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

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

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

Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last [math]\displaystyle{ \mathbb{Q}_{\ell} }[/math] means the constant sheaf on [math]\displaystyle{ X }[/math] and the rest mean that on [math]\displaystyle{ * }[/math], [math]\displaystyle{ f:X\to * }[/math], and

[math]\displaystyle{ \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), }[/math]

the above computation furnishes the [math]\displaystyle{ \ell }[/math]-adic Poincaré duality

[math]\displaystyle{ \mathrm{H}_{c}^{n}\left(X ; \mathbb{Q}_{\ell}\right)^{*} \cong \mathrm{H}^{2 d-n}(X ; \mathbb{Q}(d)) }[/math]

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  treats the topological setting
• Artin, Michael (1972). Alexandre Grothendieck. ed (in French). 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. 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, pp.10-11, https://guests.mpim-bonn.mpg.de/gallauer/docs/m6ff.pdf  gives the duality statements.