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 the category of sheaves on X. The
direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Definition
Suppose we are given a sheaf on and that we want to transport to using a
continuous map .
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 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.
The restriction maps, as well as the
functoriality of the inverse image follows from the
universal property of
direct limits.
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
- .
Properties
- 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 .
References