On equimultiple ideals
The following typifies the nature of the results: Let \(R\) be a local ring with maximal ideal \(M\). Let \(I\) and \(J\) be ideals such that \(I+J\) is \(M\)- primary. Let \(G(I) = \bigoplus^ \infty_{s=0} I^ s/I^{s+1}\) be the associated graded ring. Then: (a) \(G(I) \otimes_ R G(J)\) has dimension \(d+g\) where \(d, g\) are the analytic spreads of \(I, J\) respectively. (b) If \(I\) and \(J\) are analytically disjoint ideals, each generated by regular sequences, then the tensor ring \(G(I) \otimes_ R G(J)\) is a Cohen-Macaulay ring of dimension \(\dim R\). (c) If \(I\) and \(J\) are analytically disjoint ideals such that \(I + J = M\), with \(I\) containing \(X\) such that \(I^ 2 = IX\), and \(J\) containing \(Y\) such that \(J^ 2 = JY\) with each of \(X\) and \(Y\) being generated by regular sequences, then \(G(I) \otimes_ R G(J)\) is Cohen-Macaulay of dimension \(\dim R\).
scientific article; zbMATH DE number 567416
scientific article