Stalk of a sheaf (${\mathcal{F}}_x$)

The stalk ${\mathcal{F}}_x$ is the direct limit of $\mathcal{F}(U)$ over neighborhoods $U$ of $x$. A morphism of sheaves is an isomorphism or forms an exact sequence exactly when it does so on every stalk.

The derived costalk at $x$ is $i_x^{!}E$, where $i_x:\{x\}\hookrightarrow X$. It computes local cohomology supported at $x$. On an oriented $N$-manifold the constant sheaf has costalk $\mathbb{Q}[-N]$, whose cohomology is in degree $N$.

For presheaves of sets, $F_x={\underrightarrow{\lim }}_{U\ni x}F(U)$. This functor preserves finite limits because the neighborhood system is filtered and filtered colimits commute with finite limits in sets. Its right adjoint sends a set $S$ to the presheaf taking the value $S$ on opens containing $x$ and the singleton on other opens.

The joint stalk functor $\mathbf{Sh}(X)\to {\mathbf{Set}}^X$ is faithful: equality of germs makes two images of a section locally equal, and the sheaf uniqueness axiom makes them globally equal. Its right adjoint is the product of the set-valued skyscraper sheaves at all points. The Beck comonadicity theorem then makes the joint stalk functor comonadic.

For a point inclusion $i:\{x\}\hookrightarrow X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$ has sections $A$ on opens containing $x$ and zero on other opens. Its restriction maps are identities or maps to zero. If $A\ne 0$, its nonzero stalks occur at every point of the closure $\overline{\{x\}}$: exactly those points whose every neighbourhood contains $x$. Thus the familiar assertion that only the stalk at $x$ is nonzero requires $x$ to be a closed point. It is always flasque, giving the stated skyscraper sheaf cohomology.

For any point inclusion $i:\{x\}\hookrightarrow X$, the skyscraper sheaf $i_{\ast }A$ has restriction maps that are identities or maps to zero. It is flasque, so its higher sheaf cohomology vanishes and its global sections are $A$. This requires no assumption that $x$ is closed.

The germ $s_x$ of a section $s\in \mathcal{F}(U)$ is its image in the stalk of a sheaf ${\mathcal{F}}_x$. It is zero exactly when $s$ restricts to zero on some neighbourhood of $x$.

The support of a sheaf section is the closed setIts complement is open because a zero germ of a sheaf section is represented by a section that vanishes on a neighbourhood.