Hadamard regularization: Difference between revisions

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The '''stalk''' of a [[sheaf (mathematics)|sheaf]] is a [[mathematics|mathematical]] construction capturing the behaviour of a sheaf around a given point.
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==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 <math>\mathcal{F}</math> on that small neighborhood should be the same as the behavior of <math>\mathcal{F}</math> 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 <math>\mathcal{F}</math> at ''x'', usually denoted <math>\mathcal{F}_x</math>, is:
:<math>\mathcal{F}_x := \varinjlim_{U\ni x} \mathcal{F}(U).</math>
Here the [[direct limit]] is indexed over all the [[open set]]s containing ''x'', with order relation induced by inclusion ( <math>U<V</math>, if <math>U\supset V</math>). By definition (or [[universal property]]) of the direct limit, an element of the stalk is an equivalence class of elements <math>x_U \in \mathcal F(U)</math>, where two such sections <math>x_U</math> and  <math>x_V</math> are considered [[equivalence class|equivalent]] if the restrictions of the two sections coincide on some neighborhood of x.
 
===Alternative definition===
There is another approach to defining a stalk which 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 <math>\mathcal{F}_x</math> is the same as the [[inverse image functor|inverse image]] sheaf <math>i^{-1}\mathcal{F}</math>.  Notice that the only open sets of the one point space {''x''} are {''x''} and ∅, and there is no data over the empty set. Over {''x''}, however, we get:
:<math>i^{-1}\mathcal{F}(\{x\}) = \varinjlim_{U\supseteq\{x\}} \mathcal{F}(U) = \varinjlim_{U\ni x} \mathcal{F}(U) = \mathcal{F}_x.</math>
 
==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 group]]s or [[ring (mathematics)|rings]].
 
There is a natural morphism ''F''(''U'') → ''F''<sub>''x''</sub> 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 (mathematics)|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 <math>\underline S</math> associated to some set or group etc. ''S'' has the same set or group as stalks at every point: for any point ''x'', pick an open [[connected space|connected]] neighbourhood. The sections of <math>\underline S</math> on a connected open equal ''S'' and restriction maps are the identities. Therefore the direct limit collapses to yield ''S'' as the stalk.
 
===Sheaves of analytic functions===
 
For example, in the sheaf of [[analytic function]]s 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 function]]s 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 <math>1 + e^{-1/x^2}</math>, 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 <math>1 + e^{-1/x^2}</math> 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===
 
On an [[affine scheme]] ''X''=[[spectrum of a ring|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 of a module|localization]] ''M''<sub>''p''</sub>.
 
===Skyscraper sheaf===
 
On any topological space, the [[skyscraper sheaf]] associated to a [[closed set|closed point]] ''x'' and a group or ring ''G'' has the stalks ''0'' off ''x'' and ''G'' in ''x'' &mdash; 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 resolution]]s, used for example in [[algebraic geometry]] to get [[functor]]ial [[injective resolution]]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]]), 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 [[exact functor|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 [[presheaf|presheaves]]. However, stalks of sheaves and presheaves are tightly linked:
*Given a presheaf ''P'' and its [[sheafification]] ''F'', the stalks of ''P'' and ''F'' agree.
 
==References==
See the [[Sheaf (mathematics)#References|References in the article on sheaves]].
 
[[Category:Sheaf theory]]
 
[[de:Garbe (Mathematik)#Halme und Keime]]

Latest revision as of 08:00, 11 January 2015

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