Yes, this is true in general.
It suffices to show the stalks vanish. Pick and take an injective resolution . For any open containing , we get a chain complex
whose cohomology groups are .
Taking direct limits of these sections gives the chain complex
of stalks, which has zero cohomology in positive degrees because the original complex was a resolution. However, direct limits are exact and so we find
as desired.
Generally, cohomology tells you the obstructions to patching local solutions into global solutions, and this says that locally those obstructions vanish.
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