Cohen-Macaulay approximation and multiplicity
Let \((R,m)\) be a Gorenstein local ring. For a finite \(R\)-module \(C\), let \(\delta (C)\) denote the \(\delta\)-invariant defined in the theory of Cohen-Macaulay approximation due to M. Auslander and {R.-O. Buchweitz.} Let \(\Omega^ i (C)\) be the \(i\)-th syzygy in the minimal resolution of \(C\) and set \(\delta^ i (C)=\delta (\Omega^ i (C))\). In the present paper the author defines a new invariant index(R) as the smallest \(n\) such that \(\delta (R/m^ n) \neq 0\). It is shown that, if \(R\) is not regular, one has \[ \text{mult} (R) \geq \text{index} (R)+\text{embdim} R-\text{dim} R-1. \] It follows that if \(\text{mult} (R)= \text{index} (R)\) then \(R\) is a hypersurface ring. The main theorem of this paper says that, conversely, a hypersurface ring \(R=S/xS\) satisfies \(\text{mult} (R)=\text{index} (R)\). The proof is based on a careful study of \(\delta\)-invariants over a hypersurface ring, in particular the following theorem: Let \(R=S/xS\) with \(S\) regular and \(x \in m^ 2_ S\). Let \(I\) be an ideal of \(R\) and \(J\) be its inverse image in \(S\). Then \(x \in m_ SJ \Leftrightarrow \delta (R/I)=0 \Leftrightarrow \delta^ i (R/I)=0\) for all \(i>0\). [Reviewer's note: In subsequent work the author studies the invariant \(\text{index} (R)\) further, and conjectures that it is the minimum of \(n\) such that there exists a parameter ideal \(q\) of \(R\) containing \(m^ n\), and this conjecture was verified for homogeneous Gorenstein rings by \textit{J. Herzog} in Contemp. Math. 159, 95-102 (1994)].
scientific article; zbMATH DE number 130540
scientific article