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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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Well-definedness of the projection associated to the sheaf of germs of a presheaf

I'm currently reading Izu Vaisman's Cohomology and differential forms ($1973$) having never studied sheaf theory before, so I will briefly write down the definitions in case they don't match with ...
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Show that the ring of germs at 0 is local

I have been posed the following problem: Prove that $C_0$ (which is the ring of germs of continuous functions at $0$) is a local ring, with maximal ideal $M$ consisting of the germs of functions that ...
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If f(x) and g(x) have equal value and the same derivative at x=0, do they define the same germ at x=0?

In Wikipedia page on germ, it says that Given a point $x$ of a topological space $X$, and two maps $f,g\,:\,X\to Y$ (where $Y$ is any set), then $f$ and $g$ define the same germ at $x$ if there is a ...
Luzveraz's user avatar
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Is it possible to consistently and naturally define this subset of Hardy field?

Consider Hardy field $H$, the field of germs of functions at positive infinity. Can we define $H_I\subset H$, such that it would have the following properties: $H_I$ is an integral domain. For each ...
Anixx's user avatar
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Geometric notion of modality

I'm reading Singularity Theory (one of the authors is Arnold). I am a bit confused on the concept of modality. Is there an easy geometric description of it? The book has a relation between codimension,...
quantum's user avatar
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Addition and multiplication in $C_p^{\infty}$

I am trying to define addition and multiplication in the set of all germs of $C^{\infty}$ functions on $\mathbb{R}^n$ at a point $p$. Before continuing, I would like to pose a quick question: I know ...
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Showing that $I/I^2 \simeq T_{x_0}^{\ast} X.$

Let $X$ be a Poisson manifold with Poisson bivector field $\Pi.$ Let $x_{0} \in X$ be such that $\Pi (x_{0}) = 0.$ Let $\mathcal O (X)_{x_{0}}$ denote the ring of germs of the smooth functions on $X$ ...
Anil Bagchi.'s user avatar
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"sheaf of germs of functions" VS "sheaf of functions"?

I encountered phrases such as "sheaf of germs of continuous functions" (e.g., page 164 of the book "Foundations of Differentiable Manifolds and Lie Groups" by Warner 1983) and &...
user470904's user avatar
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156 views

Can germs be defined as a quotient of vector spaces?

Summary: Let $M$ be a smooth manifold and $p\in M$. I know of two notions of germs of functions at $p$, the more restrictive of which can be written as a quotient vector space. I am curious whether ...
WillG's user avatar
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The only maximal ideal of the set of all function germs around $p$

Here is the definition we are using for the set of all function germs around p: Now, I want to show that $m(p) := \{\bar{\phi} \in \mathcal{\varepsilon}(p)| \bar{\phi}(p) = 0\}$ is the only maximal ...
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Showing that the vector space structures induced by $\alpha$ and $\beta \alpha$ are equal(#3.3.13).

In the context of "The Tangent Space" and after defining "Germs" and to prove that the vector space structure on the tangent space does not depend on the choice of charts, here is ...
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9 votes
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Why are local rings called local?

I gather that rings of germs of functions at a point $p$ on a manifold/variety/etc. are local with the maximal ideal containing exactly the germs of functions which vanish at $p$. So in some sense, ...
Vercassivelaunos's user avatar
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Units in the ring of germs of a continuous functions at $p$.

Let $X$ be a metric space.Consider the set $A=\{(U,f): U $open set containing $p$,$f:U\to \mathbb R$ continuous$\}$.Define an equivalence relation on $A$ by $(U,f)\sim (V,g)$ if $\exists$ an open set ...
Kishalay Sarkar's user avatar
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On kernels in the category of sets (more specifically, kernels of maps between stalks)

There's a functor that taakes a presheaf $\mathcal F$ on $X$ and assigns to it the stalk at $x$, written $\mathcal F_x$. There's also a result saying that this functor is exact. In proving this, we ...
user557's user avatar
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When are germs of real valued continuous functions induced by global sections?

Let $X$ be a topological space and $\cal F$ the sheaf of continuous real valued functions on it. Somewhere I read the following definition: Let $x\in X$. Then the 'stalk' of $\cal F$ at $x$ is $\...
leoli1's user avatar
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Radius of convergence of a non-zero function with zero Taylor series

A classic example of a nonzero function with identically zero Taylor expansion is the following: \begin{equation*} f(x)= \begin{cases} e^{-\frac{1}{x^2}}\quad &\text{if $x\neq 0$}\\ ...
logarithm's user avatar
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understanding analytic continuation along a path using germ

Let $\mathbb{O}$ is the sheaf of germs of holomorphic functions on $\mathbb{C}$, $p: \mathbb{O} \rightarrow \mathbb{C}$ defined by sending the germ $f_a$ to $a \in \mathbb{C}$. Given a path $\gamma: [...
CuriousAlpaca's user avatar
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How is the Algebra $C_{p}^{\infty}(U)$ of germs of $C^\infty$ functions in $U$ at $p$ is the Same as $C_{p}^{\infty}(M)$

Hi i am reading An introduction to manifolds by Loring and have some doubts in remark 8.2. It is written that If $U$ is an open set containing $p$ in $M$ then the algebra $C_{p}^{\infty}(U)$ of germs ...
Jason's user avatar
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If $p \in U\subseteq M$ is an open subset, then $C^\infty_p(U) = C^\infty_p(M)$.

