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

\displaystyle\mathcal{O}_{M,p}(U)=\mathcal{O}_M(U)/\mathcal{Z}_{M,p}(U)

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

   

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