The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two
Let \((A,\mathfrak m)\) be a noetherian local ring, \(Q\) a parameter ideal in \(A\) and \(I=Q:\mathfrak m\). The problem of when the equality \(I^2=IQ\) holds true is discussed in the case where \(A\) is a Buchsbaum ring. The purposes of the paper are to show the following Theorem 1.1 and to give examples such that the equality stated above does not hold. Theorem 1.1: Assume that \(A\) is a Buchsbaum local ring with multiplicity two and \(\text{depth}\,A>0\). Then the equality \(I^2=IQ\) holds true for any parameter ideal \(Q\) in \(A\), where \(I=Q:\mathfrak m\). Result in section\,4: For given integers \(1 \leq d < m\), there exists a Buchsbaum local ring \((A,\mathfrak m)\) with \(\dim A = d\), \(\text{depth}\,A = d-1\) and multiplicity \(2m\) such that \(A\) contains a parameter ideal \(Q\) which is a minimal reduction of \(\mathfrak m\) and \(I^2 \neq QI\) but \(I^3 = QI^2\), where \(I = Q:\mathfrak m\).
scientific article; zbMATH DE number 2145474