On the index of reducibility of parameter ideals and Cohen-Macaulayness in a local ring
For a module \(M\) over a local ring \(A\) it is well-known that if \(M\) is Cohen-Macaulay then the index of reducibility of a submodule \(xM\) is independent of the choice of the parameter ideal \(x\) [cf. \textit{D. G. Northcott} and \textit{D. Rees}, Q. J. Math., Oxf. II. Ser. 8, 11-127 (1957; Zbl 0077.260)]. The converse is false, unless the common index is 1. The author seeks extra conditions to produce a partial converse. The main result states that, with \(n\) denoting the maximum index of \(xM\) as \(x\) varies, if (i) \(A\) is the homomorphic image of a Cohen-Macaulay ring, (ii) \(M_ P\) is Cohen-Macaulay when \(\dim M_ P<n\), and (iii) all associated prime ideals of \(M\) have the same dimension, then \(M\) is Cohen-Macaulay. It follows that \(A\) unmixed and every parameter ideal has index 2 imply \(A\) is Cohen-Macaulay. Also for constant index \(n(>2)\) if \(A\) satisfies Serre's condition \(S_{n-1}\) then \(A\) is Cohen- Macaulay. The author deduces his main theorem from a stronger more technical result which includes replacing (i) above by the existence of a dualizing complex.
scientific article; zbMATH DE number 633567
scientific article