Stalk - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

Let $\mathcal F$ be a sheaf on a topological space $X$ , and let $x\in X$ . The stalk $\mathcal F_x$ is the collection of germs of sections of $\mathcal F$ near $x$ . Concretely, a germ is represented by a pair $(U,s)$ , where $U$ is an open neighborhood of $x$ and $s\in\mathcal F(U)$ . Two representatives $(U,s)$ and $(V,t)$ define the same germ if $s$ and $t$ agree on some neighborhood of $x$ contained in $U\cap V$ .

Thus the stalk forgets behavior away from $x$ while retaining everything visible arbitrarily close to it. For the sheaf of continuous real-valued functions, $\mathcal F_x$ consists of germs of continuous functions at $x$ . For a structure sheaf on an affine scheme, the stalk at a prime $\mathfrak p$ is the local ring $A_{\mathfrak p}$ .