Proposition 2

Let $A$ be a noetherian ring. The following conditions are equivalent:

(i) $A$ is regular; (ii) every maximal ideal of $A$ is completely secant; (iii) every ideal $J$ of $A$ such that $A/J$ is regular is completely secant.

(i) $\Rightarrow$ (iii): suppose the ring $A$ regular; let $J$ be an ideal of $A$ such that $A/J$ is regular, and let $p$ be a prime ideal of $A$ containing $J$. Then the local rings $A_p$ and $A_p/J_p$ are regular, hence $J_p$ is generated by a completely secant sequence for $A_p$ (VIII, § 5, No. 3, Cor. 2 and Prop. 2), which means that $J$ is completely secant at $p$.

(iii) $\Rightarrow$ (ii): this is clear since a field is a regular ring.

(ii) $\Rightarrow$ (i): let $m$ be a maximal ideal of $A$; under assumption (ii), the maximal ideal $mA_m$ of $A_m$ is generated by a completely secant sequence for $A_m$, hence $A_m$ is regular (VIII, § 5, No. 2, Th. 1). Consequently $A$ is regular.