Direct image functor (original) (raw)

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In mathematics, the direct image functor describes how structured data assigned to one space can be systematically transferred to another space using a continuous map between them. More precisely, if we have a sheaf—an object that encodes data like functions or sections over open regions—defined on a space X, and a continuous map from X to another space Y, then the direct image functor produces a corresponding sheaf on Y. This construction is a central tool in sheaf theory and is widely used in topology and algebraic geometry to relate local data across spaces.

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.

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

f ∗ : Sh ⁡ ( X ) → Sh ⁡ ( Y ) {\displaystyle f_{*}:\operatorname {Sh} (X)\to \operatorname {Sh} (Y)} {\displaystyle 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 ) ) . {\displaystyle f_{*}F(U):=F(f^{-1}(U)).} {\displaystyle f_{*}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.

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.[1]

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.

H o m S h ( X ) ( f − 1 G , F ) = H o m S h ( Y ) ( G , f ∗ F ) {\displaystyle \mathrm {Hom} _{\mathbf {Sh} (X)}(f^{-1}{\mathcal {G}},{\mathcal {F}})=\mathrm {Hom} _{\mathbf {Sh} (Y)}({\mathcal {G}},f_{*}{\mathcal {F}})} {\displaystyle \mathrm {Hom} _{\mathbf {Sh} (X)}(f^{-1}{\mathcal {G}},{\mathcal {F}})=\mathrm {Hom} _{\mathbf {Sh} (Y)}({\mathcal {G}},f_{*}{\mathcal {F}})}.

Higher direct images

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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 _Rq f_∗.

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

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

where Hq denotes sheaf cohomology.

In the context of algebraic geometry and a morphism f : X → Y {\displaystyle f:X\to Y} {\displaystyle f:X\to Y} of quasi-compact and quasi-separated schemes, one likewise has the right derived functor

R f ∗ : D q c o h ( X ) → D q c o h ( Y ) {\displaystyle Rf_{*}:D_{qcoh}(X)\to D_{qcoh}(Y)} {\displaystyle Rf_{*}:D_{qcoh}(X)\to D_{qcoh}(Y)}

as a functor between the (unbounded) derived categories of quasi-coherent sheaves. In this situation, R f ∗ {\displaystyle Rf_{*}} {\displaystyle Rf_{*}} always admits a right adjoint f × {\displaystyle f^{\times }} {\displaystyle f^{\times }}.[2] This is closely related, but not generally equivalent to, the exceptional inverse image functor f ! {\displaystyle f^{!}} {\displaystyle f^{!}}, unless f {\displaystyle f} {\displaystyle f} is also proper.

  1. ^ "Section 26.24 (01LA): Functoriality for quasi-coherent modules—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20.
  2. ^ "Section 48.3 (0A9D): Right adjoint of pushforward—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-20.