Inverse Image Functor - Definition

Definition

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.

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 -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

.