Concept
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 . 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 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 . 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!. 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 , 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 . Let be a smooth manifold of dimension and let be the unique map which maps everything to one point. For a ring , one finds that is the shifted -orientation sheaf. On the other hand, let be a smooth -variety of dimension . If denotes the structure morphism then is the shifted canonical sheaf on . Moreover, let be a smooth -variety of dimension and a prime invertible in . Then where denotes the Tate twist. Recalling the definition of the compactly supported cohomology as and noting that below the last means the constant sheaf on and the rest mean that on , , and the above computation furnishes the -adic Poincaré duality from the repeated application of the adjunction condition.