Inverse image functor
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map , the inverse image functor is a functor from the category of sheaves on Y to that X. 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!|
|Base change theorems|
Suppose we are given a sheaf on and that we want to transport to using a continuous map .
for each open set of , we immediately run into a problem: 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 to be the sheaf associated to the presheaf:
(Here is an open subset of and the colimit runs over all open subsets of containing .)
For example, if is just the inclusion of a point of , then is just the stalk of at this point.
When dealing with morphisms of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of -modules, where is the structure sheaf of . Then the functor 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 , 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 . This implies that there are natural unit and counit morphisms and . These morphisms yield a natural adjunction correspondence:
However, the morphisms and are almost never isomorphisms. For example, if denotes the inclusion of a closed subset, the stalk 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 .