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) → Fx 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.
Read more about this topic: Sheaf (mathematics)
Famous quotes containing the word stalks:
“Cut grass lies frail:
Brief is the breath
Mown stalks exhale,”
—Philip Larkin (19221985)