Image functors for sheaves

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} \rightarrow f_f^{}\mathcal{G} and f^{}f_\mathcal{F} \rightarrow \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.

Base Change

Another useful property of the image functors is base change. Given continuous maps f:X \rightarrow Z and g:Y \rightarrow Z, which induce morphisms \bar f:X\times_Z Y \rightarrow Y and \bar g:X\times_Z Y \rightarrow X, there exists a canonical isomorphism R \bar f_* R\bar g^! \cong Rf^! Rg_*.

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∗ F → 0 Its Verdier dual reads :i∗Ri! F → F → Rj∗j∗ F → i∗Ri! F[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

• treats the topological setting
• {{cite book | editor1-link = Alexandre Grothendieck | editor2-link = Jean-Louis Verdier
• is another reference for the étale case.