Kernels of quasi-coherent sheaf morphisms ($\ker (\tilde{M}\to \tilde{N})=\widetilde{\ker (M\to N)}$)

On an affine scheme, a morphism between quasi-coherent sheaves is determined by the induced map on global sections, because its maps on principal open subsets are their localizations. Exactness of localization gives the displayed identity. Thus the kernel sheaf of a morphism between quasi-coherent sheaves is quasi-coherent on an arbitrary scheme, by applying the affine argument locally. No finite generation is needed.