stalk
Let $F$ be a presheaf^{} over a topological space^{} $X$ with values in an abelian category^{} $\mathcal{A}$, and suppose direct limits^{} exist in $\mathcal{A}$. For any point $p\in X$, the stalk ${F}_{p}$ of $F$ at $p$ is defined to be the object in $\mathcal{A}$ which is the direct limit of the objects $F(U)$ over the directed set of all open sets $U\subset X$ containing $p$, with respect to the restriction^{} morphisms^{} of $F$. In other words,
$${F}_{p}:=\underset{U\ni p}{\underset{\u27f6}{lim}}F(U)$$ 
If $\mathcal{A}$ is a category consisting of sets, the stalk ${F}_{p}$ can be viewed as the set of all germs of sections^{} of $F$ at the point $p$. That is, the set ${F}_{p}$ consists of all the equivalence classes^{} of ordered pairs^{} $(U,s)$ where $p\in U$ and $s\in F(U)$, under the equivalence relation $(U,s)\sim (V,t)$ if there exists a neighborhood^{} $W\subset U\cap V$ of $p$ such that ${\mathrm{res}}_{U,W}s={\mathrm{res}}_{V,W}t$.
By universal properties^{} of direct limit, a morphism $\varphi :F\u27f6G$ of presheaves over $X$ induces a morphism ${\varphi}_{p}:{F}_{p}\u27f6{G}_{p}$ on each stalk ${F}_{p}$ of $F$. Stalks are most useful in the context of sheaves, since they encapsulate all of the local data of the sheaf at the point $p$ (recall that sheaves are basically defined as presheaves which have the property of being completely characterized by their local behavior). Indeed, in many of the standard examples of sheaves that take values in rings (such as the sheaf ${\mathcal{D}}_{X}$ of smooth functions, or the sheaf ${\mathcal{O}}_{X}$ of regular functions), the ring ${F}_{p}$ is a local ring, and much of geometry is devoted to the study of sheaves whose stalks are local rings (socalled “locally ringed spaces”).
We mention here a few illustrations of how stalks accurately reflect the local behavior of a sheaf; all of these are drawn from [1].

•
A morphism of sheaves $\varphi :F\u27f6G$ over $X$ is an isomorphism^{} if and only if the induced morphism ${\varphi}_{p}$ is an isomorphism on each stalk.

•
A sequence^{} $F\u27f6G\u27f6H$ of morphisms of sheaves over $X$ is an exact sequence^{} at $G$ if and only if the induced morphism ${F}_{p}\u27f6{G}_{p}\u27f6{H}_{p}$ is exact at each stalk ${G}_{p}$.

•
The sheafification^{} ${F}^{\prime}$ of a presheaf $F$ has stalk equal to ${F}_{p}$ at every point $p$.
References
 1 Robin Hartshorne, Algebraic Geometry^{}, Springer–Verlag New York Inc., 1977 (GTM 52).
Title  stalk 

Canonical name  Stalk 
Date of creation  20130322 12:37:15 
Last modified on  20130322 12:37:15 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  9 
Author  djao (24) 
Entry type  Definition 
Classification  msc 54B40 
Classification  msc 14F05 
Classification  msc 18F20 
Related topic  Sheaf 
Related topic  LocalRing 