Čech resolution on a semi-separated scheme (${\mathcal{C}}^p={\prod }_{i_0<\cdots <i_p}(j_{i_0\cdots i_p}{)}_{\ast }(\mathcal{F}{|}_{U_{i_0\cdots i_p}})$)

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.