Theorem 1

Consider the following conditions: (i) The sequence $x$ is M-regular (X, p. 158). (ii) The sequence $x$ is completely secant for M (X, p. 157, def. 2). (iii) One has $H_1(x, M) = 0$. (iv) The homomorphism $\alpha_M^x : (A[X_1, ..., X_n]/\mathfrak{d}) \otimes_A M \to \bigoplus_{r \geq 0} x^r M$ is bijective. (v) The homomorphism $\beta_M^x : (A/x)[X_1, ..., X_n] \otimes_A M \to \bigoplus_{r \geq 0} (x^r M/x^{r+1} M)$ is bijective.

Then one has the implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\Leftrightarrow$ (iv) $\Rightarrow$ (v). If for $1 \leq i \leq n$ the A-module $M/(x_1 M + \cdots + x_{i-1} M)$ is separated for the $x$-adic topology, the conditions (i) to (v) are equivalent.

The theorem will be proved in Nos 8 and 9.