Given a sheaf F defined on a topological space X and a continuous map f: X → Y, we can define a new sheaf f∗F on Y, called the direct image sheaf or the pushforward sheaf of F along f, such that the global sections of f∗F is given by the global sections of F. This assignment gives rise to a functor f∗ from the category of sheaves on X to the category of sheaves on Y, which is known as the direct image functor.
The direct image functor sends a sheaf F on X to its direct image presheaf f∗F on Y, defined on open subsets U of Y by This turns out to be a sheaf on Y, and is called the direct image sheaf or pushforward sheaf of F along f. Since a morphism of sheaves φ: F → G on X gives rise to a morphism of sheaves f∗(φ): f∗(F) → f∗(G) on Y in an obvious way, we indeed have that f∗ is a functor.
If Y is a point, and f: X → Y the unique continuous map, then Sh(Y) is the category Ab of abelian groups, and the direct image functor f∗: Sh(X) → Ab equals the global sections functor.
Similarly, if f: (X, OX) → (Y, OY) is a morphism of ringed spaces, we obtain a direct image functor f∗: Sh(X,OX) → Sh(Y,OY) from the category of sheaves of OX-modules to the category of sheaves of OY-modules.
Moreover, if f is now a morphism of quasi-compact and quasi-separated schemes, then f∗ preserves the property of being quasi-coherent, so we obtain the direct image functor between categories of quasi-coherent sheaves.
They are called higher direct images and denoted Rq f∗.
[2] This is closely related, but not generally equivalent to, the exceptional inverse image functor