nLab
stalk

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Topos Theory

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Definition

Recall that a point xx of a topos EE is a geometric morphism

x:SetE. x : Set \to E \,.

The stalk at xx of an object eEe \in E is the image of ee under the corresponding inverse image morphism

x *:ESet x^* : E \to Set

i.e.

stalk x(e):=x *(e). stalk_x(e) := x^*(e) \,.

Special case of sheaf topoi

If EE is the category of sheaves on the category of open subsets Op(X)Op(X) of a topological space XX

E=Sh(X), E = Sh(X) \,,

then the topos points of EE come precisely from the ordinary points

(x:*X)Top, (x : {*} \to X) \in Top \,,

of the space XX, where the direct image morphism

x *:SetSh(X) x_* : Set \to Sh(X)

sends every set to the sheaf which is the constant functor on that set. By the general Kan extension formula for the inverse image (see there) one finds in this case for any sheaf FSh(X)F \in Sh(X) the stalk

stalk x(F) =colim (*x 1(V))(const *,x 1)F(V) =colim VX|xVF(V). \begin{aligned} stalk_x(F) & = colim_{({*} \to x^{-1}(V)) \in (const_{*}, x^{-1}) } F(V) \\ &= colim_{V \subset X | x \in V} F(V) \end{aligned} \,.

So for sheaves on (open subsets of) topological spaces the stalk at a given point is the colimit over all values of the sheaf on open subsets containing this point.

By the general definition of colimits in Set described at limits and colimits by example, the elements in this colimit can in turn be described as equivalence classes represented pairs (z,V)(z, V) with xVx \in V zF(V)z \in F(V), where the equivalence relation says that two such pairs (z 1,V 1)(z_1, V_1) and (z 2,V 2)(z_2, V_2) coincide if there is a third pair (z,U)(z,U) with UV 1U \subset V_1 and UV 2U \subset V_2 such that z=z 1| U=z 2| Uz = z_1|_U = z_2|_U.

for F=C()F = C(-) a sheaf of functions on XX, such an equivalence class, hence such an element in a stalk of FF is called a function germ.

Testing sheaf morphisms on stalks

For EE a topos with enough points, the behaviour of morphisms f:ABf : A \to B in EE can be tested on stalks:

Theorem

A morphism f:ABf : A \to B of sheaves on XX is a

if and only if every induced map of stalk sets stalk x(f):stalk x(A)stalk x(B)stalk_x(f) : stalk_x(A) \to stalk_x(B) is, for all xXx \in X

Proof

The statement for isomorphisms follows from the identification of sheaves with etale spaces (e.g. section II, 6, corollary 3 in MacLane-Moerdijk, Sheaves in Geometry and Logic). The statement for epimorphisms/monomorphisms is proposition 6 there.

Example

Let XX be a smooth manifold and let Ω n(X)\Omega^n(X) and Z n+1(X)Z^{n+1}(X) be the sheaves of differential nn-forms and that of closed differential (n+1)(n+1)-forms on XX, respectively, for some nn \in \mathbb{N}. Let

d:Ω n(X)Z n+1 d : \Omega^n(X) \to Z^{n+1}

be the morphism of sheaves that is given on each open subset by the deRham differential.

Then:

  • for UXU \subset X the map d U:Ω n(U)Z n+1(U)d_U : \Omega^n(U) \to Z^{n+1}(U) need not be epi, since not every closed form is exact;

  • but by the Poincare lemma every closed form is locally exact, so that for each xXx \in X the map of stalks d x:stalk x(Ω n(X))stalk x(Z n+1(X))d_x : stalk_x(\Omega^n(X)) \to stalk_x(Z^{n+1}(X)) is an epimorphism.

Accordingly, the morphism d:Ω n(X)Z n+1(X)d : \Omega^n(X) \to Z^{n+1}(X) is an epimorphism of sheaves.

This kind of example plays a crucial role in the computation of abelian sheaf cohomology, see the examples listed there.

Examples

For a locally ringed topos with structure sheaf 𝒪\mathcal{O}, the stalk of the multiplicative group 𝔾 m\mathbb{G}_m at a point xx is the multiplicative group 𝒪 x ×\mathcal{O}_x^\times in the stalk local ring of the structure sheaf. (e.g. Milne, example 6.13)

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

Revised on September 24, 2014 18:14:51 by fischermasamune? (108.88.37.146)