In mathematics, the inverse image functor is a covariant construction of sheaves. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Suppose we are given a sheaf G on Y and that we want to transport G to X using a continuous map f : X → Y .
We will call the result the inverse image or pullback sheaf f − 1 G . If we try to imitate the direct image by setting
f − 1 G ( U ) = G ( f ( U ) ) for each open set U of X , we immediately run into a problem: f ( U ) is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we define f − 1 G to be the sheaf associated to the presheaf:
U ↦ lim → V ⊇ f ( U ) G ( V ) . (Here U is an open subset of X and the colimit runs over all open subsets V of Y containing f ( U ) .)
For example, if f is just the inclusion of a point y of Y , then f − 1 ( F ) is just the stalk of F at this point.
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.
When dealing with morphisms f : X → Y of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of O Y -modules, where O Y is the structure sheaf of Y . Then the functor f − 1 is inappropriate, because in general it does not even give sheaves of O X -modules. In order to remedy this, one defines in this situation for a sheaf of O Y -modules G its inverse image by
f ∗ G := f − 1 G ⊗ f − 1 O Y O X .
While f − 1 is more complicated to define than f ∗ , the stalks are easier to compute: given a point x ∈ X , one has ( f − 1 G ) x ≅ G f ( x ) . f − 1 is an exact functor, as can be seen by the above calculation of the stalks. f ∗ is (in general) only right exact. If f ∗ is exact, f is called flat. f − 1 is the left adjoint of the direct image functor f ∗ . This implies that there are natural unit and counit morphisms G → f ∗ f − 1 G and f − 1 f ∗ F → F . These morphisms yield a natural adjunction correspondence: H o m S h ( X ) ( f − 1 G , F ) = H o m S h ( Y ) ( G , f ∗ F ) .
However, these morphisms are almost never isomorphisms. For example, if i : Z → Y denotes the inclusion of a closed subset, the stalk of i ∗ i − 1 G at a point y ∈ Y is canonically isomorphic to G y if y is in Z and 0 otherwise. A similar adjunction holds for the case of sheaves of modules, replacing i − 1 by i ∗ .
