Exercise 6, § 9

Let A be a ring, $x = (x_1, ..., x_n)$ a sequence of elements of A, x the ideal $\sum_i x_i$ A. Let

be an exact sequence of A-modules. We suppose that the sequence x is completely transversal for M and that the A-modules $M''/(x_1, M'' + ... + x_{i-1}, M'')$ are separated for the x-adic topology ($1 \leq i \leq n$). Prove that the following conditions are equivalent: α) The sequence x is completely transversal for M''. β) The homomorphism $\bigoplus_r (x^r M'/x^{r+1} M') \to \bigoplus_r (x^r M/x^{r+1} M)$ induced by i is injective. γ) The homomorphism $M'/xM' \to M/xM$ induced by i is injective.