I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent $A$-module $M$, its localization $M_{f}$ at $f\in A$ is a coherent $A_{f}$-module. But I don't see how to prove this---I cannot relate the map $A^p_f\to M_f$ to some appropriate map $A^p\to M$ for instance. Would someone help me with this problem?
Cf. We follow the definition of stacks project on coherence, i.e., the module itself is finitely generated, and any finitely generated submodule of it is finitely presented.
If $N'$ is a finitely generated $S^{-1}A$-submodule of $S^{-1}M$, then there is a finitely generated $A$-submodule $N$ of $M$ such that $N'=S^{-1}N$. (Suppose that $N'$ is generated by $x_1/s_1,\dots,x_n/s_n$. Set $N=Ax_1+\cdots+Ax_n$. Then $N'=S^{-1}N$.) Since $N$ is finitely presented, by localizing at $S$ we get that $N'$ is also finitely presented.