Inverse image functor
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.
|Image functors for sheaves|
|direct image f∗|
|inverse image f∗|
|direct image with compact support f!|
|exceptional inverse image Rf!|
Suppose given a sheaf on Y and that we want to transport to X using a continuous map f : X → Y. We will call the result the inverse image or pullback sheaf . If we try to imitate the direct image by setting for each open set U of X, we immediately run into a problem: f(U) is not necessarily open. The best we can do is to approximate it by open sets, and even then we will get a presheaf, not a sheaf. Consequently we define to be the sheaf associated to the presheaf:
(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 is just the stalk of at this point.
When dealing with morphisms f : X → Y of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of -modules, where is the structure sheaf of Y. Then the functor f−1 is inappropriate, because (in general) it does not even give sheaves of -modules. In order to remedy this, one defines in this situation for a sheaf of -modules its inverse image by
- While is more complicated to define than f∗, the stalks are easier to compute: given a point , one has .
- is an exact functor, as can be seen by the above calculation of the stalks.
- is (in general) only right exact. If is exact, f is called flat.
- is the left adjoint of the direct image functor f∗. This implies that there are natural unit and counit morphisms and . These morphisms yield a natural adjunction correspondence:
However, these morphisms are almost never isomorphisms. For example, if denotes the inclusion of a closed subset, the stalks of at a point is canonically isomorphic to if is in and otherwise. A similar adjunction holds for the case of sheaves of modules, replacing by .