Locally vanishing principle for sheaf cohomology ($\xi \in H^i(X,\mathcal{F}),i>0$)

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 $1,\dots ,i-1$, 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.