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
:R''f''^!: 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 R''f''_! of the direct image with compact support. Its existence follows from certain properties of R''f''_! and general theorems about existence of adjoint functors, as does the unicity.

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

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 R''f''^!, 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_ /math> is the shifted $\Lambda$- orientation sheaf.

On the other hand, let $X$ be a smooth $k$-variety of dimension $d$. If $f: X \rightarrow \operatorname(k)$ denotes the structure morphism then f^ k \cong \omega_ /math> 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_ \cong \mathbb_(d) d/math> where $(d)$ denotes the Tate twist.

Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last $\mathbb_$ means the constant sheaf on $X$ and the rest mean that on $*$, $f:X\to *$, and
:: \mathrm_^(X)^ \cong \operatorname\left(f_! f^ \mathbb_ \mathbb_\right) \cong \operatorname\left(\mathbb_, f_ f^ \mathbb_ nright),
the above computation furnishes the $\ell$-adic Poincaré duality
:: $\mathrm_^\left(X ; \mathbb_\right)^ \cong \mathrm^(X ; \mathbb(d))$
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.

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Sheaf theory