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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78 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 ...
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0answers
73 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
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1answer
78 views

Isomorphism between the space of linear operator and matrices for finite dimensional spaces

Prove that $\operatorname{Lin}(U,V)$ is isomorphic to the space of $m$ by $n$ matrices, where $\dim(U)=n$ and $\dim(V)=m$. Thanks so much for your enlightment.
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1answer
159 views

Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
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1answer
48 views

Systematizing graph morphisms

Trying to systematize possible notions of graph morphisms I came about the following classification: A morphism $f$ which sends a graph $G$ to another graph $G'$ is – first of all – ...
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2answers
127 views

An example of a monomorphism that is not an equalizer in “Abstract and Concrete Categories — The Joy of Cats”

I am sorry, maybe i am just confused or have not understood some definition, but i do not understand the following remark in Abstract and Concrete Categories -- The Joy of Cats on page 117: ...
4
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1answer
36 views

How to denote an 'atomic' morphism in category?

I want to distinguish between two disjoint classes of morphisms in a category: (1) those morphisms that are composed of other morphisms (other than identities) and could conceivably be factored into a ...
2
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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 ...
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1answer
76 views

Isomorphisms of $\mathbb P^1$

Prove that every isomorphism of $\mathbb P^1$ (over an algebrically closed field $\mathbb K$) is of the form $$ \phi(x_0: x_1) = (ax_0+bx_1 : cx_0 + dx_1) $$ where $\begin{pmatrix} a & b ...
3
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1answer
94 views

Morphisms between quasiprojective varieties preserve irreducibility

Let $X,Y$ be two quasiprojective varieties and $\phi \colon X \to Y$ a surjective morphism. Let $Z \subset Y$ a closed set such that $\phi^{-1}(Z)$ is irreducible. Prove that $Z$ is irreducible. ...
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6answers
532 views

What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
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4answers
140 views

To what extent are morphisms required to be functions?

Just beginning category theory, and I am looking for clarification on the precise nature of morphisms. The most familiar categories, e.g. $\mathbf{Top}$, have morphisms that are functions in the ...
4
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1answer
67 views

If $M$ is either Noetherian or Artinian and $M^{(n)}\cong M^{(m)}$, then $m=n$.

The question is: Prove that if $_RM$ is either artinian or noetherian and if $m,n\in\mathbb{N}$ with $M^{(m)}\cong M^{(n)}$, then $m=n$. Here, $M^{(m)}$ denotes the direct sum of $m$ times $M$. I ...
4
votes
2answers
85 views

Problem 3.1.2 in Liu — Omission in problem statement?

Exercise 3.1.2 in Liu's Algebraic Geometry and Arithmetic Curves is as follows. Let $f:X\rightarrow Y$ be a morphism of schemes. For any scheme $T$, let $f(T):X(T)\rightarrow Y(T)$ denote the ...
4
votes
1answer
80 views

Finite fiber of scheme morphism is zero-dimensional?

Let $X$ and $Y$ be locally Noetherian schemes and $f:X\rightarrow Y$ be an étale morphism of finite type. Let $x\in X$ and $y=f(x)$. I would like to know why the fiber $X_y$ is a zero-dimensional ...
3
votes
1answer
87 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} ...
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0answers
86 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 ...
5
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1answer
86 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 ...
7
votes
2answers
148 views

Inverse of open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that ...
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2answers
77 views

Why is this composition of scheme morphisms proper?

I am learning about proper morphisms from Liu's book. I have a question about the proof of the Lemma 3.17 on page 104. Let $A$ and $B$ be rings and suppose $\operatorname{Spec} B$ is proper over ...
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4answers
737 views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
7
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1answer
117 views

Category with endomorphisms only

How is called a category with endomorphisms only? How is called a subcategory got from an other category by removing all morphisms except of endomorphisms?
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3answers
167 views

Does every category have a functor?

Is there any one (or more) categories that doesn't have a functor? Functors go between categories, so is there any category that only has an identity functor but no other functor that maps it to ...
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0answers
62 views

Homomorphical Equivalence is NP-complete

Two graphs $G,H$ are homomorphically equivalent if there are exists a homomorphism from $G$ to $H$ and a homomorphism from $H$ to $G$. The task is to prove that this decision problem ...
3
votes
3answers
93 views

Given a functor between categories, how to denote a morphism between particular objects of that category

I have a very common situation, for which I need both: (1) notation; and, if available, (2) a general relative term. Let's say that: there is a functor between categories, $f:C_1\to C_2$, $c_1$ is ...
8
votes
1answer
75 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
3
votes
2answers
320 views

Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
3
votes
1answer
61 views

Every injective function is an inclusion (up to a unique bijection)

Let $X$ be a set and let $A$ be a subet of $X$. Let $i:A\longrightarrow X$ be the usual inclusion of $A$ in $X$. Then $i$ is an example of an injective function. I want to show that every ...
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votes
3answers
91 views

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb{Z}_p \oplus \mathbb{Z}_p$ into $\mathbb{Z}_p$.

