Exercise 6, § 2

The hypotheses of the preceding exercise are retained; for every finitely generated A-module N such that $N/xN$ is of finite length, we denote by $\chi(x, N)$ the integer $\chi(K_\bullet(x, N))$.

a) For every submodule $M'$ of $M$, one has $\chi(x, M) = \chi(x, M') + \chi(x, M/M')$.

b) If $x_1 M = 0$, prove that $\chi(x, M)$ is zero.

c) We denote by $x'$ the sequence $(x_2, \ldots, x_n)$; prove the equality

(use exercise 8 of A, X, p. 207).

d) For $1 \leqslant i \leqslant n$, we denote by $K_i$ the kernel of the homothety of ratio $x_i$ in $M/(x_1 M + \ldots + x_{i-1} M)$. Prove the equality

e) Let $p_1, \ldots, p_n$ be integers $> 1$, $x^P$ the sequence $(x_1^{p_1}, \ldots, x_n^{p_n})$. Prove the equality $\chi(x^P, M) = p_1 \ldots p_n \chi(x, M)$ (treat first the case $n = 1$, then reduce to it with the aid of Cor. 2 of A, X, p. 157).