Proposition 5

Let $x=(x_1,\ldots,x_n)$ be a sequence of elements of $A$. If for $i=1,\ldots,n$, the homothety of ratio $x_i$ in the $A$-module $M/(x_1M+\cdots+x_{i-1}M)$ is injective, the sequence $x$ is completely secant for $M$.

A sequence satisfying the conditions of the proposition is said to be regular for M, or M-regular. We shall note that this property depends in general on the order of the $x_i$; for example the sequence $(1, 0)$ is always M-regular, whereas the sequence $(0, 1)$ is so only if M is zero. On the other hand, the fact that a sequence is completely secant does not depend on the order of the terms.

Let us prove the proposition by induction on $n$, the case $n = 0$ being immediate. Put $x' = (x_1, ..., x_{n-1})$; if the sequence $x$ is M-regular, the sequence $x'$ is M-regular and multiplication by $x_n$ in $M/(x')M$ is injective; by the induction hypothesis, we have $H_i(x', M) = 0$ for $i > 0$; it then follows from cor. 3 of X, p. 157, that $H_i(x, M) = 0$ for $i > 0$.