Exercise 9, § 9

Let A be a ring, M an A-module, k an integer $\geq 1$. One says that a family x of elements of A is k-secant for M if one has $H_i(x, M) = 0$ for $1 \leq i \leq k$. A family x is therefore completely secant if and only if it is k-secant for every $k \geq 1$. a) Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of A-modules. Now prove that if the family x is k-secant for M’ and M”, it is so for M. b) If x is k-secant for M and if N is a flat A-module, x is k-secant for $M \otimes_A N$. c) Let $a_1, ..., a_n$ be integers $\geq 1$. Now prove that a sequence $(x_1, ..., x_n)$ is k-secant for M if and only if the same is true of the sequence $(x_1^{a_1}, ..., x_n^{a_n})$. d) For every pair of integers $k, n$ with $1 \leq k < n$, give an example of a ring A, of an A-module M and of a sequence $(x_1, ..., x_n)$ of elements of A which is k-secant but not $(k+1)$-secant for M (use X, p. 159, remark 4).