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.

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March 24, 2011 - Posted by John Armstrong | Differential Topology, Topology

5 Comments »

Are you missing an equals or defined to be sign in the first displayed formula (the quotient)?

Comment by Robert | March 25, 2011 | Reply
sorry, fixed

Comment by John Armstrong | March 26, 2011 | Reply
Just checking that ‘algebra’ in par. 2 is something meeting this definition: http://en.wikipedia.org/wiki/Associative_algebra

which seems to fit, but as things get more complicated, uncertainties multiply …

Comment by Avery D Andrews | March 27, 2011 | Reply
Yes: like a ring, but built on a vector space, not just a set.

Comment by John Armstrong | March 27, 2011 | Reply
[...] we take a manifold with structure sheaf . We pick some point and get the stalk of germs of functions at . This is a real algebra, and we define a “tangent vector at [...]

Pingback by Tangent Vectors at a Point « The Unapologetic Mathematician | March 29, 2011 | Reply

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