An algorithm for the Traverso-Swan theorem on seminormal rings
By Traverso-Swan's theorem a reduced ring \(A\) is seminormal if and only if the canonical map \(\text{Pic } A \rightarrow \text{Pic } A[x]\) is an isomorphism. The necessity could follow also from an algorithm presented in this paper. Let \(C\) be a reduced ring and \(f_1,\ldots, f_n\), \(g_1,\ldots, g_n\) polynomials in \(C[x]\) such that \(\sum_i f_ig_i=1\), \(f_1(0)=g_i(0)=1\), \(f_i(0)=g_i(0)=0\) for \(i>1\). Let \(A\) be the subring of \(C\) generated by the coefficients of the polynomials \(m_{ij}=f_ig_j\), \(1\leq i,j\leq n\) and \(B\) be the subring of \(C\) generated by the coefficients of the polynomials \(f_i,g_i\), \(1\leq i\leq n\). Then \(B\) is the seminormal closure of \(A\) in \(B\).
scientific article; zbMATH DE number 5353584