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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 colimit 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 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

.

• 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 ^* is (in general) only right exact. If f ^* is exact, f is called flat.
• 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. A similar adjunction holds for the case of sheaves of modules, replacing f ^{- 1} by f^∗.

• Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, MR842190, ISBN 978-3-540-16389-3. See section II.4.