Further examples of stalks

Question

I am currently learning about stalks for the first time. In my exploration online about the topic, I routinely run into the same three examples:

• In a constant sheaf associated with an abelian group $A$, any stalk is isomorphic to $A$.
• In the sheaf of real-valued continuous functions, stalks are all germs at the given point.
• In the sheaf of complex-analytic functions, stalks are all germs at the given point.

I understand why these examples are ubiquitous. Stalks are about local behavior, and germs are an example of functions locally behaving the same being identified. Great!

That said, I would love to see more examples of stalks outside of these.

Answer 1

Here are some examples that might be helpful to calculate to get more experience/intuition:

• Let $X$ be a topological space, and let $x\in X.$ Let $i : \{x\}\to X$ be the inclusion of $x$ into $X.$ Let $A$ be any set, and let $\underline{A}$ be the constant sheaf with value $A$ on $\{x\}.$ Then $i_\ast\underline{A}$ is called a skyscraper sheaf, and $\left(i_\ast\underline{A}\right)_y = A$ if $y\in\overline{\{x\}},$ and $\left(i_\ast\underline{A}\right)_y = \{\ast\}$ otherwise.
• Let $X$ be a topological space, $Z\subseteq X$ be a closed subset, $i : Z\to X$ be the inclusion of $Z$ into $X,$ and $\mathcal{F}$ a sheaf (of sets) on $Z.$ Then $(i_\ast\mathcal{F})_x = \mathcal{F}_x$ if $x\in Z,$ and $(i_\ast\mathcal{F})_x = \{\ast\}$ otherwise.
• Let $f : X\to Y$ be a continuous map of topological spaces, and let $\mathcal{F}$ be a sheaf on $Y.$ Then for any $x\in X,$ $\left(f^{-1}\mathcal{F}\right)_x\cong\mathcal{F}_{f(x)}.$
• You can also check that if $\mathcal{F}$ is a sheaf on $X,$ and $i : \{x\}\to X$ is the inclusion of a point, then $i^{-1}\mathcal{F}$ is the constant sheaf associated to $\mathcal{F}_x$ on $\{x\}.$
• You can find an example of a continuous map of spaces $f : X\to Y,$ a point $x\in X,$ and a sheaf $\mathcal{F}$ on $X$ such that $\left(f_\ast\mathcal{F}\right)_{f(x)}\not\cong\mathcal{F}_x.$
• If $\mathcal{F}$ is a quasicoherent sheaf on a scheme $X,$ then $\mathcal{F}_x = M_\frak{p},$ where $\operatorname{Spec}A = U\subseteq X$ is an affine open containing $x,$ $\mathfrak{p}\subseteq A$ is the prime ideal of $A$ corresponding to $x,$ and $M = \Gamma(U,\mathcal{F}).$
• A special case of the last example is when $\mathcal{F} = \mathcal{O}_X$ is the structure sheaf of $X.$ In this case, $\mathcal{O}_{X,x} = A_\mathfrak{p},$ where $x, A,$ and $\mathfrak{p}$ are as before.