The direct image functor is left exact, but usually not right exact. Hence one can consider the right derived functors of the direct image. They are called higher direct images and denoted Rq f∗.
One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, Rq f∗(F) is the sheaf associated to the presheaves
Read more about this topic: Direct Image Functor
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