Proposition 10

*Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, $N$ a finitely generated B-module and $y = (y_1, \ldots, y_s)$ a sequence of elements of $m_B$. Let $\mathfrak{y}$ denote the ideal of $B$ generated by this sequence. The following conditions are equivalent:

(i) the A-module $N/\mathfrak{y}N$ is flat and the sequence $y$ is N-regular; (ii) the A-module $N$ is flat and the sequence $y$ is $(\kappa_A \otimes_A N)$-regular.*

When these conditions are satisfied, for every A-module $M$, the sequence $y$ is $M \otimes_A N$-regular.

Let us prove the equivalence of (i) and (ii) by induction on $s$. The case $s = 0$ being evident, suppose $s \geqslant 1$; let $y'$ denote the sequence $(y_1, \ldots, y_{s-1})$ and $\mathfrak{y}'$ the ideal of $B$ generated by it. By lemma 2 applied to the B-module $N/\mathfrak{y}'N$ and to the element $y_s$ of $B$, condition (i) is equivalent to

(i') the A-module $N/\mathfrak{y}'N$ is flat, the sequence $y'$ is N-regular, and the homothety of ratio $y_s$ in $\kappa_A \otimes_A (N/\mathfrak{y}'N) = (\kappa_A \otimes_A N)/\mathfrak{y}'(\kappa_A \otimes_A N)$ is injective.

This condition is equivalent to (ii) by the induction hypothesis.

The last assertion follows likewise by induction on $s$ from the last assertion of lemma 2.