Proposition 11

Let M be an A-module, N a finitely generated A-module and $u : M \to N$ a homomorphism. For u to be surjective, it is necessary and sufficient that, for all $m \in S_2$, the homomorphism $M/mM \to N/mN$ derived from $u$ by taking quotients be surjective.

It follows from Theorem 1 that, for $u$ to be surjective, it is necessary and sufficient that $u_m : M_m \to N_m$ be surjective for all $m \in \Omega$. As $A$ is a local ring and $N_m$ is a finitely generated $A_m$-module, this amounts to saying that the homomorphism $u'_m : M_m/mM_m \to N_m/mN_m$, obtained by taking quotients, is surjective (no. 2, Corollary 1 to Proposition 4); but $M_m/mM_m$ (resp. $N_m/mN_m$) is identified with $M/mM$ (resp. $N/mN$) (Proposition9), whence the proposition.