Image functors for sheaves

In mathematics, especially in sheaf theory, a domain applied in areas such as topology, logic and algebraic geometry, there are four image functors for sheaves which belong together in various senses.

Given a continuous mapping "f": "X" → "Y" of topological spaces, and the category "Sh"(–) of sheaves of abelian groups on a topological space. The functors in question are

* direct image "f"_∗ : "Sh"("X") → "Sh"("Y")
* inverse image "f"^∗ : "Sh"("Y") → "Sh"("X")
* direct image with compact support "f"_! : "Sh"("X") → "Sh"("Y")
* exceptional inverse image "Rf"^! : "D"("Sh"("Y")) → "D"("Sh"("X")).

The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes.

Adjointness

The functors are adjoint to each other as depicted at the right, where, as usual, $F\; leftrightarrows\; G$ means that "F" is left adjoint to "G" (equivalently "G" right adjoint to "F"), i.e. :"Hom"("F"("A"), "B") ≅ "Hom"("A", "G"("B"))for any two objects "A", "B" in the two categories being adjoint by "F" and "G".

For example, "f"^∗ is the left adjoint of "f"_*. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms $mathcal\{G\}\; ightarrow\; f\_*f^\{*\}mathcal\{G\}$ and $f^\{*\}f\_*mathcal\{F\}\; ightarrow\; mathcal\{F\}$ for $mathcal\; G$ on "Y" and $mathcal\; F$ on "X", respectively. However, these are "almost never" isomorphisms - see the localization example below.

Verdier duality

Verdier duality gives another link between them: morally speaking, it exchanges "∗" and "!", i.e. in the synopsis above it exchanges functors along the diagonals. For example the direct image is dual to the direct image with compact support. This phenomenon is studied and used in the theory of perverse sheaves.

Localization

In the particular situation of a closed subspace "i": "Z" ⊂ "X" and the complementary open subset "j": "U" ⊂ "X", the situation simplifies insofar that for "j"^∗="j"^! and "i"_!="i"_∗ and for any sheaf "F" on "X", one gets exact sequences:0 → "j"_!"j"^∗ "F" → "F" → "i"_∗"i"^∗ → 0Its Verdier dual reads:"i"_∗"Ri"^! "F" → "F" → "Rj"_∗"j"^∗ → "i"_∗"Ri"^! [1] ,a distinguished triangle in the derived category of sheaves on "X".

The adjointness relations read in this case:$i^*\; leftrightarrows\; i\_*=i\_!\; leftrightarrows\; i^!$ and:$j\_!\; leftrightarrows\; j^!=j^*\; leftrightarrows\; j\_*$.

References

* | year=1986 treats the topological setting
* treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.
* is another reference for the étale case.