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137
votes
2answers
4k views

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
30
votes
4answers
782 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 $\mathbb R[Y]/((Y^2+1)^2)$.
20
votes
3answers
943 views

Intuition for étale morphisms

Currently working on algebraic surfaces over the complex numbers. I did a course on schemes but at the moment just work in the language of varieties. Now i encounter the term "étale morphism" every ...
19
votes
2answers
536 views

Is every group the automorphism group of a group?

Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$? EDIT: It has been pointed out that the answer ...
12
votes
1answer
191 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 ...
12
votes
1answer
350 views

Working with Morphisms in Local Coordinates

In light of the holiday, I would like to air a grievance. I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods. Let me explain what I mean with ...
10
votes
6answers
734 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 ...
10
votes
1answer
303 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 ...
9
votes
2answers
432 views

Morphisms in the category of natural transformations?

I am learning the basics of category theory, so this question is probably obvious to anyone who knows the subject. The resources I've seen all take the following approach: 0) A category is a ...
8
votes
1answer
85 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 $Tx=x^{-1}~\forall~x\in G.$
8
votes
1answer
84 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 ...
8
votes
1answer
157 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 ...
7
votes
1answer
148 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?
7
votes
3answers
245 views

Category theory without codomains?

A surjection is a function whose range equals its codomain. Thus, the distinction between functions and surjections requires the notion of a codomain. Similarly, a bijection is an injection whose ...
7
votes
2answers
208 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 ...
7
votes
1answer
512 views

History behind Exact Sequences.

I am very much interested in listening to the history behind the exact sequence. We know that the exact sequence is sequence of objects with morphisms such that image of one morphism equals to the ...
7
votes
1answer
67 views

Space of morhisms of representations, its dimension in special case

The symmetric group $S_n$ acts linearly on $\mathbb{C}^n$, hence it brings up to the representation in $\Lambda^m\mathbb{C}^n$. The goal is to evaluate the dimension of morphisms ...
6
votes
3answers
436 views

What are all the homomorphisms between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$?

Any homomorphism $φ$ between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$ is completely defined by $φ(1)$. So from $$0 = φ(0) = φ(18) = φ(18 \cdot 1) = 18 \cdot φ(1) = 15 \cdot φ(1) + 3 \cdot ...
6
votes
2answers
259 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
6
votes
3answers
145 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 ...
6
votes
1answer
96 views

How to use flatness here?

Let $X\to S$ be a scheme. Definition: A relative effective Cartier divisor on $X/S$ is a closed subscheme $D\subset X$ such that the ideal sheaf $I$ of $D$ is invertible and $D\to S$ is flat. Let ...
5
votes
5answers
1k views

In a ring homomorphism we always have $f(1)=1$? [duplicate]

Possible Duplicate: the image of $1$ by a homomorphism between unitary rings I'm studying the Atiyah's commutative algebra book and I realized that in the beginning of the book, the author ...
5
votes
2answers
85 views

Differences between nilpotent and pointwise nilpotent endomorphisms.

Consider an endomorphism of a module $f:M\rightarrow M$. We have that $f$ is pointwise nilpotent if $\forall x\in M,\ \exists n,\ n\in \mathbb N$ such that $f^{n}(x)=0$. I already know that the ...
5
votes
1answer
95 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 ...
5
votes
2answers
189 views

Where is the symmetric group hidden in the Yoneda lemma?

In extension to the question Yoneda-Lemma as generalization of Cayley`s theorem?, can someone point out to me where, in the categorical notation and analyzation of the Cayley's theorem, the symmetrc ...
5
votes
1answer
74 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
5
votes
1answer
86 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 ...
5
votes
1answer
60 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
93 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 ...
4
votes
2answers
87 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
2answers
186 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 ?
4
votes
2answers
204 views

homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.

Let $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ be a homomorphism of the multiplicative group of complex numbers to the multiplicative group of real numbers. I need to show that the kernel of $f$ must ...
4
votes
2answers
80 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 ...
4
votes
3answers
168 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 ...
4
votes
1answer
212 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?
4
votes
2answers
106 views

Generalizing central automorphism group condition to endomorphisms

Given a group $G$, we can define its central automorphism group by $$\operatorname{Aut}_c(G)= C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)) = \{ \phi\in\operatorname{Aut}(G) : \phi(g)g^{-1} \in ...
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
1answer
86 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 ...
4
votes
1answer
139 views

On automorphisms group of order $p^n$

Let $G$ be a finite group, such that $\mid Aut(G)\mid=p^n$. Then prove $G$ is p-group or $G\cong P\times C_{2}$, where $P$ is a p-group. Thank you
4
votes
1answer
217 views

When is the image of a group morphism a normal subgroup?

Let $f : G \to G'$ be a group morphism. I need to find a necessary and sufficient condition such that $\operatorname{Im}(f)$ is a normal subgroup of $G'$.
4
votes
2answers
146 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
79 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 ...
4
votes
1answer
37 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 ...
4
votes
0answers
51 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, ...
3
votes
3answers
536 views

Examples of categories where epimorphism does not have a right inverse, not surjective

Epimorphism is defined as following: $f \in \operatorname{Hom}_C(A,B)$ is epimorphism if $\forall Z. \forall h', h'' \in \operatorname{Hom}_C(B, Z)$ the following holds: $h' f = ...
3
votes
2answers
186 views

Basic questions about definitions in category theory

I'm just getting started in category theory, and I'm not understanding the basic definitions. For example, a common example of a category is a poset. So suppose I have a trivial poset $P$ of 10 ...
3
votes
3answers
73 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
3
votes
3answers
94 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 ...
3
votes
2answers
389 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
2answers
91 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 ...