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Questions tagged [germs]

In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets

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Real advantage in considering germs of smooth functions

I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, ...
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Corollary of the Malgrange Preparation Theorem

Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that $$f(0,0)=0,\ \frac{\partial f}{\partial t} (0,0) = 0,\ldots, \frac{\partial^{k-1} f}{\partial t^{k-1}} (0,0) = 0,\ \...
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Germs: Why is it sensible to define a function on a collection of equivalence classes by its action on each element?

I am following Loring W. Tu in his second edition of 'An introduction to manifolds'. Here is a pdf-copy of the book. On page 87 he defines $C^\infty_p(M)$ as the set of germs of $C^\infty$-functions ...
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Equivalence of definitions for ring of germs $C_p^{\infty}(\mathbb R^n)$

I want to show the characterizations of $C_p^{\infty}(\mathbb R^n)$ are equivalent: $$A= \{[f] | \text{smooth} \ f: \mathbb R^n \to \mathbb R \}$$ $$B= \{[f] | \text{smooth} \ f: U_p \to \mathbb R \}...
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Which is the definition of the set of germs $C_p^{\infty}(\mathbb R^n)$? Does $C^{\infty}(U)$ consist of germs or functions?

Which one is the set known as $C_p^{\infty}(\mathbb R^n)$? The set of germs of smooth real-valued functions defined on $\mathbb R^n$ The set of germs of smooth real-valued functions defined on a ...
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Difference between germs of holomorphic functions and the functions themselves?

I'm learning about the space of germs of holomorphic functions. As far as I understand we define the space $\mathcal{O}_x$ to be the set of "germs" i.e. equivalence classes of functions that coincide ...
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Different definitions of derivation at a point

I have come across the following two different definitions for "derivation at a point" (for both definitions, assume M is a smooth manifold and $p \in M$): Def 1: A derivation at p is an $\mathbb{R}...
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help with the proof - regular points of a union of sets

I have such a proof from a leture which I don't understand. $M$ - complex manifold. Let $\left\{ A_j\right\}_{j\in J}$ be a nonempty, locally finite family of analytic subset of $M$, such that for $...
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60 views

$\mathcal{O}_{\mathbb{A}^n,x}\simeq \mathbb{C}[X_1,…,X_n]_{m(x)}$

I try to understand the proof of the following proposition: For every point $x=(x_1,...,x_n)\in \mathbb{A}^n$ there exists a natural isomorphism $$\mathcal{O}_{\mathbb{A}^n,x}\simeq \mathbb{...
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Does curvature affect these “infinitesimal isometry groups”?

This is a follow-up to a previous question, where it was concluded that at any point of Euclidean space, the "infinitesimal isometry group" is $O(n) \times \mathbb{R}^n$ and the "pointed infinitesimal ...
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Basic Jets and Germs calculations (Catastophe Theory)

I am a bit confused on these ideas, though they should be simple calculations. I have an exam for this in 2/3weeks and I need more examples. If you can explain 3 or more (or point me in the direction ...
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Question on germs of $C^1$ functions defined on a neighbourhood of $0$ in $\mathbb{R}$

Let $C^1_0\mathbb{R}$ denote the algebra of germs of $C^1$-functions in a neighbourhood of $0$ in $\mathbb{R}$. Furthermore let $I$ be the subalgebra of germs in $C^1_0\mathbb{R}$ that vanish at $...
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Two functions that have the same germ have the same derivation

Is this proof okay? (EDIT: is the claim even true?) I claim: $f$ and $g$ have the same germ at $p$ if and only if they have the same derivations at the same point. ($f$ and $g$ has the same germ at $...
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$(z_1,\dotsc,z_n)$ is a prime ideal of $\mathcal{O}_{\mathbb{C}^n,0}$: why?

I am studying the preliminary part of a course in Complex Geometry, and it deals with analytic sets and holomorphic functions, at the germ level. At some point, it says that the ideals $I_k:=(z_1,\...
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Is there a relationship between germs and Taylor coefficients?

For the definition of germ, please see below. I am having some difficulty internalizing the concept of germ due to an inability to think of concrete examples, which led to me having the following ...
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Question about the proof of the Malgrange Preparation Theorem

I have a question about the proof of what is sometimes called the generalized Malgrange preparation theorem. This proof is in both Brocker and Lander's "Differentiable Germs and Catastrophes" and ...
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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 ...
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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 ...
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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, ...
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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 $\...
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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 ...
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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 $$ \tau \circ f = g \circ \sigma$$ ...
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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 $\...
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478 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 ...
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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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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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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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355 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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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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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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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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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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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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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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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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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 ...