= Exact direct image from an affine open in a semi-separated scheme
{title2=$j_*:\operatorname{QCoh}(U)\to\operatorname{QCoh}(X)$}
If $U$ is an <affine open subscheme> of a <semi-separated scheme>, its <open immersion> $j:U\hookrightarrow X$ has exact <direct image> on <quasi-coherent sheaves>. Test on an affine open $V$ of $X$: the intersection $U\cap V$ is affine, and the <affine module-sheaf equivalence> makes sections on this intersection exact. The direct image is also quasi-coherent, since this open immersion is an <affine morphism>.
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