One-dimensional almost Gorenstein rings
Let \(A\) be an one-dimensional local Cohen-Macaulay ring with finite closure \(\bar A\). It is well-known that \(\ell (\bar A/A) \geq \ell (A/C) + \text{type}(A)-1\), where \(C = A:\bar A\). If \(A\) is a Gorenstein ring, then \(\text{type}(A) = 1\) and \(\ell(\bar A/A) = \ell(A/C)\). For this reason the authors call \(A\) an almost Gorenstein ring if \(\ell(\bar A/A) = \ell(A/C) + \text{type}(A) - 1\). They study this class of local rings under the assumption that \(A\) has a canonical ideal \(K\) such that \(A \subseteq K \subseteq \bar A\). If \(A\) is analytically irreducible and residual, one can assign with \(A\) a numerical semigroup \(v(A)\). \textit{E. Kunz} [Proc. Am. Math. Soc. 25, 748-751 (1970; Zbl 0197.31401)] proved that \(A\) is Gorenstein if and only if \(v(A)\) is symmetric. Inspired of this result, the authors show that almost Gorenstein rings and some of their properties can be similarly characterized in terms of \(v(A)\).
scientific article; zbMATH DE number 1013931
scientific article