For question about morphism between groups, ring, topological spaces, vector space, categories, etcs... Please also use the correspondent tags (e.g. (group-theory), (ring-theory)) in order to precise the involved structure.

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2answers
45 views

group,subgroup and isomorphism

I study group theory now but I could not understand isomorphisms very well. In the book that I study I have seen that; $\mathbb{Z}_6=\{0,1,2,3,4,5\}$ is given and $H=\{0,2\}$ is a subgroup of the ...
5
votes
1answer
58 views

Relative effective Cartier divisors

I have two different definitions of a relative effective Cartier divisor. The first one is a bit outdated and defines the notion over analytic spaces, in the following way: Definition 1: Let $X$ be ...
5
votes
1answer
89 views

Bijection abstract simplicial complex

Given two compact Hausdorff spaces $X$ and $Y$ and $h \colon X \to Y$ a homeomorphism, how can I prove that $h_{\mathfrak{A}} : N(\mathfrak{A}) \to N(h(\mathfrak{A}))$ is a bijection where ...
2
votes
1answer
71 views

Finite fiber- unramified morphisms

I'm in trouble understandig the proof of Proposition 3.2 Chapter 1 of Milne's Book "Étale Cohomology". Let $f:Y\rightarrow X$ be locally of finite-type. The following are equivalent. $(a)$ f is ...
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1answer
41 views

How do I show that an endomorphism is self-adjoint if and only if $\langle u, Tu \rangle \in \mathbb{R}$ for all $u \in \mathbb{V}$

Let $$(V,\langle \cdot , \cdot \rangle)$$ be a complex vector space. Let $T \in \mathcal{L}(V)$ be an endomorphism. Now I want to show, that $T \in \mathcal{L}(V)$ is self-adjoint if and only if ...
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1answer
41 views

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$.

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$, and justify your answer. Hint: Say $f(x)= x^3-2$. How many roots of $f$ are in $K$? For this I know $x^3-2$ has 1 real root, ...
2
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0answers
27 views

Question on morphism locally of finite type

The exercise 3.1 in GTM 52 by Hartshorne require to prove that $f:X \longrightarrow Y$ is locally of finite type iff for every open affine subset $V=\text{Spec}B$, $f^{-1}(V)$ can be covered by open ...
2
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0answers
59 views

Homeomorphism form $(-1,1)$ to $\mathbb{R}$

I want to show that every open intervall $(a,b)$ is homeomorph to $\mathbb{R}$. On $(a,b)$ I chose the relative topology $\mathcal{T}_{(a,b)}$ and on $\mathbb{R}$ the trivial topology $\mathcal{T}$ ...
2
votes
0answers
59 views

Epimorphism (abelian group)

Let $(G,\cdot), (H,*)$ Groups and $f: G\rightarrow H$ an Epimorphism. Show that: If G is an abelian group, then H is also an abelian group. Is the reversal of this proposition also true? My idea: ...
2
votes
0answers
79 views

Every $\mathbb{P}^n$-bundle is a $\mathbb{P}(\mathscr{E})$

I am working on exercise II.7.10(c) in Hartshorne's Algebraic geometry, which asks: Let $X$ be a noetherian regular scheme. Show that every $\mathbb{P}^n$-bundle $P$ over $X$ is isomorphic to ...
2
votes
0answers
41 views

Being a morphism of quasiprojective varieties is a local property

Let $X,Y$ be quasiprojective varieties and $\phi \colon X \to Y$ be a map. Suppose there exists a cover $\mathscr U = \{U_i\}_{i \in I}$ of open subsets $U_i \subset X$ such that for every $i \in ...
2
votes
0answers
273 views

Isomorphic measure-preserving systems: circle and torus

According to Definition 2.7 in Ergodic Theory: with a view towards Number Theory, the systems $(X, \mathcal{B}_X, \mu, T)$ and $(Y, \mathcal{B}_Y, \nu, S)$ are isomorphic when there is a $X' \in ...
2
votes
0answers
42 views

Can we embed K(X_eta) canonically in K(X)

Let $f:X\longrightarrow S$ be a morphism of schemes. Assume $X$ and $S$ are integral. Let $\eta$ be the generic point of $S$ and let $X_\eta\longrightarrow \textrm{Spec} \ K(S)$ be the induced ...
2
votes
0answers
103 views

