Inverse image functor
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In mathematics, the inverse image functor is a contravariant 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.
Given a continuous map of topological spaces 
, the inverse image functor f - 1 associates to any sheaf 
on Y its inverse image 
, which is a sheaf on X.
It is defined to be the sheaf associated to the presheaf
(U is an open subset of X and the limit runs over all open subsets V of Y containing f(U)).
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.
When dealing with locally ringed spaces, for example schemes in algebraic geometry, one also uses the inverse image of sheaves of modules. Let 
be a sheaf of 
-modules. Its inverse image as an 
-module is then defined by
.
[edit] Properties
- While f-1 is more complicated to define than f * , the stalks are easier to compute: given a point
, one has 
.
- f - 1 is an exact functor, as can be seen by the above calculation of the stalks.
- f - 1 is the left adjoint of the direct image functor f * . This implies that there are natural unit and counit morphisms
and 
. However, these 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 y is in Z and 0 otherwise.
