Proposition 4

Let $n$ be an integer $\geq 0$. The following conditions are equivalent:

(i) $\operatorname{dh}(A)\leq n$,

(ii) for every $A$-module M, we have $\mathrm{dp}_A(M)\leq n$,

(ii′) for every finitely generated $A$-module M, we have $\mathrm{dp}_A(M)\leq n$,

(iii) for every exact sequence

where the $P_i$ are projective, $K$ is projective,

(iv) for every exact sequence

where the $I^i$ are injective, $N$ is injective,

(v) every $A$-module possesses an injective resolution of length $\leq n$.

The equivalence of conditions (i), (ii) and (iii) follows from prop. 1. We have obviously (ii) $\Rightarrow$ (ii’). It is therefore enough to proving (ii’) $\Rightarrow$ (iv) $\Rightarrow$ (v) $\Rightarrow$ (i).

(ii’) $\Rightarrow$ (iv): with the notation of (iv), let $K$ be the kernel of $I^0 \to I^1$. By X, p. 128, cor. 4, we have for every $A$-module $M$ an isomorphism $\mathrm{Ext}_A^1(M, N) \to \mathrm{Ext}_A^{n+1}(M, K)$; it then follows from (ii’) that $\mathrm{Ext}_A^1(M, N)$ is zero for every finitely generated $A$-module $M$. By X, p. 93, prop. 11, this implies that $N$ is injective, whence (iv).

(iv) $\Rightarrow$ (v): let $M$ be an $A$-module. Applying (iv) to the exact sequence

of X, p. 52, we conclude that $K^{n-1}(M)$ is injective, whence (v).

(v) $\Rightarrow$ (i): this follows from X, p. 100, th. 1.