Exceptional inverse image functor

In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.

Definition

Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor

Rf^!: D(Y) → D(X)

where D(–) denotes the derived category of sheaves of abelian groups or modules over a fixed ring.

It is defined to be the right adjoint of the total derived functor Rf_! of the direct image with compact support. Its existence follows from certain properties of Rf_! and general theorems about existence of adjoint functors, as does the unicity.

The notation Rf^! is an abuse of notation insofar as there is in general no functor f^! whose derived functor would be Rf^!.

Examples and properties

• If f: X → Y is an immersion of a locally closed subspace, then it is possible to define

:f^!(F) := f^∗ G,

where G is the subsheaf of F of which the sections on some open subset U of Y are the sections s ∈ F(U) whose support is contained in X. The functor f^! is left exact, and the above Rf^!, whose existence is guaranteed by abstract nonsense, is indeed the derived functor of this f^!. Moreover f^! is right adjoint to f_!, too.

• Slightly more generally, a similar statement holds for any quasi-finite morphism such as an étale morphism.
• If f is an open immersion, the exceptional inverse image equals the usual inverse image.

Duality of the exceptional inverse image functor

Let be a smooth manifold of dimension and let be the unique map which maps everything to one point. For a ring , one finds that is the shifted -orientation sheaf.

On the other hand, let be a smooth -variety of dimension . If denotes the structure morphism then is the shifted canonical sheaf on .

Moreover, let be a smooth -variety of dimension and a prime invertible in . Then where denotes the Tate twist.

Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last means the constant sheaf on and the rest mean that on , , and

:

the above computation furnishes the -adic Poincaré duality

:

from the repeated application of the adjunction condition.

References

• treats the topological setting
• treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.

• gives the duality statements.