A differential criterion for complete intersections (Q1363140)

Let \(A\) be a noetherian ring whose maximal spectrum has dimension at most 1. For instance, \(A\) can be a noetherian local ring or an order in a number field. Let \(B\) be a finite projective \(A\)-algebra that becomes étale over the total ring of quotients of \(A\). In this note it is shown that \(B\) is of the form \(A[X_1,\dots,X_n]/(f_1,\dots,f_n)\) if and only if the Fitting ideal \(\text{Fit}_B(\Omega_{B/A})\) of the module of differentials of \(B\) over \(A\) is free of rank 1 as a \(B\)-module. In particular, the ring of integers in a number field \(K\) is of the form \(\mathbb{Z}[X_1,\dots,X_n]/(f_1,\dots,f_n)\) if and only if the different of \(K\) over \(\mathbb{Q}\) is a principal ideal.