A construction of Gorenstein rings (Q858723): Difference between revisions

Let \(R\) be a local Cohen-Macaulay ring and \(M\) be an \(R\)-module. Recall that the trivial extension, \(R\ltimes M\), of \(R\) by \(M\), is the \(R\)-algebra \(R\oplus M\) whose multiplication makes \((1,M)\) an ideal which squares to zero. A theorem of \textit{I. Reiten} [Proc. Am. Math. Soc. 32, 417--420 (1972; Zbl 0235.13016)], says that \(R\ltimes M\) is a Gorenstein ring if and only if \(M\) is a canonical module of \(R\). The present paper considers a similar construction \(R\bowtie I\), where \(I\) is an ideal of \(R\). (The ring \(R\bowtie I\) is reduced whenever \(R\) is reduced, so \(\bowtie\) is definitely different than \(\ltimes\).) The result in the present paper is parallel to Reiten's result: the ring \(R\bowtie I\) is Gorenstein if and only if \(I\) is isomorphic to a canonical module of \(R\). The construction is applied to rings of the form \(R=k[[t_1,\dots,t_n]]/(P_1\cap \dots \cap P_h)\), where each \(P_i\) is a prime ideal of height \(n-1\). Such rings are called algebroid curves with \(h\) branches. It is shown that \(R\bowtie I\) is an algebroid curve with \(2h\) branches.