Exercise 15, § 1

Let $A$ be a local noetherian ring, $M$ a finitely generated $A$-module. Prove the inequalities

(Prove the first inequality by induction on the integer $g = \mathrm{grade}(M)$, using Cor. 2 of Prop. 13, No. 7 in the case $g = 0$ and then by considering an $A$-regular element of $\mathrm{Ann}(M)$. For the second, observe that we have $\mathrm{grade}(M) \leqslant \mathrm{prof}(A_p)$ for every $p \in \mathrm{Supp}(M)$.)