Corollary 3

Let $A$ be a noetherian ring, $J$ an ideal of $A$ and $\hat{A}$ the separated completion of $A$ for the $J$-adic topology.

a) In order that the ring $\hat{A}$ be regular, it is necessary and sufficient that, for every maximal ideal $m$ of $A$ containing $J$, the ring $A_m$ be regular.

b) If the ring $A$ is regular, the ring $\hat{A}$ is regular. If the ring $\hat{A}$ is regular and the ideal $J$ contained in the radical of $A$, the ring $A$ is regular.