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.
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!.
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.
Let
X
d
f:X → *
Λ
f!Λ=\omegaX,[d]
Λ
On the other hand, let
X
k
d
f:X → \operatorname{Spec}(k)
f!k\cong\omegaX[d]
X
Moreover, let
X
k
d
\ell
k
f!Q\ell\congQ\ell(d)[2d]
(d)
Recalling the definition of the compactly supported cohomology as lower-shriek pushforward and noting that below the last
Q\ell
X
*
f:X\to*
| n | |
| H | |
| c |
(X)*\cong\operatorname{Hom}\left(f!f*Q\ell[n],Q\ell\right)\cong\operatorname{Hom}\left(Q\ell,f*f!Q\ell[-n]\right),
\ell
| n | |
| H | |
| c |
\left(X;Q\ell\right)*\congH2(X;Q(d))