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

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

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

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

${\displaystyle \mathrm {H} _{c}^{n}(X)^{*}\cong \operatorname {Hom} \left(f_{!}f^{*}\mathbb {Q} _{\ell }[n],\mathbb {Q} _{\ell }\right)\cong \operatorname {Hom} \left(\mathbb {Q} _{\ell },f_{*}f^{!}\mathbb {Q} _{\ell }[-n]\right),}$

the above computation furnishes the ${\displaystyle \ell }$-adic Poincaré duality

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

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.