The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.
Motivation and definition
Sheaves are defined on open sets, but the underlying topological space consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point of . Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of , the behavior of the sheaf on that small neighborhood should be the same as the behavior of at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
The precise definition is as follows: the stalk of at , usually denoted , is:
Here the direct limit is indexed over all the open sets containing , with order relation induced by reverse inclusion By definition (or universal property) of the direct limit, an element of the stalk is an equivalence class of elements , where two such sections and are considered equivalent if the restrictions of the two sections coincide on some neighborhood of .
Alternative definition
There is another approach to defining a stalk that is useful in some contexts. Choose a point of , and let be the inclusion of the one point space into . Then the stalk is the same as the inverse image sheaf . Notice that the only open sets of the one point space are and , and there is no data over the empty set. Over , however, we get:
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 for any open set containing : it takes a section in 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 .
Examples
Constant sheaves
The constant sheaf associated to some set (or group, ring, etc). has the same set or group as stalks at every point: for any point , pick an open connected neighborhood. The sections of on a connected open equal and restriction maps are the identities. Therefore, the direct limit collapses to yield as the stalk.
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 locally 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 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, 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 from its germ. Even if we know in advance that 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, could equal the constant function with value 1. We cannot even reconstruct on a small open neighborhood containing the origin, because we cannot tell whether the bump of fits entirely in or whether it is so large that is identically one in .
On the other hand, germs of smooth functions can distinguish between the constant function with value one and the function , because the latter function is not identically one on any neighborhood of the origin. This example shows that germs contain more information than the power series expansion of a function, because the power series of is identically one. (This extra information is related to the fact that the stalk of the sheaf of smooth functions at the origin is a non-Noetherian ring. The Krull intersection theorem says that this cannot happen for a Noetherian ring.)