= Locally vanishing principle for sheaf cohomology
{title2=$\xi\in H^i(X,\mathcal F),\quad i>0$}
A positive-degree <sheaf cohomology> class restricts to zero on some neighbourhood of every point, because a representing cocycle in a <flasque resolution> is locally a boundary. If a basis closed under finite intersections has vanishing local cohomology in degrees $1,\ldots,i-1$, the neighbourhoods can be chosen in that basis so that the image of the class in $H^i(X,{}_U\mathcal F)$ is zero. Here ${}_U\mathcal F$ is the <direct image from an open restriction>. Lower local vanishing makes the edge map $H^i(X,{}_U\mathcal F)\to H^i(U,\mathcal F|_U)$ injective. Local vanishing of a class alone does not imply global vanishing.
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