Direct image functor

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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.

Contents

[edit] Definition

Image functors for sheaves

direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!

f^* \leftrightarrows f_*
(R)f_! \leftrightarrows (R)f^!

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

f_*: Sh(X) \to Sh(Y)

sends a sheaf F on X to its direct image presheaf

f_*F : U \mapsto F(f^{-1}(U)),

which turns out be a sheaf on Y. This assignment is functorial, 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.

[edit] Example

If Y is a point, then the direct image equals the global sections functor.

[edit] Variants

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

[edit] Higher direct images

The direct image functor is left exact, but usually not right exact. Hence on can consider the right derived functors of the direct image. They are called higher direct images and denoted Rq f∗.

One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, Rq f∗ is the sheaf associated to the presheaves

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

[edit] Properties

  • The direct image functor is right adjoint to the inverse image functor.
  • If f is the inclusion of a closed subspace X ⊂ Y then f∗ is exact. Actually, in this case f∗ is an equivalence between sheaves on X and sheaves on Y supported on X.

[edit] Reference

This article incorporates material from Direct image (functor) on PlanetMath, which is licensed under the GFDL.