Stalk (sheaf)

There is another approach to defining a stalk that is useful in some contexts. Choose a point ${\displaystyle x}$ of ${\displaystyle X}$, and let ${\displaystyle i}$ be the inclusion of the one point space ${\displaystyle \{x\}}$ into ${\displaystyle X}$. Then the stalk ${\displaystyle {\mathcal {F}}_{x}}$ is the same as the inverse image sheaf ${\displaystyle i^{-1}{\mathcal {F}}}$. Notice that the only open sets of the one point space ${\displaystyle \{x\}}$ are ${\displaystyle \{x\}}$ and ${\displaystyle \emptyset }$, and there is no data over the empty set. Over ${\displaystyle \{x\}}$, however, we get:

${\displaystyle i^{-1}{\mathcal {F}}(\{x\})=\varinjlim _{U\supseteq \{x\}}{\mathcal {F}}(U)=\varinjlim _{U\ni x}{\mathcal {F}}(U)={\mathcal {F}}_{x}.}$

The constant sheaf ${\displaystyle {\underline {S}}}$ associated to some set (or group, ring, etc). ${\displaystyle S}$ has the same set or group as stalks at every point: for any point ${\displaystyle x}$, pick an open connected neighborhood. The sections of ${\displaystyle {\underline {S}}}$ on a connected open equal ${\displaystyle S}$ and restriction maps are the identities. Therefore, the direct limit collapses to yield ${\displaystyle S}$ as the stalk.

For example, in the sheaf of analytic functions 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 locally 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!)

In contrast, for the sheaf of smooth functions on a smooth manifold, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. For example, let ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ 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, ${\displaystyle f}$ 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 ${\displaystyle f}$ from its germ. Even if we know in advance that ${\displaystyle f}$ 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, ${\displaystyle f}$ could equal the constant function with value 1. We cannot even reconstruct ${\displaystyle f}$ on a small open neighborhood ${\displaystyle U}$ containing the origin, because we cannot tell whether the bump of ${\displaystyle f}$ fits entirely in ${\displaystyle U}$ or whether it is so large that ${\displaystyle f}$ is identically one in ${\displaystyle U}$.

On the other hand, germs of smooth functions can distinguish between the constant function with value one and the function ${\displaystyle 1+e^{-1/x^{2}}}$, 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 ${\displaystyle 1+e^{-1/x^{2}}}$ 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.)

On an affine scheme ${\displaystyle X=\mathrm {Spec} (A)}$, the stalk of a quasi-coherent sheaf ${\displaystyle {\mathcal {F}}}$ corresponding to an ${\displaystyle A}$-module ${\displaystyle M}$ in a point ${\displaystyle x}$ corresponding to a prime ideal ${\displaystyle p}$ is just the localization ${\displaystyle M_{p}}$.

On any topological space, the skyscraper sheaf associated to a closed point ${\displaystyle x}$ and a group or ring ${\displaystyle G}$ has the stalks ${\displaystyle 0}$ off ${\displaystyle x}$ and ${\displaystyle G}$ on ${\displaystyle x}$—hence the name skyscraper. This idea makes more sense if one adopts the common visualisation of functions mapping from some space above to a space below; with this visualisation, any function that maps ${\displaystyle G\to x}$ has ${\displaystyle G}$ positioned directly above ${\displaystyle x}$. The same property holds for any point ${\displaystyle x}$ 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 resolutions, used for example in algebraic geometry to get functorial injective resolutions of sheaves.