= Kernels of quasi-coherent sheaf morphisms
{title2=$\ker(\widetilde M\to\widetilde N)=\widetilde{\ker(M\to N)}$}
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.
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