Exercise 16, § 8

Let $A$ be a right and left Noetherian local ring and $\mathfrak{m}$ its maximal ideal; denote by $k_s$ (resp. $k_d$) the left a-module (resp. right) $A/m$. Let $M$ be a left a-module of finite type and $n$ an integer. (a) Prove that one has $\mathrm{dp}_A(M) \leq n$ if and only if $\mathrm{Tor}_{n+1}^{A}(k_d, M) = 0$ (use exercise 4, p. 185). (b) Prove that one has $\mathrm{dh}(A) = \mathrm{dp}_A(k_s)$; in order that $\mathrm{dh}(A) \leq n$, it is necessary and sufficient that one have

(c) Let $x$ be an element of the center of $A$, belonging to $\mathfrak{m}$, such that the homothety of ratio $x$ in $M$ is injective. Prove that one has $\mathrm{dp}_A(M/xM) = \mathrm{dp}_A(M) + 1$ (cf. also exercise 14).