Given two quasi - coherent $\mathcal{O}_X$ - modules $\mathcal{F}$ and $\mathcal{G}$ on a scheme $X$ and a morphism $f:\mathcal{F} \rightarrow \mathcal{G}$ and a point $x\in X$ such that $f_x:\mathcal{F}_x\rightarrow \mathcal{G}_x$ is an isomorphism, I wish to prove the following:
If $\mathcal{F}$ is locally of finite type and $\mathcal{G}$ is locally of finite presentation, then there is an open neighborhood $U$ of $x$ such that $f|_U:\mathcal{F}|_U\rightarrow \mathcal{G}|_U$ is an isomorphism.
I think I am close to solving it myself, but I am missing a crucial step in my argument.
Assume $X = \rm{Spec}(A)$ is affine. Then $\mathcal{F}$ and $\mathcal{G}$ are associated to $A$ - modules $M$ and $N$ of finite type and finite presentation respectively. I want to show that $\rm{ker}(f)$ and $\rm{coker}(f)$ are locally of finite type. Because both $\rm{ker}(f)$ and $\rm{coker}(f)$ are quasi - coherent and thus it follows from $\rm{ker}(f)_x = 0 = \rm{coker}(f)_x$ that there is an open nbdh $U$ of $x$ such that $\rm{ker}(f)|_U = \rm{coker}(f)|_U = 0$ (this has been shown before).
But I don't know how to show that $\rm{ker}(f)$ and $\rm{coker}(f)$ are locally of finite type. Is that even true? And is my idea basically correct?
Thank you very much for any comment!