\) is a polynomial on the indeterminate \(t\). In this case, \(\mu(\operatorname{Der}_k(R))\) is less than or equal the multiplicity of \(R\) (which is defined by the author as the multiplicity of the value semigroup of \(R\)).\N\NIn the third section, the author proves that if \(R=k[\![t^n,t^{n+1},\dots,t^{2n-3},t^{2n-2}+t^{2n-1}]\!]\) is Gorenstein, then \(\operatorname{Der}_k(R)\) is generated by \(\left\{ t^i\frac{\partial}{\partial t} : i\in\{n+1,\dots,2n\}\right\}\).\N\NThe fourth section is devoted to the study of the derivation module of rings of the form \(R=k[\![t^{n_1},\dots, t^{n_{k-1}},t^{n_k}+t^c]\!]\), with \(n_1,\dots,n_k,c\) positive integers such that (1) \(n_1< \dots < n_k < c\), and (2) the value semigroup of \(R\) is minimally generated by \(\{n_1,\dots,n_k\}\) and has Frobenius number \(c\). In this setting \(\operatorname{Der}_k(R)\) is minimally generated by \(\left\{t^{n_i+1}\frac{\partial}{\partial t} : i\in\{1,\dots,k\} \right\}\cup \left\{ t^{c+1}\frac{\partial}{\partial t}\right\}\).\N\NThe fith section studies a particular case of a complete intersection of the form \(k[\![_1,\dots,x_n ]\!]/I\) with \(I\) generated by polynomials of the form \(f+1\) with \(f\) homogeneus and fulfilling some extra properties.\N\NThe last section is devoted to semigroup rings of numerical semigroups, that is, \(R=k[\![S]\!]\), with \(S\) a numerical semigroup. The author studies the cases when \(S\) is a maximal embedding dimension numerical semigroup and \(S\) is irreducible. Let \(S\) be a numerical semigroup with multiplicity \(m\), and let \(M=S\setminus\{0\}\) be its maximal ideal. The author seems to have missed the fact that the results presented related to \(M^{-1}=S-M=\{x\in \mathbb{Z} : x+M\subseteq S\}\), are particular instances of the fact that \(\operatorname{PF}(S)=-m+\operatorname{Maximals}_{\le_S}(\operatorname{Ap}(S,m))\) (see Chapter 1 in [\textit{J. C. Rosales} and \textit{P. A. García-Sánchez}, Numerical semigroups. Dordrecht: Springer (2009; Zbl 1220.20047)]), where \(a\le_S b\), for \(a\) and \(b\) integers, if \(b-a\in S\), and \(\operatorname{PF}(S)\) is the set of integers \(z\) such that \(z+M\subseteq S\), with \(M=S\setminus\{0\}\) (whith the author's notation, \(M^{-1}=S\cup \operatorname{PF}(S)\)). For the case \(S\) is of maximal embedding dimension, \(\operatorname{PF}(S)=-m+(\operatorname{Ap}(S,m)\setminus\{0\})\); for the case \(S\) is symmetric \(\operatorname{PF}(S)=\{\operatorname{F}(S)\}\); and for the case \(S\) is pseudo-symmetric \(\operatorname{PF}(S)=\{\operatorname{F}(S)/2,\operatorname{F}(S)\}\) (see for instance Chapters 2 and 3 in [\textit{J. C. Rosales} and \textit{P. A. García-Sánchez}, Numerical semigroups. Dordrecht: Springer (2009; Zbl 1220.20047)]).