Exact direct image from an affine open in a semi-separated scheme ($j_{\ast }:\mathrm{QCoh}(U)\to \mathrm{QCoh}(X)$)

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.