Sheaf (mathematics) - Stalks of A Sheaf

Stalks of A Sheaf

The stalk of a sheaf captures the properties of a sheaf "around" a point x ∈ X. Here, "around" means that, conceptually speaking, one looks at smaller and smaller neighborhood of the point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.

The stalk is defined by

the direct limit being over all open subsets of X containing the given point x. In other words, an element of the stalk is given by a section over some open neighborhood of x, and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism F(U) → F_x takes a section s in F(U) to its germ. This generalises the usual definition of a germ.

A different way of defining the stalk is

where i is the inclusion of the one-point space {x} into X. The equivalence follows from the definition of the inverse image.

In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. They also find use in constructions such as Godement resolutions.