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Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
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1answer
56 views

Help with understanding the notation $\mathbb{C}\{f\}$

I am reading an article "Relative Cohomology and volume forms" of J. P. Francoise. Here the author considers the germ of a function $f\colon(\mathbb{C}^n,0) \to (\mathbb{C},0)$, and he speaks of the ...
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24 views

What does it mean to restrict a function germ to a set germ?

Two sets $S$ and $T$ define the same germ at a point $\xi$ in a topological space $M$ if there is a neighbourhood $U$ of $\xi$ such that $S \cap U = T \cap U$. Two functions $f,g : M \rightarrow \Bbb ...
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59 views

How special are the polynomials amongst the smooth functions?

This is a naive question, so perhaps the answer will be made obvious by the right remark. On a smooth manifold, there is no notion of polynomial (apart from constants). I would like to know if, ...
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1answer
57 views

Irreducibility of holomorphic functions in a neighborhood of a point

Let $D \subset \mathbb C^n$ be a domain and let $f \in \mathscr O(D)$, $f \not\equiv 0$ be a holomorphic function. Define $$ V_f = \bigl\{ z \in D : f(z) = 0 \bigr\}. $$ Let $p \in V_f$. Suppose ...
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43 views

Germs of $C^\infty$ functions near $0$ vs. germs of infinitely differentiable functions at $0$

I was reading Jean Dieudonné's nice counterexample about permutations of regular sequences (see here), then the following question came to my mind: What is the difference between the ring of germs ...
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17 views

The germ of finite order

Let Diff$^{\omega}(\mathbb{R}^2,0)$ be a group of analytic germs of $\mathbb{R}^2$, if $g\in$ Diff$^{\omega}(\mathbb{R}^2,0) $ has finite order,i.e. $\exists n$ s.t. $g^n=$id, does it imply $g$ just ...
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1answer
135 views

These germs make me sick!

I need a "mini-crashcours" concerning the space of germs of continuous functions in order to solve an exercise which requires me to show that limits in this space aren't always unique. We have ...
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2answers
313 views

A question about the strict transform on blow-ups

I arrived at the following phrase at a material that I'm reading: Let $\pi :N'\rightarrow N$ be the blow-up of center $P$. For a given $a\in\mathcal{O}$ and $P'\in\pi^{-1}(P)$, the strict ...
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1answer
50 views

A question in germs and multiplicity of zeroes.

Suppose that I have $N$ a bidimensional analytical manifold, $\mathcal{F}$ a foliation in $N$, and let $P\in N$. Being $\mathcal{O}$ the local ring of germs of holomorphic functions in $P$, and ...
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98 views

Why are germs of functions important?

Why is it necessary to define germs of functions (in my case, for foliations, but my question is in general)? does any inconsistency arises if instead of using a germ in some context, I use ...
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1answer
106 views

germ finitely determined

Does anyone know any result on finitely determined germs to help me prove that the germ $f(x,y)=x^3+ xy^3$ is $4$- determined? I tried using the definition of germs finitely determined, which is:$f: ...
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77 views

Calculating germs for complete holomorphic function

I'm trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where ...
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263 views

Nicer Description of Germs of Continuous Functions

I may be asking something a little out of my comfort zone at this moment so bear with me. Before I begin let me provide some background for the interested outsider: Let $X$ be a topological space ...