Exercise 4, § 8

Let M be an A-module. One calls injective dimension of M, and denotes it by $\mathrm{di}_A(M)$, the greatest lower bound in $\overline{\mathbf{Z}}$ of the lengths of the injective resolutions of M. If n is an integer $\geqslant 0$, now prove that the following conditions are equivalent: α) $\mathrm{di}_A(M) \leqslant n$; β) $\mathrm{Ext}_A^r(N, M) = 0$ for every A-module N and every integer $r > n$; γ) $\mathrm{Ext}_A^{n+1}(N, M) = 0$ for every A-module N ; δ) $\mathrm{Ext}_A^{n+1}(N, M) = 0$ for every monogenous A-module N ; ε) for every exact sequence $0 \to M \to I^0 \to \cdots \to I^{n-1} \to N \to 0$, where the $I^k$ are injective, N is injective.