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.

  • $\begingroup$ Important example in algebraic geometry: Let $X$ be a scheme, let $\mathcal{F}$ be a quasi-coherent sheaf on $X,$ and let $x\in X.$ If $U = \operatorname{Spec}A$ is an affine open of $X$ containing $x$ and $x$ corresponds to a prime ideal $\mathfrak{p}\subseteq A,$ then $\mathcal{F}_x \cong \mathcal{F}(U)_{\mathfrak{p}}$ (the stalk at $x$ is the localization of the $A$-module $\mathcal{F}(U)$ at $\mathfrak{p}$). In particular, if $\mathcal{O}_X$ is the structure sheaf of $X,$ then $\mathcal{O}_{X,x}\cong A_{\mathfrak{p}}.$ $\endgroup$ – Stahl Aug 31 '18 at 22:34
  • $\begingroup$ To my mind, examples 2 and 3 are almost tautological - a stalk of a sheaf is pretty much defined to be the set of germs of sections of the sheaf at the point. (Though certainly, if you've seen the concept of "germ" before, then it could be useful to point out that stalks of sheaves are a direct generalization.) $\endgroup$ – Daniel Schepler Aug 31 '18 at 22:54
  • $\begingroup$ @DanielSchepler Exactly my frustration! $\endgroup$ – Santana Afton Aug 31 '18 at 22:58
  • $\begingroup$ Of course, if $x$ is an isolated point, then $\mathscr{F}_x \simeq \Gamma(\{ x \}, \mathscr{F})$. $\endgroup$ – Daniel Schepler Sep 1 '18 at 0:01

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.
  • $\begingroup$ These look great! What is this notation for a sheaf: both $f_*\mathcal{F}$ and $f^{-1}\mathcal{F}$? $\endgroup$ – Santana Afton Sep 1 '18 at 13:37
  • $\begingroup$ $f_\ast\mathcal{F}$ is the pushforward of $\mathcal{F}$ along $f,$ AKA the direct image of $\mathcal{F},$ defined by $f_\ast\mathcal{F}(U) = \mathcal{F}(f^{-1}(U)).$ $f^{-1}\mathcal{F}$ is the inverse image sheaf or pullback of $\mathcal{F},$ defined as the sheafification of $U\mapsto \underset{V\supseteq f(U)\textrm{ open}}{\varinjlim}\mathcal{F}(V).$ $\endgroup$ – Stahl Sep 1 '18 at 14:22

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