The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.
Contents
Motivation and definition
Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point x of X. Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of x, the behavior of the sheaf
The precise definition is as follows: the stalk of
Here the direct limit is indexed over all the open sets containing x, with order relation induced by reverse inclusion (
Alternative definition
There is another approach to defining a stalk that is useful in some contexts. Choose a point x of X, and let i be the inclusion of the one point space {x} into X. Then the stalk
Remarks
For some categories C the direct limit used to define the stalk may not exist. However, it exists for most categories which occur in practice, such as the category of sets or most categories of algebraic objects such as abelian groups or rings, which are namely cocomplete.
There is a natural morphism F(U) → Fx for any open set U containing x: it takes a section s in F(U) to its germ, that is, its equivalence class in the direct limit. This is a generalization of the usual concept of a germ, which can be recovered by looking at the stalks of the sheaf of continuous functions on X.
Examples
Germs are more useful for some sheaves than for others.
Constant sheaves
The constant sheaf
Sheaves of analytic functions
For example, in the sheaf of analytic functions on an analytic manifold, a germ of a function at a point determines the function in a small neighborhood of a point. This is because the germ records the function's power series expansion, and all analytic functions are by definition equal to their power series. Using analytic continuation, we find that the germ at a point determines the function on any connected open set where the function can be everywhere defined. (This does not imply that all the restriction maps of this sheaf are injective!)
Sheaves of smooth functions
In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. For example, let f : R → R be a bump function which is identically one in a neighborhood of the origin and identically zero far away from the origin. On any sufficiently small neighborhood containing the origin, f is identically one, so at the origin it has the same germ as the constant function with value 1. Suppose that we want to reconstruct f from its germ. Even if we know in advance that f is a bump function, the germ does not tell us how large its bump is. From what the germ tells us, the bump could be infinitely wide, that is, f could equal the constant function with value 1. We cannot even reconstruct f on a small open neighborhood U containing the origin, because we cannot tell whether the bump of f fits entirely in U or whether it is so large that f is identically one in U.
On the other hand, germs of smooth functions can distinguish between the constant function with value one and the function
Quasi-coherent sheaves
On an affine scheme X=Spec A, the stalk of a quasi-coherent sheaf F corresponding to an A-module M in a point x corresponding to a prime ideal p is just the localization Mp.
Skyscraper sheaf
On any topological space, the skyscraper sheaf associated to a closed point x and a group or ring G has the stalks 0 off x and G in x — whence the name skyscraper. The same property holds for any point x if the topological space in question is a T1 space, since every point of a T1 space is closed. This feature is the basis of the construction of Godement resolutions, used for example in algebraic geometry to get functorial injective resolutions of sheaves.
Properties of the stalk
As outlined in the introduction, stalks capture the local behaviour of a sheaf. As a sheaf is supposed to be determined by its local restrictions (see gluing axiom), it can be expected that the stalks capture a fair amount of the information that the sheaf is encoding. This is indeed true:
In particular:
Both statements are false for presheaves. However, stalks of sheaves and presheaves are tightly linked: