# 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 ...
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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 ...
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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,...
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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 $X$ ...
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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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### 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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### 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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### 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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