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb{Z}_p \oplus \mathbb{Z}_p$ into $\mathbb{Z}_p$. How can I able to solve this problem?can anyone help me please.
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2answers
114 views

Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$

I'm trying to prove that Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$ (p prime). I know that Aut $\mathbb Z_p$ has $p-1$ elements because $\mathbb Z_p$ has $p-1$ possiblities of generators, so intuitively ...
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3answers
139 views

Ring homomorphism question.

If $R$ is a ring, show that there is exactly one ring homomorphism $\phi: \mathbb{Z} \to R$. I can't grasp the idea that there can be only one ring homomorphism. Aren't there many (at least more than ...
3
votes
1answer
84 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
8
votes
1answer
76 views

Prove that $Tx=x^{-1}~\forall~x\in G.$ [duplicate]

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$
3
votes
2answers
561 views

If $\phi:G\to\bar{G}$ is an isomorphism and if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$.

If $\phi:G\to\bar{G}$ is an isomorphism and if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$. I am struggling with getting started with the problem. I know ...
8
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1answer
154 views

Path Connectedness and continuous bijections

Mathoverflow. Are there any two topological spaces $X$ and $Y$ such that they are path connected and such that there exist continuous bijections $X\rightarrow Y$ and $Y\rightarrow X$, but and yet ...
5
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1answer
92 views

A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields? My guess is yes, because every morphism ...
1
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1answer
112 views

How I can find the inverse of an isomorphism?

The motivation of this question can be found in Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not? My question is: How I can find the inverse of the ...
1
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0answers
120 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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1answer
89 views

Problem of monomorphism of rings [duplicate]

Let $A$ a ring, for each monomorphism $f:A^m \rightarrow A^n$, I don't know how to prove that $m\leq n$. I can't start the problem, I have no idea, help me please.
2
votes
1answer
168 views

Image of a morphism of varieties

Suppose $A$ and $B$ are two algebraic varieties, and $f:A\to B$ is a morphism of algebraic varieties. I guess it is true that $\text{im}(f)$ is itself an algebraic variety. But how to prove it?
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1answer
179 views

If each component of a Cartesian product is homeomorphic to another space, are the Cartesian products homeomorphic

Assume we are given a space $A$ with a metric $d$. Assume $A = A_1 \times A_2 \times A_3 \cdots$, ie. $A$ is a Cartesian product of spaces $A_i$, where $i \in I$. $I$ is countable or countably ...
2
votes
1answer
42 views

What is the name for the intermediary object(s) of functional composition?

Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$. What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming ...
10
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1answer
257 views

Question on a homomorphism of a set G.

I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19 Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
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3answers
132 views

How can I conclude that an automorphism is the identity map?

Let $\phi$ be an automorphism on a group $G$. If $\phi$ maps any one non-identity element in $G$ to itself, is $\phi$ necessarily the identity map? What if $G$ is cyclic?
3
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1answer
278 views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
5
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1answer
82 views

Morphism between matrices and linear equations

I'm currently a beginner at linear algebra. So, in some books I see authors start defining linear equations and then they define matrices and, supposedly, the definition of associative matrix is to ...
0
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5answers
174 views

Show that $S_2$ is isomorphic to $Z_2$

Show that $S_2$ is isomorphic to $Z_2$ I know that $S_2$ has two elements $\sigma_1$ and $\sigma_2$...and $Z_2$ has two elements 0 and 1. But isomorphism is a bijective function, right? So doesn't ...
0
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1answer
50 views

Extensions of $\sigma:\mathbb{Q}(\xi)\rightarrow\mathbb{Q}(\xi), \ \sigma(\xi)=\xi^{2}$ to $\mathbb{Q}(\xi)(\sqrt[3]{2})$

Let $L$ be the splitting field of $T^{3}-2$ and let $\sigma:\mathbb{Q}\left(\xi\right)\rightarrow\mathbb{Q}\left(\xi\right)$ be a morphism defined by $\sigma\left(\xi\right)=\xi^{2}$. Find all ...
0
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1answer
62 views

Extensions of $\mathbb{Q}\left(\sqrt{2}\right)\rightarrow\mathbb{Q}\left(\sqrt{2}\right),\ \sqrt{2}\mapsto -\sqrt{2}.$

I have to solve an exercise that requires to find all extensions $\hat{\sigma}:\mathbb{Q}\left(\sqrt{2},\xi\right)\rightarrow\mathbb{Q}\left(\sqrt{2},\xi\right)$ (with $\xi$ being a third root of the ...
4
votes
0answers
50 views

Extending a homeomorphism between two curves [duplicate]

Possible Duplicate: Is there “essentially only 1” Jordan arc in the plane? Let $\gamma : [0,1] \to \mathbb{R}^2$ be a simple curve with $\gamma(0)=(0,0)$ and $\gamma(1) =(1,0)$. Clearly, ...