nLab stalk

Definition

Recall that a point $x$ of a topos $E$ is a geometric morphism

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

The stalk at $x$ of an object $e \in E$ is the image of $e$ under the corresponding inverse image morphism

$$x^* \,\colon\, E \to Set$$

i.e.

$$stalk_x(e) \,\coloneqq\, x^*(e) \,.$$

Special case of sheaf topoi

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

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

then the topos points of $E$ come precisely from the ordinary points

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

of the space $X$, where the direct image morphism

$$x_* \colon Set \to Sh(X)$$

sends every set to the skyscraper sheaf at this point of $X$. By the general Kan extension formula for the inverse image (see there) one finds in this case for any sheaf $F \in Sh(X)$ the stalk

$$\begin{aligned} stalk_x(F) & = \underset{ ({*} \to x^{-1}(V)) \in (const_{*}, x^{-1}) }{colim} F(V) \\ & = \underset{ { V \subset X } \atop { x \in V } }{ colim } F(V) \end{aligned} \,.$$

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 $(z, V)$ with $x \in V$ $z \in F(V)$, where the equivalence relation says that two such pairs $(z_1, V_1)$ and $(z_2, V_2)$ coincide if there is a third pair $(z,U)$ with $U \subset V_1$ and $U \subset V_2$ such that $z = z_1|_U = z_2|_U$.

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

Testing sheaf morphisms on stalks

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

Theorem

A morphism $f : A \to B$ of sheaves on $X$ is a

• monomorphism

• resp. epimorphism

• resp. isomorphism

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

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

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

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

Then:

• for $U \subset X$ the map $d_U : \Omega^n(U) \to Z^{n+1}(U)$ 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 $x \in X$ the map of stalks $d_x : stalk_x(\Omega^n(X)) \to stalk_x(Z^{n+1}(X))$ is an epimorphism.

Accordingly, the morphism $d : \Omega^n(X) \to Z^{n+1}(X)$ 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 $\mathcal{O}$, the stalk of the multiplicative group $\mathbb{G}_m$ at a point $x$ is the multiplicative group $\mathcal{O}_x^\times$ in the stalk local ring of the structure sheaf. (e.g. Milne, example 6.13)

Related concepts

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

• James Milne, section 6 of Lectures on Étale Cohomology

• Ofer Gabber, Shane Kelly, Points in algebraic geometry, J. Pure App. Alg. 219:10 (2015) 4667-4680 doi