Exceptional inverse image functor
# 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 $f^{!}\mathbb {Q} _{\ell }\cong \mathbb {Q} _{\ell }(d)[2d]$ 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

       $\mathrm {H} _{c}^{n}\left(X;\mathbb {Q} _{\ell }\right)^{*}\cong \mathrm {H} ^{2d-n}(X;\mathbb {Q} (d))$

from the repeated application of the adjunction condition.

