Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings
This article extends earlier work of \textit{F. Delgado de la Mata} [Manuscr. Math. 59, 347-374 (1987; Zbl 0611.14025)] to characterize the Gorenstein property for a one-dimensional local analytically reduced Cohen Macaulay ring \((R, {\mathfrak m}, k)\). Associated to such a ring one has \(h\) discrete pseudo-valuation rings \(V_ i\), and in turn regular maximal ideals \({\mathfrak n}_ i\) of \(S\), the integral closure of \(R\) in its total ring of quotients. Let \(f_ i = [S/{\mathfrak n}_ i : k]\). For each \(i \in \{1, \dots, h\}\) and multivector \(\alpha \in \mathbb{Z}^ h\) there is \(k\)-vector space \(C (\alpha, i)\) defined in terms of the valuations; likewise \(\gamma \in \mathbb{Z}^ h\) is associated to the conductor \({\mathfrak f} = (R:_ S S)\). The main result, reformulating the classical equality \(\ell_ R (S/{\mathfrak f}) = 2 \ell_ R (R/{\mathfrak f})\), is that \(R\) is Gorenstein if and only if \(\dim C (\alpha,i) + \dim C (\gamma - \alpha - e_ i,k) = f_ i\) for all \(\alpha\) and \(i\). If all \(f_ i = 1\), one calls \(R\) residually rational. With this additional hypothesis, the authors define a semigroup \(\Gamma\) of values of \(R\), and show that if \(\Gamma\) is symmetric, then \(R\) is Gorenstein (and conversely, if \(k\) is infinite).
one- dimensional local analytically reduced Cohen Macaulay ring
scientific article; zbMATH DE number 624946
scientific article