Stalk (sheaf) explained
In mathematics, 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:
l{F}x:=\varinjlimU\nil{F}(U).
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:
i-1l{F}(\{x\})=\varinjlimU\supseteq\{x\
} \mathcal(U) = \varinjlim_ \mathcal(U) = \mathcal_x.
Remarks
For some categories C the direct limit used to define the stalk may not exist. However, it exists for most categories that 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
associated to some set,
, (or group, ring, etc) is a sheaf for which
for all
in
.
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 that 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.)
Quasi-coherent sheaves
, the stalk of a
quasi-coherent sheaf
corresponding to an
-module
in a point
corresponding to a
prime ideal
is just the localization
.
Skyscraper sheaf
and a group or ring
has the stalks
off
and
on
- hence the name
skyscraper. This idea makes more sense if one adopts the common visualisation of functions mapping from some space above to a space below; with this visualisation, any function that maps
has
positioned directly above
. The same property holds for any point
if the topological space in question is a
T1 space, since every point of a T
1 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:
- A morphism of sheaves is an isomorphism, epimorphism, or monomorphism, respectively, if and only if the induced morphisms on all stalks have the same property. (However it is not true that two sheaves, all of whose stalks are isomorphic, are isomorphic, too, because there may be no map between the sheaves in question.)
In particular:
- A sheaf is zero (if we are dealing with sheafs of groups), if and only if all stalks of the sheaf vanish. Therefore, the exactness of a given functor can be tested on the stalks, which is often easier as one can pass to smaller and smaller neighbourhoods.
Both statements are false for presheaves. However, stalks of sheaves and presheaves are tightly linked:
and its sheafification
, the stalks of
and
agree. This follows from the fact that the sheaf
is the image of
through the left adjoint
} \to Sh(X) (because the sheafification functor is left adjoint to the inclusion functor
} ) and the fact that left adjoints preserve colimits.
Reference
- Book: [{{Google books|3rtX9t-nnvwC&q|keywords="Stalk"|plainurl=yes}} Algebraic Geometry ]. 9780387902449 . Hartshorne . Robin . 1977 . 10.1007/978-1-4757-3849-0.
- Book: 10.1017/CBO9780511661761 . [{{Google books|JOglpKIw6kIC|keywords="Stalk"|plainurl=yes}} Sheaf Theory ]. 1975 . Tennison . B. R. . 9780521207843.
External links