List of large classes of functors providing morphisms

I recently posted this question on mathoverflow and it was closed as being too localized. I am hoping to more precisely say what I mean here. I recently learned, through my Topology coursework, that ...
1
vote
0answers
32 views

Riesz homomorphism (Banach-Stone theorem)

Let $X,Y$ be compact Haussdorff spaces, and consider $$J:C(X)\to C(Y) $$ a bounded linear bijection such that $J(f\cdot g) = J(f)J(g)$. I know that if $T:C(X)\to C(Y)$ is a linear bijection, being ...
1
vote
0answers
33 views

Do normal group endomorphisms form a normal submonoid?

What it says on the tin. A group endomorphism $v\colon G\to G$ is called normal if $v(aba^{-1})=av(b)a^{-1}$ for all $a,b\in G$. Equivalently, the map $g\mapsto v(g^{-1})g$ is a group homomorphism. ...
1
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0answers
59 views

Functor whose values on morphisms are monomorphisms

Is there a name for a functor whose values on morphisms are monomorphisms?
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0answers
88 views

Number of kernals of all $\mathbb{Z}_n$ to $\mathbb{Z}_m$ homomorphisms?

How many subgroups $K \le \mathbb{Z}_n$ are there with $K =\ker(\phi)$ for some homomorphism $\phi\colon\mathbb{Z}_n \rightarrow \mathbb{Z}_m$? Stuck and need a hint. Have so far that there are ...
1
vote
0answers
125 views

Order of kernel of a homomorphism

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
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0answers
46 views

if a map is bijective, then it has inverse. is the inverse also bijective?

I asked this question because I want to know if given two groups that are isomorphic will always have the same order or not.
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0answers
14 views

Hypergraph notation and hypergraph morphisms

There are two parts to my question. The first part is about notation for hypergraphs. The sconed is about the notion of morphisms for hypergraphs. For the notation part, the context is that I make ...
0
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0answers
22 views

Endomorphisms of Groups - Book Recommendation

Which books dealing with group theory have considerable material on endomorphisms? The books I have seen usually have something on homomorphisms, isomorphisms, and automorphisms, but very little on ...
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0answers
19 views

Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
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0answers
142 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
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0answers
20 views

Two Homomorphisms Question

As an exercise in automata and formal languages, we got two question which I would like to share with and ask you if I am on the right lane a) Give all homomorphism $\varphi:\mathbb{N} \rightarrow ...
0
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0answers
25 views

Classification of endomorphisms in infinite dimension

I'm interested in the following question : let $E$ be an infinite dimensional vector space and $u$ some endomorphism of $E$. We consider endomorphisms equivalent to $u$, that is, of the form $\alpha ...
0
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0answers
36 views

Homomorphism on the group of isometries

Prove that the map $f: M \rightarrow \{1,r\}$ defined by $t_a \rho_{\theta} \mapsto 1$, $t_a \rho_{\theta}r \mapsto r$ is a homomorphism. M denotes the set of isometries of the plane; r the reflexion ...
0
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0answers
82 views

Morphisms in Bourbaki “Theory of Sets”

In Bourbaki "Theory of Sets" there is notion of "morphisms" and different kind of morphisms such as "initial morphisms". These are defined in terms of order theory. It seems that Bourbaki treatment ...
0
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0answers
82 views

Verifying that a function is a morphism by checking a generating set

In some categories, to verify that a map $f : X \to Y$ is a morphism, it suffices to check only a generating set for $Y$ (or, rather, a generating set for some structure on $Y$ such as a topology or a ...
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0answers
158 views

Homomorphisms and Automorphisms between cyclic groups of prime order

Let $A = B \times C$ where $B$ and $C$ are cyclic of order $p$ and $p^2$ respectively, where $p$ a prime. How many endomorphisms are there? How many of these endomorphisms are automorphisms?
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0answers
69 views

Applied Math question: Is there a way to use the set of GPS satellite positions as a lattice?

Okay, I am wondering if it is possible to take a satellite constellation, say the gps satellites, and use the satellites' positions (at time $t$) as points (shown below) and use them as a lattice. ...