Theorem 1

Let $M, N$ be two A-modules, $u : M \to N$ an A-homomorphism and, for all $m \in \Omega$, let $u_m : M, \to N_m$ be the corresponding $A_m$-*homomorphism* (§ 2, no. 2, Remark 5). For $u$ to be injective (resp. surjective, bijective, zero), it is necessary and sufficient that, for all $m \in \Omega$, $u_m$ be injective (resp. surjective, bijective, zero).

To say that, for all $m \in \Omega$, $u_m$ is injective (resp. surjective, bijective, zero) is equivalent to saying that the homomorphism $\bigoplus_m u_m : \bigoplus M, \to \bigoplus N$, has the same property. But $\bigoplus_m M_m = M \otimes_A E, \bigoplus_m N_m = N \otimes_A E$ and $\bigoplus_m u_m = u \otimes 1$, where $E = \bigoplus_m \mathbf{A}_m$; as $E$ is faithfully flat (Proposition 10), the theorem follows from Chapter I, § 3, no. 1, Proposition 1 (c) and Proposition 2.