Proposition 19

Let $A$ be a ring and $S$ a multiplicative subset of $A$. (i) If $M$ and $N$ are two $A$-modules and $M$ is finitely generated (resp. finitely presented), the canonical homomorphism (Chapter I, §2, no. 10, formula (10))

is injective (resp. bijective). (ii) If $M', N'$ are two $S^{-1}A$-modules, the canonical bijection

is bijective.

As $S^{-1}A$ is a flat $A$-module, (i) is a particular case of Chapter I, §2, no. 10, Proposition 11. On the other hand, we have already remarked that every $A$-homomorphism of $S^{-1}A$-modules is necessarily an $S^{-1}A$-homomorphism, in the course of proving Proposition 3 of no. 2; whence (ii).