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 $$X$$ be a smooth manifold of dimension $$d$$ and let $$f: X \rightarrow *$$ be the unique map which maps everything to one point. For a ring $$\Lambda$$, one finds that $$f^{!} \Lambda=\omega_{X, \Lambda}[d]$$ is the shifted $$\Lambda$$-orientation sheaf.

On the other hand, let $$X$$ be a smooth $$k$$-variety of dimension $$d$$. If $$f: X \rightarrow \operatorname{Spec}(k)$$ denotes the structure morphism then $$f^{!} k \cong \omega_{X}[d]$$ is the shifted canonical sheaf on $$X$$.

Moreover, let $$X$$ be a smooth $$k$$-variety of dimension $$d$$ and $$\ell$$ a prime invertible in $$k$$. Then $$f^{!} \mathbb{Q}_{\ell} \cong \mathbb{Q}_{\ell}(d)[2 d]$$ where $$(d)$$ denotes the Tate twist.

Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last $$\mathbb{Q}_{\ell}$$ means the constant sheaf on $$X$$ and the rest mean that on $$*$$, $$f:X\to *$$, and
 * $$\mathrm{H}_{c}^{n}(X)^{*} \cong \operatorname{Hom}\left(f_! f^{*} \mathbb{Q}_{\ell}[n], \mathbb{Q}_{\ell}\right) \cong \operatorname{Hom}\left(\mathbb{Q}_{\ell}, f_{*} f^{!} \mathbb{Q}_{\ell}[-n]\right),$$

the above computation furnishes the $$\ell$$-adic Poincaré duality
 * $$\mathrm{H}_{c}^{n}\left(X ; \mathbb{Q}_{\ell}\right)^{*} \cong \mathrm{H}^{2 d-n}(X ; \mathbb{Q}(d))$$

from the repeated application of the adjunction condition.