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

Image functors for sheaves
direct image $f_{*}$
inverse image $f^{*}$
direct image with compact support $f_{!}$
exceptional inverse image $Rf^{!}$
$f^{}\leftrightarrows f_{}$
$(R)f_{!}\leftrightarrows (R)f^{!}$
Base change theorems
v t e

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.

References

Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190 treats the topological setting
Artin, Michael (1972). Alexandre Grothendieck; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3. Lecture notes in mathematics (in French). Vol. 305. Berlin; New York: Springer-Verlag. pp. vi+640. doi:10.1007/BFb0070714. ISBN 978-3-540-06118-2. treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.
Gallauer, Martin, An Introduction to Six Functor Formalisms (PDF), pp.10-11 gives the duality statements.

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