Gorenstein curves and symmetry of the semigroup of values
Let \({\mathfrak O}\) be the local ring of a reduced algebroid curve over an algebraically closed field K, and let \(p_ 1,...,p_ d\) be its minimal primes. Then \({\mathfrak O}_ i={\mathfrak O}/p_ i\) is an integral domain and its normalization \(\bar{\mathfrak O}_ i\) is a discrete valuation ring; let \(v_ i\) be the corresponding valuation. The natural injection \({\mathfrak O}\to {\mathfrak O}_ 1\times...\times {\mathfrak O}_ d\) and the \(v_ i\) define an additive subsemigroup S of \({\mathbb{N}}^ d\). The conductor \(\{z\in \bar {\mathfrak O}: z\bar {\mathfrak O}\subset {\mathfrak O}\}\), where \(\bar{\mathfrak O}\) is the integral closure of \({\mathfrak O}\) in its total quotient ring, is a non-zero principal ideal \(\zeta\) \(\bar{\mathfrak O}\) in \(\bar{\mathfrak O}=\bar{\mathfrak O}_ 1\times...\times \bar{\mathfrak O}_ d\), and if \(\zeta =(z_ 1,...,z_ d)\) and \(\delta =(v_ 1(z_ 1),...,v_ d(z_ d))\) then \(\delta+{\mathbb{N}}^ d\subset S\). Set \(\tau=\delta - (1,...,1)\). For \(\alpha\in {\mathbb{Z}}^ d\) set \(\Delta_ i(\alpha)=\{\beta \in S:\quad \beta_ i=\alpha_ i\) and \(\beta_ j>\alpha_ j\) for \(j\neq i\}\) and \(\Delta(\alpha)= \Delta_ 1(\alpha)\cup...\cup \Delta_ d(\alpha)\). The main result of this paper says that \({\mathfrak O}\) is Gorenstein if and only if the semigroup S satisfies the following ``symmetry'' condition: for \(\alpha \in {\mathbb{Z}}^ d\), one has \(\alpha\in S\) if and only if \(\Delta(\tau -\alpha)=\emptyset\). This generalizes the well known result for \(d=1\), due to \textit{E. Kunz} [Proc. Am. Math. Soc. 25, 748-751 (1970; Zbl 0197.314)]. The proof is considerably more complicated than the case of \(d=1\).
scientific article; zbMATH DE number 4132313