Exercise 7, § 8

Let M be an A-module. The flat dimension of M is called, and is denoted by $\mathrm{dpl}_A(M)$, the greatest lower bound in $\overline{\mathbf{Z}}$ of the lengths of the flat resolutions of M. a) If n is an integer $\geqslant 0$, prove that the following conditions are equivalent: α) $\mathrm{dpl}_A(M) \leqslant n$. β) $\mathrm{Tor}_r^A(N, M) = 0$ for every right a-module N and every integer $r > n$. γ) $\mathrm{Tor}_{n+1}^A(N, M) = 0$ for every right a-module N. δ) $\mathrm{Tor}_{n+1}^A(N, M) = 0$ for every monogeneous right a-module N. ε) For every exact sequence $0 \to K \to P_{n-1} \to \cdots \to P_0 \to M \to 0$ where the $P_k$ are flat, K is flat. b) Prove that $\mathrm{dpl}_A(M) \leqslant \mathrm{dp}_A(M)$; there is equality if A is Noetherian and M is of finite type. c) Let $(N_\alpha)_{\alpha \in I}$ be an inductive system of A-modules and N its inductive limit. Prove the inequality