Proposition 3

Let $A$ be a noetherian ring and $J$ an ideal of $A$ such that $A/J$ is regular. The following conditions are equivalent:

(i) the ideal $J$ is completely secant; (ii) for every prime (resp. maximal) ideal $p$ of $A$ containing $J$, the ring $A_p$ is regular; (iii) the separated completion of $A$ for the $J$-adic topology is regular.

According to Th. 1, condition (i) means that for every prime (resp. maximal) ideal $p$ of $A$ containing $J$, the ideal $J_p$ of the local ring $A_p$ is generated by a completely secant sequence for $A_p$. Since $A_p/J_p$ is regular by hypothesis, this latter condition is equivalent to the regularity of $A_p$ (VIII, § 5, No. 3, Prop. 2 and its Cor. 2); this proves the equivalence of (i) and (ii). The equivalence of (ii) and (iii) follows from § 4, No. 2, Cor. 3 of Prop. 4.