The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
2answers
36 views

Ring of germs of smooth functions on $\mathbb{R}^{n}$ in $0$.

First of all, I'm quite new to this theory, so it may be very dumb questions. Sorry for that. Let $R$ be the ring of germs of $C^{\infty}$ functions on $\mathbb{R}^{n}$ in $0$. Let $K$ be the ...
5
votes
2answers
66 views

Germs and local ring.

I'm having trouble understanding the following argument (which I believe to be somewhat incomplete or flawed). Let $A=C(X)$ be the set of continuous functions from the topological space $X$ to the ...
2
votes
0answers
33 views

on the infinite power of the maximal ideal

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat functions, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
2
votes
1answer
66 views

How to show that the stalk over p is a vector space

We were given this information: "A set $U \subseteq \mathbb{R}^n$ is open if for every $p \in U$ there exists an $\epsilon =\epsilon(p) > 0$ so that for every $y \in \mathbb{R}^n$ such that ...
0
votes
1answer
51 views

Value of the derivative of a function at a point depends only on the germ at that point

Suppose that f : I → R is a $C^∞$ function defined on an open subset I ⊆ R. How can I show that for $a \in I$ the value $f^n (s)$, n = 1, 2, 3, . . . of the derivative of $f$ of order n at s depends ...
0
votes
2answers
96 views

Stuck on this proof that $ord(f) = ord(g)$

Let $f, g: \mathbb R \to \mathbb R$ be smooth maps such that $f(a) = g(a') = 0$ and let $\tau, \sigma : \mathbb R \to \mathbb R$ be diffeomorphisms such that $$ \tau \circ f = g \circ \sigma$$ ...
1
vote
1answer
39 views

Finding explicit discrete valuation of ring of germs of analytic functions on $\mathbb{C}$

I found interesting problem set http://www.math.lsa.umich.edu/~kesmith/593hmwk2-2014.pdf and I noted Problem 3-3. And I found another version: Let $\mathcal{U}$ be the subset of all open sets of ...
0
votes
0answers
16 views

question on singularity theory and normal forms of local function germs

Hello I realize this might be a difficult topic but I was just trying to practice for my advanced class on singularities I wanted to compute normal forms for corank 2 singularities of the function ...
2
votes
1answer
115 views

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 ...
3
votes
1answer
60 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 ...
0
votes
0answers
47 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 ...
1
vote
0answers
66 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, ...
4
votes
1answer
80 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 ...
2
votes
0answers
58 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 ...
6
votes
1answer
153 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 ...
7
votes
2answers
512 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 ...
2
votes
1answer
53 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 ...
6
votes
1answer
113 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 ...
6
votes
1answer
109 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: ...
1
vote
0answers
83 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 ...
3
votes
0answers
329 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 ...