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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0
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2answers
95 views

$k$-algebra homomorphisms

I would like to ask if the following is true: Let $A$ be a $k$-algebra where $k$ is any field. If we have a $k$-algebra homomorphism $f:A\rightarrow k$, does it follow that $\ker(f)$ is a maximal ...
2
votes
2answers
73 views

Proving the Frobenius map is an endomorphism

I have prime $p$, and $K$ a field such that $p \cdot 1 = 1+1+\cdots+1 = 0$. I am asked to prove that $F: K \rightarrow K$, $a \mapsto a^p$ is a ring homomorphism. I can prove this for ...
0
votes
1answer
44 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, ...
0
votes
1answer
172 views

Isomorphism of R-modules

Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$?
3
votes
3answers
219 views

Isomorphisms and the Fundamental Homomorphism Theorem

Let $$ R=\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} : a,b∈ℝ\right\}⊂M_2(ℝ) $$ and $$ I=\left\{ \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}: b∈ℝ\right\}. $$ Identify the ...
0
votes
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 ...
3
votes
1answer
160 views

Automorphism of $\mathbb{Z}\rtimes\mathbb{Z}$

I'm looking for a description of $\mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z})$. I've tried an unsuccessfully combinatorical approac, does anymore have some hints? Thank you.
0
votes
1answer
92 views

Finding homomorphisms and kernels from a given ring R

Give the following rings $R$ and ideals $I$ find a ring $S$ and a homomorphism $f:R \rightarrow S$ with kernel $I$ i) $R=\mathbb{Q} [x], I=(x^{2}-2)R$ ii)$R=\mathbb{Z}[i], I=2R$ (Gaussian Integers) ...
0
votes
1answer
66 views

number of automorphisms for group in order 169

Let $G$ be a group with order 169. Prove number of automorphisms is at least 143. I thought that 169 is 13 squared so maybe G isomorphic to $ Z_{169} $ but I dont have any idea. How can I solve ...
0
votes
1answer
119 views

Hypercube and dihedral group

Let $G_n$ denote the subgroup of the orthogonal group $O_n$ of elements that send the hypercube to itself, the group of symmetries $C_n$, including the orientation-reversing symmetries. It would ...
0
votes
0answers
40 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 ...
2
votes
1answer
48 views

Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
2
votes
1answer
67 views

Exponentials “commutes” explicitly

i have a question about exponentials in a category $\Bbb{A}$. I have to prove that the following holds: $C^{A\times B}\cong(C^A)^B$. Therefore i have to give two arrows. This can be done on an ...
1
vote
0answers
59 views

Functor whose values on morphisms are monomorphisms

Is there a name for a functor whose values on morphisms are monomorphisms?
2
votes
3answers
65 views

A notation for a morphism in a thin category

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$. Is there a standard notation for this morphism (given $A$ and $B$)?
1
vote
2answers
174 views

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$?

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$? If not, what is a good counterexample? If yes, how can we prove that there's a ...
1
vote
1answer
42 views

Morphisms of complexes chain [closed]

I have a small question: Why is the following true? "If we have a continuous mapping between two topological spaces $f:X\rightarrow Y$, we can associate a morphism of chain complexes $f_*\colon ...
1
vote
2answers
124 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
3
votes
2answers
102 views

How can I quantify over the class of all cardinalities?

I'd like to quantify over all cardinalities of sets. My end goal is to make a category-theoretic arguement: For all cardinalities of sets, in the category of sets with maps as morphisms: the ...
2
votes
1answer
67 views

On Lemma 4.1 of Hartshorne's algebraic geometry text

I'm in the process of teaching myself algebraic geometry from Hartshorne. Lemma 4.1 says that if we let $X$ and $Y$ be two varieties, and let $\phi$ and $\psi$ be two morphisms from $X$ to $Y$, and ...
0
votes
3answers
100 views

From where comes the horizonal composition in a 2-category?

Viewing a 2-category as a category enriched in $\mathsf{Cat}$, I can see from where comes the vertical composition: morphisms of a 2-category are objects of $\mathsf{Cat}$ and 2-morphisms of this ...
1
vote
1answer
55 views

Replacing a morphism with composition with this morphism

I have a certain category. I feel it is better to study the functor $x\mapsto f\circ x$ (where $\circ$ is the composition in my category) than the morphism $f$ itself. How is it called when a ...
4
votes
2answers
205 views

The isomorphisms between two vector spaces

Let $V$ and $W$ be two vector spaces over real number field, if they are isomorphic as vector spaces over rational number field, are they isomorphic as real vector spaces ?
2
votes
0answers
93 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
1answer
113 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.
12
votes
1answer
206 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 ...
1
vote
1answer
92 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 – ...
4
votes
2answers
162 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
votes
1answer
38 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
votes
0answers
42 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 ...
4
votes
1answer
84 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
votes
1answer
171 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. ...
11
votes
6answers
928 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 ...
1
vote
4answers
154 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
votes
1answer
69 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
89 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
97 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
104 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} ...
1
vote
0answers
93 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
votes
1answer
94 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
275 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 ...
4
votes
2answers
90 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 ...
31
votes
4answers
837 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
votes
1answer
153 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?
4
votes
3answers
170 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 ...
2
votes
1answer
77 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
95 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
85 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
556 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
87 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 ...