Exercise 18, § 8

a) Let $(M_\alpha)_{\alpha \in I}$ be an inductive system of $A$-modules relative to a countable set $I$, $M$ its limit, $n$ an integer $\geqslant 0$. Now prove that if $\mathrm{dp}_A(M_\alpha) \leqslant n$ for every $\alpha \in I$, one has $\mathrm{dp}_A(M) \leqslant n + 1$ (reduce to the case $n = 0$ and $I = \mathbf{N}$, then write an exact sequence $0 \to \bigoplus_n M_n \to \bigoplus_n M_n \to M \to 0$).

b) Let $P$ be a flat $A$-module admitting a presentation $A^{(S)} \to A^{(T)} \to P \to 0$, where the set $S$ is countable. Deduce from a) that one has $\mathrm{dp}_A(P) \leqslant 1$.

c) Suppose that every left ideal of the ring $A$ is generated by a countable set of elements. Prove that one has $\mathrm{dp}_A(M) \leqslant \mathrm{dpl}_A(M) + 1$ for every $A$-module $M$ generated by a countable family of elements. Deduce the inequalities $\mathrm{dh}(A) \leqslant \mathrm{td}(A) + 1$ and $|\mathrm{dh}(A) - \mathrm{dh}(A^\circ)| \leqslant 1$. (Use exercise 12, p. 181, exercise 8, p. 202, as well as b).)