= Čech lifting below the first possible local cohomology degree
{c}
{title2=$0\to\check H^i(\mathcal U,\mathcal F)\to H^i(X,\mathcal F)\to\prod_aH^i(U_a,\mathcal F)$}
For a finite cover whose nonempty intersections have zero <sheaf cohomology> in degrees $1,\ldots,i-1$, the displayed sequence is exact for $i>0$. Apply the <Čech cochain complex> to a <flasque resolution>. Local primitives and successive primitives of their overlap differences reduce a class with zero restrictions to a Čech cocycle with values in the original <sheaf>. Vanishing in degree $i$ on all intersections is not needed.
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