of U and X over Y. Properties Forming sheaf categories and direct image functors itself defines a functor from the category of topological spaces to the category of categories: given continuous maps f: X → Y and g: Y → Z, we have (gf)∗=g∗f∗. The direct image functor is right adjoint to the inverse image functor, which means that for any continuous and sheaves respectively on X, Y, there is a natural isomorphism: . If f is the inclusion of a closed subspace X ⊆ Y then f∗ is exact. Actually, in this case f∗ is an equivalence between the category of sheaves on X and the category of sheaves on Y supported on X. This follows from the fact that the stalk of is if and zero otherwise (here the closedness of X in Y is used). If f is the morphism of affine schemes determined by a ring homomorphism , then the direct image functor f∗ on quasi-coherent sheaves identifies with the restriction of scalars functor along φ. Higher direct images 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, the sheaf Rq f∗(F) is the sheaf associated to the presheaf , where Hq denotes sheaf cohomology. In the context of algebraic geometry and a morphism of quasi-compact and quasi-separated schemes, one likewise has the right derived functor as a functor between the (unbounded) derived categories of quasi-coherent sheaves. In this situation, always admits a right adjoint . This is closely related, but not generally equivalent to, the exceptional inverse image functor , unless is also proper. See also Proper base change theorem References , esp. section II.4 Sheaf theory Theory of continuous functions