The conductor of one-dimensional Gorenstein rings in their blowing-up
Let (A,M) be a one-dimensional Cohen-Macaulay local ring with multiplicity \(e>1\), Hilbert function H with \(H(n)=\lambda (M^ n/M^{n+1})\) and first neighbourhood ring B obtained by blowing up M. The note contributes to the study if the conductor \(I=Ann_ A(B/A)\) of A in B in terms of two indices \(\nu =Min\{n| H(n)=e\}\) and \(\mu =Max\{n| H(n)=\left( \begin{matrix} n+r\\ r\end{matrix} \right)\}\), where \(r+1=\lambda (M/M^ 2)\) is the embedding dimension. It is shown that \(M^{e-1}\subseteq M^{\nu}\subseteq I\subseteq M^{\mu}\) and in the extreme cases \((i)\quad I=M^{\mu}\) if, and only if, \(e=\left( \begin{matrix} \nu +r\\ r\end{matrix} \right)\) so \(\mu =\nu\) and H is determined by r and e, and \((ii)\quad M^{e-1}implies\) \(r=1\), the converse of a result of Northcott and Matlis [see \textit{E. Matlis}, ``1-dimensional Cohen-Macaulay rings'', Lect. Notes Math. 327 (1973; Zbl 0264.13012)]. In the special case of A Gorenstein the authors prove that \(I\subseteq M^ 2\) or \(I=M\) and \(r=1\); moreover \(I+Ax=Ax:_ AM\) for any superficial element x and if in addition the associated graded ring is Cohen-Macaulay (Gorenstein) then \(I\subseteq M^{\nu}+Ax\) \((I=M^{\nu}).\) The reader should beware of proofs containing misprints.
scientific article; zbMATH DE number 4165032
scientific article