One dimensional local rings of maximal and almost maximal length
Let \((A,{\mathfrak m})\) be a reduced excellent Cohen-Macaulay local ring with infinite residue field. Assume that \(A\) has a canonical module \(\omega\). Let \(C\) be the conductor of \(A\) in its integral closure \(\overline A\). It is well known that if \(A\) is Gorenstein then \(\ell(\overline A/A)=\ell(A/C)\), where \(\ell=\text{length}_ A\). In general, one has the inequality \((*)\) \(\ell(\overline A/A)\leq\ell(A/C)t(A)\), where \(t(A)\) is the Cohen-Macaulay type of \(A\). Thus \((*)\) is an equality if \(A\) is Gorenstein. The converse is false. The authors investigate rings for which \((*)\) is an equality, called rings of maximal length, and also rings for which the equality \(\ell(\overline A/A)=\ell(A/C)t(A)-1\) holds, called rings of almost maximal length. The following are among the several nice results proved: (1) With a suitable choice of \(\omega\), \(A\) has maximal length if and only if \(\omega/C\omega\) is \(A/C\)-free. (2) \(A\) has maximal length if and only if either \(A\) is Gorenstein or there exists an \(x\in {\mathfrak m}\) such that \({\mathfrak m}=(x,C)\) and \(C=x^ p\overline A\) for some \(p\). (3) Suppose \(\overline A\) is local, \(\text{mult}(A)\leq 3\) and \({\mathfrak m}=(x,C)\) for some \(x\in{\mathfrak m}\). Then \(A\) has maximal or almost maximal length. (4) \(A\) has almost maximal length and \(\text{mult}(A)=1+t(A)\) if and only if there exists a transversal \(x\) of \({\mathfrak m}\) such that \({\mathfrak m}=(x,C)\) and \(\ell(C/x^ p\overline A)=1\), where \(p\) is the least integer with \(x^ p\in C\). These results generalize more specific results for semigroup rings obtained earlier by the first author and \textit{F. Curtis} [Semigroup Forum 42, 219-235 (1991; Zbl 0722.20046)].
scientific article; zbMATH DE number 165841
scientific article