Integral closure and generic elements
Let \((R,m)\) be a formally equidimensional local ring with \(depth\;R\geq 2\), \(I=(a_1,\dots,a_n)\) an \(m\)-primary ideal of \(R\) and \(T=R[x_1,\dots,x_n]/(\Sigma_{i=1}^n a_ix_i)\). Then the integral closure of \(IT\) in \(T\) is generated by the integral closure of \(I\) in \(R\). Now let \((A,m)\) be a regular local ring, \(I\subset A\) an ideal such that the embedding dimension of \(A/I\) is at most two and \(\bar I\) the integral closure of \(I\) in \(A\). Then \(I^{n-1}{\bar I}\) is the integral closure of \(I^n\) for all \(n\geq 1\).
scientific article; zbMATH DE number 5888484