Proposition 6

Let $\rho : A \to B$ be a homomorphism of noetherian rings, and $J$ an ideal of $B$. The following conditions are equivalent:

(i) the $A$-module $B/J$ is flat and the ideal $J$ is completely secant;

(ii) for every $q \in V(J)$, the $A$-module $B_q$ is flat and, for every $A$-algebra $A'$ such that the ring $A' \otimes_A B$ is noetherian, the ideal $J(A' \otimes_A B)$ of $A' \otimes_A B$ is completely secant;

(iii) for every maximal ideal $n$ of $B$ containing $J$, the $A$-module $B_n$ is flat and the ideal $J(\kappa(\rho^{-1}(n)) \otimes_A B_n)$ of $\kappa(\rho^{-1}(n)) \otimes_A B_n$ is completely secant.

By Th. 1 of No. 2, condition (i) is equivalent to the following condition (i') (resp. (i'')):

(i') (resp. (i'')) for every prime ideal (resp. maximal ideal) $q$ of $B$ containing $J$, the $A_{\rho^{-1}(q)}$-module $B_q/J_q$ is flat and the ideal $J_q$ of $B_q$ is generated by a $B_q$-regular sequence.

(i') $\Rightarrow$ (ii): assume that (i') holds. Let $A'$ be an $A$-algebra such that the ring $B' = A' \otimes_A B$ is noetherian. Let $q$ be a prime ideal of $B$ containing $J$; put $p = \rho^{-1}(q)$. Since $B'_q$ is identified with $A'_p \otimes_{A_p} B_q$, it follows from the implication (iv) $\Rightarrow$ (ii) of Prop. 5 (No. 5) that the ideal $JB'_q$ of $B'_q$ is generated by a $B'_q$-regular sequence and that $B_q$ is flat over $A_p$, hence over $A$. Moreover, for every prime ideal $r$ of $B'$ containing $JB'$, the inverse image $q$ of $r$ in $B$ contains $J$, and $B'_r$ is identified with a ring of fractions of $B'_q$. Hence $JB'_r$ is a completely secant ideal of $B'_r$, and $JB'$ is a completely secant ideal of $B'$.

(ii) $\Rightarrow$ (iii): this is trivial.

(iii) $\Rightarrow$ (i''): suppose that (iii) is satisfied. Let $n$ be a maximal ideal of $B$ containing $J$; put $m = \rho^{-1}(n)$. Let $x$ be a sequence of elements of $J_n$ whose images in $J_n/nJ_n$ form a basis of this $\kappa(n)$-vector space. The sequence $x$ generates $J_n$; by Prop. 1 of No. 1, it is $(\kappa(m) \otimes_A B_n)$-regular. Since the $A_m$-module $B_n$ is flat by assumption, it follows from the implication (iii) $\Rightarrow$ (iv) of Prop. 5 that condition (i'') is satisfied.