The stalk of a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
is a
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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 set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s containing ''
'', with order relation induced by reverse inclusion By definition (or
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
) of the direct limit, an element of the stalk is an equivalence class of elements
, where two such sections
and
are considered
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
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
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
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
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
or most categories of algebraic objects such as
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s or
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, which are namely
cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one ...
.
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
Germ or germs may refer to:
Science
* Germ (microorganism), an informal word for a pathogen
* Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually
* Germ layer, a primary layer of cells that forms during embryo ...
, 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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 function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s on an
analytic manifold
In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic ge ...
, 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
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
expansion, and all analytic functions are by definition locally equal to their power series. Using
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
, 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
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. For example, let
be a
bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bum ...
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
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
. The
Krull intersection theorem In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
says that this cannot happen for a Noetherian ring.)
Quasi-coherent sheaves
On an
affine scheme , the stalk of a
quasi-coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
''
'' corresponding to an
-module
in a point ''
'' corresponding to a
prime ideal is just the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
.
Skyscraper sheaf
On any topological space, the
skyscraper sheaf associated to a
closed point ''
'' and a group or ring
has the stalks ''0'' off ''
'' and ''
'' in ''
''—hence the name
skyscraper. 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
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
ial
injective resolution In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to def ...
s 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
In mathematics, the gluing axiom is introduced to define what a sheaf (mathematics), sheaf \mathcal F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor
::(X) \rightarrow C
to a cate ...
), 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
,
epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms ,
: g_1 \circ f = g_2 \circ f ...
, or
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a 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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
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:
*Given a presheaf
and its
sheafification , the stalks of
and
agree. This follows from the fact that the sheaf
is the image of
through the
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
(because the sheafification functor is left adjoint to the inclusion functor
) and the fact that left adjoints preserve colimits.
External links
stalkin
nLab
Sheaf theory
de:Garbe (Mathematik)#Halme und Keime