Suppose we are given a sheaf \mathcal{G} on Y and that we want to transport \mathcal{G} to X using a
continuous map f\colon X\to Y. We will call the result the
inverse image or
pullback sheaf f^{-1}\mathcal{G}. If we try to imitate the
direct image by setting :f^{-1}\mathcal{G}(U) = \mathcal{G}(f(U)) for each open set U of X, we immediately run into a problem: f(U) 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 f^{-1}\mathcal{G} to be the
sheaf associated to the presheaf: :U \mapsto \varinjlim_{V\supseteq f(U)}\mathcal{G}(V). (Here 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 f^{-1}(\mathcal{F}) is just the
stalk of \mathcal{F} 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\colon X\to Y of
locally ringed spaces, for example
schemes in
algebraic geometry, one often works with
sheaves of \mathcal{O}_Y-modules, where \mathcal{O}_Y is the structure sheaf of Y. Then the functor f^{-1} is inappropriate, because in general it does not even give sheaves of \mathcal{O}_X-modules. In order to remedy this, one defines in this situation for a sheaf of \mathcal O_Y-modules \mathcal G its inverse image by :f^*\mathcal G := f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X. == Properties ==