Resumen de A differential criterion for complete intersections

Bart de Smit

• 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 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.