Germs of Functions
Let’s take the structure sheaves we defined last time and consider the stalks at a point . It turns out that since we’re working with sheaves of -algebras, we can sort of shortcut the messy limit process.
As before, given some open neighborhood of , we let be the algebra of smooth functions — as smooth as is itself — on . Now we define to be the ideal of those functions which vanish on some neighborhood of . Then we define the quotient
Notice that we have effectively pushed our limiting process into the definition of the ideal , where for each open neighborhood of we get an ideal of functions vanishing on . The ideal we care about is the union over all such neighborhoods , and the process of taking this union is effectively a limit.
Anyhow, there’s still the possibility that this depends on the from which we started. But this is actually not the case; we get a uniquely defined algebra no matter which neighborhood of we start from.
Indeed, I say that there is an isomorphism . In the one direction, this is simply induced by the restriction map — if two functions are equal on some neighborhood of in , then they’re certainly equal on some neighborhood of in . And this restriction is just as clearly injective, since if two functions are equivalent in then they must agree on some neighborhood of , which means they were already equivalent in .
The harder part is showing that this map is surjective, and thus an isomorphism. But given , let be an open neighborhood of whose closure is contained in — we can find one since must contain a neighborhood of homeomorphic to a ball in , and we can certainly find within such a neighborhood. Anyhow, we know that there exists a bump function which is identically on and supported within . We can thus define a smooth function on all of by setting inside and elsewhere. Since and agree on the neighborhood of , they are equivalent in , and thus every equivalence class in has a representative coming from .
We write the stalk as , or sometimes if the manifold 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 , but is behavior in an “infinitesimal neighborhood” around . 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.