Direct image functor

More formally, 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.

Definition

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

${\displaystyle f_{\ast }:\mathrm{Sh}(X)\to \mathrm{Sh}(Y)}$

sends a sheaf F on X to its direct image presheaf f_∗F on Y, defined on open subsets U of Y by

${\displaystyle f_{\ast }F(U):=F(f^{-1}(U)).}$

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.

Example

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.

Variants

If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if f: (X, O_X) → (Y, O_Y) is a morphism of ringed spaces, we obtain a direct image functor f_∗: Sh(X,O_X) → Sh(Y,O_Y) from the category of sheaves of O_X-modules to the category of sheaves of O_Y-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.