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 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. Let f: X → Y be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. 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. 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.