Proposition 1

Let A be a ring, M an A-module and n an integer ≥ 0. The following conditions are equivalent:

(i) one has di_A(M) ≤ n (in other words, M possesses an injective resolution of length ≤ n) ; (ii) for every A-module N and every integer i > n, one has Ext_A^i(N, M) = 0 ; (iii) for every ideal a of A, one has Ext_A^{n+1}(A/a, M) = 0 ; (iv) for every exact sequence of A-modules

where the $I^i$ are injective, the A-module Q is injective.

(i) ⇒ (ii) : this follows from A, X, p. 100, Th. 1. (ii) ⇒ (iii) : this is clear. (iii) ⇒ (iv) : in the situation of (iv), for every A-module N one has an isomorphism of Ext_A^1(N, Q) onto Ext_A^{n+1}(N, M) (A, X, p. 128, Cor. 4) ; under assumption (iii), the A-module Ext_A^1(A/a, Q) is zero for every ideal a of A, and Q is injective (A, X, p. 93, Prop. 11). (iv) ⇒ (i) : consider the exact sequence (A, X, p. 52)

if condition (iv) is satisfied, the A-module $K^{n-1}(M)$ is injective, whence (i).

Recall (A, X, p. 138, Def. 2) that the homological dimension of the ring $A$, denoted by $\mathrm{dh}(A)$, is the least upper bound in $\overline{\mathbf{Z}}$ of the set of integers $n$ for which there exist two $A$-modules $M$ and $N$ such that $\mathrm{Ext}^n_A(M, N)$ is nonzero. It is therefore also the least upper bound of the set of projective dimensions (or injective dimensions) of all the $A$-modules; when $A$ is noetherian, it is enough to restrict oneself to finitely generated $A$-modules (A, X, p. 139, Cor.).