Semi-separated scheme

A scheme is semi-separated when the intersection of any two affine open subsets is affine, equivalently when its diagonal is affine. This condition makes affine covers acyclic for quasi-coherent sheaves.

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.