Proposition 1

Let $A$ be a ring, $J$ an ideal of $A$, generated by a finite sequence $x = (x_1, \ldots, x_r)$ of elements of $A$. The following conditions are equivalent:

(i) the sequence $x$ is completely secant for $A$ ; (ii) (resp. (ii')) for every prime ideal (resp. maximal ideal) $p \in V(J)$, the image of $x$ in $A_p$ is completely secant for $A_p$ ; (iii) (resp. (iii')) for every prime ideal (resp. maximal ideal) $p \in V(J)$, the ideal $J$ is completely secant at $p$ and the image of $x$ in the $\kappa(p)$-vector space $\kappa(p) \otimes_A J$ forms a basis of it.

When $A$ is noetherian, these conditions are also equivalent to:

(iv) for every integer $n \geq 0$, the $A/J$-module $J^n / J^{n+1}$ is free, and the images of the monomials $x_1^{\alpha_1} \ldots x_r^{\alpha_r}$ of total degree $n$ form a basis of it.

Let $p$ be a prime ideal of $A$; let us denote by $x_p$ the image of the sequence $x$ in $A_p$. For every integer $i \geq 0$, the $A_p$-module $H_i(x_p, A_p)$ is isomorphic to $(H_i(x, A))_p$ (A, X, p. 151, 2)) ; it is zero if $p$ does not contain $J$, since then one has $J_p = A_p$ (A, X, p. 148, cor. 2). The equivalence of (i), (ii) and (ii') follows from this and from II, § 3, No. 3, cor. 2 of th. 1. On the other hand, the equivalence of (ii) and (iii), and that of (ii') and (iii'), follow from § 1, No. 3, cor. 2 of th. 1.

The implication (i) $\Rightarrow$ (iv) is a consequence of th. 1 of A, X, p. 160. Finally, condition (iv) implies by localization the analogous condition for the $(A_p/J_p)$-modules $J_p^n / J_p^{n+1}$ for every $p \in V(J)$, and the latter entails (ii) by cor. 1 of A, X, p. 160.