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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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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Are sheaves determined by their stalks? [duplicate]
Let $X$ be a scheme (or just a topological space) and $F$ and $G$ be two sheaves on them.
If $F_x=G_x$ for all $x \in X$ i.e $F$ and $G$ are equal at stalks at every point, is $F \cong G$ ?
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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 $X$ ...
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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 &...
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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 ...
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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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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, ...
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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 ...
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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 ...
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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 $\...
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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 $x\neq 0$}\\
...
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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: [...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $M_n \subset \mathcal{E}_n$ ...
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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 ...
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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 ...
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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 $\...
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Book on differential geometry which uses germs
I would like to start learning differential geometry and I find the concept of germs very beautiful and enlightening. Unfortunately, I have Lee's Introduction to Smooth Manifolds, which only mentions ...
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$\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 $...
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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 ...
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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 $k$.
...
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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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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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$\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 "...
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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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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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