The Unapologetic Mathematician

Mathematics for the interested outsider

Germs of Functions

Let’s take the structure sheaves we defined last time and consider the stalks at a point p\in M. It turns out that since we’re working with sheaves of \mathbb{R}-algebras, we can sort of shortcut the messy limit process.

As before, given some open neighborhood U of p, we let \mathcal{O}_M(U) be the algebra of smooth functions — as smooth as M is itself — on U. Now we define \mathcal{Z}_{M,p}(U) to be the ideal of those functions which vanish on some neighborhood of p. Then we define the quotient


Notice that we have effectively pushed our limiting process into the definition of the ideal \mathcal{Z}_{M,p}(U), where for each open neighborhood V\subseteq U of p we get an ideal of functions vanishing on V. The ideal we care about is the union over all such neighborhoods V, and the process of taking this union is effectively a limit.

Anyhow, there’s still the possibility that this depends on the U from which we started. But this is actually not the case; we get a uniquely defined algebra \mathcal{O}_{M,p}=\mathcal{O}_{M,p}(U) no matter which neighborhood U of p we start from.

Indeed, I say that there is an isomorphism \mathcal{O}_{M,p}(M)\to\mathcal{O}_{M,p}(U). In the one direction, this is simply induced by the restriction map \mathcal{O}_M(M)\to\mathcal{O}_M(U) — if two functions are equal on some neighborhood of p in M, then they’re certainly equal on some neighborhood of p in U. And this restriction is just as clearly injective, since if two functions are equivalent in \mathcal{O}_{M,p}(U) then they must agree on some neighborhood of p, which means they were already equivalent in \mathcal{O}_{M,p}(M).

The harder part is showing that this map is surjective, and thus an isomorphism. But given U, let V be an open neighborhood of p whose closure is contained in U — we can find one since U must contain a neighborhood of p homeomorphic to a ball in \mathbb{R}^n, and we can certainly find V within such a neighborhood. Anyhow, we know that there exists a bump function \phi which is identically 1 on V and supported within U. We can thus define a smooth function g\in\mathcal{O}_M(M) on all of M by setting g(q)=\phi(q)f(q) inside U and g(q)=0 elsewhere. Since f and g agree on the neighborhood V of p, they are equivalent in \mathcal{O}_{M,p}(U), and thus every equivalence class in \mathcal{O}_{M,p}(U) has a representative coming from \mathcal{O}_{M,p}(M).

We write the stalk as \mathcal{O}_{M,p}, or sometimes \mathcal{O}_p if the manifold M is clear from context, and we call the equivalence classes of functions in this algebra “germs” of functions. Thus a germ subsumes not just the value of a function at a point p, but is behavior in an “infinitesimal neighborhood” around p. Some authors even call the structure sheaf of a manifold — especially a complex analytic manifold (which we haven’t really discussed yet) — the “sheaf of germs” of functions on the manifold, which is a little misleading since the germs properly belong to the stalks of the sheaf. Luckily, this language is somewhat outmoded.

March 24, 2011 Posted by | Differential Topology, Topology | 5 Comments