Direct image functor

In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.

Definition

Let f: X → Y be a continuous mapping of topological spaces, and Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image functor

f_{*}:\operatorname {Sh} (X)\to \operatorname {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

f_{*}F(U):=F(f^{-1}(U)),

which turns out to be a sheaf on Y, also called the pushforward sheaf of F along f.

f_∗ is a functor, i.e. a morphism of sheaves φ: F → G on X gives rise to a morphism of sheaves f_∗(φ): f_∗(F) → f_∗(G) on Y.

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.

A similar definition applies to sheaves on topoi, such as étale sheaves. Instead of the above preimage f⁻¹(U) the fiber product of U and X over Y is used.

Properties

Forming sheave categories and direct image functors is itself 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 $f:X\to Y$ and sheaves ${\mathcal {F}},{\mathcal {G}}$ respectively on X, Y, there is a natural isomorphism:

\mathrm {Hom} _{\mathbf {Sh} (X)}(f^{-1}{\mathcal {G}},{\mathcal {F}})=\mathrm {Hom} _{\mathbf {Sh} (Y)}({\mathcal {G}},f_{*}{\mathcal {F}})

.

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 $(f_{*}{\mathcal {F}})_{y}$ is ${\mathcal {F}}_{y}$ if $y\in X$ and zero otherwise (here the closedness of X in Y is used).

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 R^q f_∗.

One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, the sheaf R^q f_∗(F) is the sheaf associated to the presheaf

U\mapsto H^{q}(f^{-1}(U),F)

(where H^q denotes sheaf cohomology.)

References

Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190, esp. section II.4