= Čech resolution on a semi-separated scheme
{c}
{title2=$\mathcal C^p=\prod_{i_0<\cdots<i_p}(j_{i_0\cdots i_p})_*(\mathcal F|_{U_{i_0\cdots i_p}})$}
For a finite affine cover of a <semi-separated scheme>, every finite intersection is affine. The augmented sheaf <Čech cochain complex> is exact: near each point, an open of the cover contains the neighbourhood under consideration, and inserting its index contracts the complex. Its terms are acyclic for a <quasi-coherent sheaf>, by affine cohomology vanishing and the affine-intersection property. Alternatively, apply the section-level <Čech cochain complex> to a <flasque resolution>; the two computations of the resulting <double complex> identify <Čech cohomology> with <sheaf cohomology>. Stalks of direct images from opens outside those opens need not vanish, so exactness must not be justified by treating these direct images as <extensions by zero>.
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