Sheaf (mathematics) - Images of Sheaves

Images of Sheaves

The definition of a morphism on sheaves makes sense only for sheaves on the same space X. This is because the data contained in a sheaf is indexed by the open sets of the space. If we have two sheaves on different spaces, then their data is indexed differently. There is no way to go directly from one set of data to the other.

However, it is possible to move a sheaf from one space to another using a continuous function. Let f : X → Y be a continuous function from a topological space X to a topological space Y. If we have a sheaf on X, we can move it to Y, and vice versa. There are four ways in which sheaves can be moved.

• A sheaf on X can be moved to Y using the direct image functor or the direct image with proper support functor .
• A sheaf on Y can be moved to X using the inverse image functor or the twisted inverse image functor .

The twisted inverse image functor is, in general, only defined as a functor between derived categories. These functors come in adjoint pairs: and are left and right adjoints of each other, and and are left and right adjoints of each other. The functors are intertwined with each other by Grothendieck duality and Verdier duality.

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted and it is distinct from . See inverse image functor.