Direct image functor

In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. 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. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi coherent sheaves and étale sheaves on a scheme.

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.

If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. 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. A similar definition applies to sheaves on topoi, such as étale sheaves. There, instead of the above preimage f−1(U), one uses the fiber product of U and X over Y.