In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves, 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

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

is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf to be the sheaf associated to the presheaf:

is an open subset and the colimit.)

is just the stalk 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 \mathcal{O}_Y-modules its inverse image by

1. .

Properties

• , the stalks.

• is an exact functor, as can be seen by the above calculation of the stalks.

• is exact, f is called flat.

• is the left adjoint of the direct image functor. These morphisms yield a natural adjunction correspondence:

• .

are almost never denotes the inclusion of a closed subset.

References

• . See section II.4.