Let $M$ be a (smooth) manifold and $p \in M$. We define $C^\infty_p(M)$ to be the set of equivalence classes of smooth functions $M \to \mathbb{R}$ that are identified when they agree on some ...
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We have a germ that contains finite, infinite, oscillating and infinitesimal parts. How can we separate them?

Suppose a germ of a function at a point (from one direction) or at infinity has finite, infinite, oscillating and infinitesimal parts. Is it possible to separate it into all four, so that The ...
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Ring structure on a space of functions between vector spaces?

In this Wikipedia article about jets, in the section about rigorous definitions, for the algebro-geometric definition, they take the vector space $C^\infty_p(\mathbb R^n,\mathbb R^m)$ of germs of ...
Vercassivelaunos's user avatar
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Meaning of "equality of functions as map germs along a curve"

I am trying to understand a notion given in this paper, summarized as the following: Let $c \colon I \to \mathbb{R}^3$ be a smooth, regular, arc-length parametrized curve with image $C$, where $I$ is ...
Mathdealer's user avatar
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What is a good reference to cite for the definition of forms in terms of germs of functions?

I would like to edit the Wikipedia articles [[Differential (infinitesimal)]] and [[Differential form]] to mention the definition of 1-forms in terms of germs of functions. None of the appropriate ...
shmuel's user avatar
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Problem 2.2 in Loring Tu's Introduction to Manifolds

another seemingly innocent problem in Loring Tu's Introduction to Manifolds is 2.2 (Algebra structure in $C_p^\infty$) that says Define carefully addition, multiplication, and scalar multiplication ...
l4teLearner's user avatar
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Quotient of Ideals in the Algebra of Function Germs

Let $\mathcal{E}_n$ be the algebra of germs at the origin of smooth functions $f : \mathbb{R}^n \supset U \to \mathbb{R}$, where $U$ is an open set containing $0$, and let $M_n \subset \mathcal{E}_n$ ...
Nick A.'s user avatar
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$\mathscr{O}_{X,a}\cong \mathscr{O}_{\mathbb{A}^n,a}/ I(X)\mathscr{O}_{\mathbb{A}^n,a}$

This problem is from Gathmann's notes problem 3.23 https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/alggeom-2014.pdf Let $X ⊂ \mathbb{A}^n$ be an affine variety, and let $a ∈ X$. Show ...
HCS's user avatar
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Is a germ equivalent to an infinite jet?

Not all smooth functions are analytic, as it is well known, so they in general cannot be represented as a power series. If we restrict our attention to analytic functions, then a specification of the ...
Bence Racskó's user avatar
3 votes
2 answers
261 views

Confusion about the notation of directional derivative

From An Introduction to Manifolds by Tu: $(1)$ Let $D_v = \sum v^i\frac{\partial}{\partial x^i}|_p$ where $v = [v^1, \dots, v^n]$ is a vector in $\Bbb R^n$ and $p = (p^1, \dots, p^n)$ a point in $\...
Oliver G's user avatar
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1 vote
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Book on differential geometry which uses germs

I would like to start learning differential geometry and I find the concept of germs very beautiful and enlightening. Unfortunately, I have Lee's Introduction to Smooth Manifolds, which only mentions ...
bd99's user avatar
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$\mathcal{C}^r$ topology in the germ space.

I'm reading the article "Local and simultaneous structural stability of certain diffeomorphisms - Marco Antônio Teixeira", and on the first page says "Denote $G^r$ the space of germs of involution at $...
Matheus Manzatto's user avatar
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Existence of a global involution extension.

I'm studying the paper "Local and simultaneous structural stability of certain diffeomorphisms. - Marco Antonio Teixeira". At the beginning of the paper, the author gives the following ...
Matheus Manzatto's user avatar
3 votes
1 answer
225 views

Stalks, Germs and Localisation

I've been trying to prove the following proposition: Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety over an algebraically closed field $k$. ...
Dave's user avatar
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11 votes
2 answers
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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,\ \...
Matheus Manzatto's user avatar
4 votes
3 answers
357 views

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 ...
Mikkel Rev's user avatar
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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 ...
user avatar
4 votes
0 answers
340 views

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 ...
TheMountainThatCodes's user avatar
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808 views

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}...
wanderingmathematician's user avatar
1 vote
1 answer
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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 $...
Yelon's user avatar
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$\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{...
Problemsolving's user avatar
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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 "...
Chill2Macht's user avatar
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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 ...
mathsislife's user avatar
1 vote
1 answer
41 views

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 $...
TheGeekGreek's user avatar
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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 $...
user8948's user avatar
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1 answer
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$(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,\...
MickG's user avatar
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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 ...
Chill2Macht's user avatar
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1 vote
1 answer
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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 ...
Justthisguy's user avatar
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4 votes
2 answers
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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 ...
MoebiusCorzer's user avatar
6 votes
2 answers
1k 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 ...
Hermès's user avatar
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2 votes
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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, ...
Dmitry Kerner's user avatar
3 votes
1 answer
107 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 $\...
John's user avatar
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