Proposition 6

a) If $\sum_{i \in I} x_i A = A$, the family $(x_i)_{i \in I}$ is completely secant for M. b) Let $x = (x_1, ..., x_n)$ be a sequence of elements of A. Let $(a_{ij}) \in \mathrm{GL}_n(A)$; put

If the sequence $x$ is completely secant for M, the same is true of the sequence $(y_1, ..., y_n)$. c) Let $k_1, ..., k_n$ be integers $\geqslant 1$; in order that the sequence $(x_1^{k_1}, ..., x_n^{k_n})$ be completely secant for M, it is necessary and sufficient that the sequence $x$ be completely secant for M. Assertion a) follows from cor. 1, p. 148. Let us prove b). Let $f : A^n \to A^n$ be the isomorphism defined by the matrix $^t(a_{ij})$; it follows from X, p. 151, that $1_M \otimes \Lambda(f)$ is an isomorphism of the complex $K.(y, M)$ onto the complex $K.(x, M)$, whence b). To prove c), it is obviously enough to prove that if $k$ is an integer $\geqslant 1$, the sequence $(x_1, ..., x_{n-1}, x_n^k)$ is completely secant for M if and only if the same is true of the sequence $x$. Let $x' = (x_1, ..., x_{n-1})$; by cor. 3, p. 157, the first condition (resp. the second) means that the homothety of ratio $x_n^k$ (resp. $x_n$) is bijective in $H_i(x', M)$ for $i \geqslant 1$ and injective in $M/(x')M$. These two conditions are clearly equivalent, whence c).