Intereting Posts

Complex Polynomial with roots in uppar half plane.
Convergence and limit of a recursive sequence
The famous root of $x(1 – x^{12})^2(1 – x^{24})^2 = (1 – x^6)^7(1 – x^8)^4$
In a graph, the vertices can be partitioned $V=V_1\cup V_2$ so that at most half of all edges run within each part?
Example of a finite non-commutative ring without a unity
Find $\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^x\right)}{\ln\left(1+2^x\right)}}$
Showing a bijection between the set of ringhomomorphisms $\mathbb{Z} \to R$ and the set of $x \in R$ for which $x^2+1=0$
Line bundles of the circle
What's an intuitive way to think about the determinant?
Equivalent definitions of Lebesgue Measurability (Rudin and Royden)
To show that the complement of the kernel of an unbounded linear functional is path connected
Are there infinitely many rational outputs for sin(x) and cos(x)?
Why is the error function defined as it is?
How Do You Actually Do Your Mathematics?
Are Singleton sets in $\mathbb{R}$ both closed and open?

Is the ring $\mathcal{O}$ of germs of $C^{\infty}$ functions defined on the neighborhoods of $0\in\mathbb{R}$ the localization of the ring of $C^{\infty}$ functions on $\mathbb{R}$ at the maximal ideal $\mathfrak{m}$ of those functions which vanish at the origin? I can construct an injective map from $C^{\infty}_\mathfrak{m}$ to $\mathcal{O}$, but I’m having trouble showing surjectivity.

I know that this is true for regular functions (those which are locally quotient of polynomials) over an algebraically closed field.

One big difference I see between regular functions and $C^{\infty}$ functions is that $\mathcal{O}$ is not integral domain.

- How to construct a simple complex torus of dimension $\geq 2$?
- Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}$?
- Is such a field element an element of a subring?
- Associative ring with identity, inverses, divisors of zero and Artinianity
- Pushforward of pullback of a sheaf
- All tree orders are lattice orders?
- A group of order 30 has a normal 5-Sylow subgroup.
- Are “$n$ by $n$ matrices with rank $k$” an affine algebraic variety?
- Condition to be a group.
- Possible Irreducible but NOT prime!

Suppose $\xi$ is a germ, and let $\phi:U\to\mathbb R$ be a function defined on some open set $U$ containing $0$ which represents $\xi$. Pick open sets $V$, $W$ such that $0\in V\subseteq\bar V\subseteq W\subseteq \bar W\subseteq U$, and pick a smooth function $\psi:\mathbb R\to\mathbb R$ such that $\psi|_V\equiv1$ and $\psi|_{\mathbb R\setminus W}\equiv0$. Then $\psi\phi$, which in principle is defined only on $U$, can be extended to all of $\mathbb R$ by zero and remain smooth. The germ of $\psi\phi$ is still $\xi$.

This argument can be embellished to prove surjectivity.

- Does square difference prove that 1 = 2?
- How can people understand complex numbers and similar mathematical concepts?
- If $a^2$ divides $b^2$, then $a$ divides $b$
- Let $z_k = \cos\frac{2k\pi}n + i\sin\frac{2k\pi}n$. Show that $\sum_{k=1}^n|z_k-z_{k-1}|<2\pi$.
- Preservation of Lipschitz Constant by Convolutions
- Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$
- Formal power series coefficient multiplication
- Stochastic Integral
- Intersection distributing over an infinite union, union over an infinite intersection
- Homology of the loop space
- Expected value of average of Brownian motion
- An integral with $2017$
- $\mathbb Z$ is not finitely generated?
- How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
- Definition of $d (P (x ,y )dx)$