Tagged Questions

A sequence of morphisms where the image of one is the kernel of the next. It is useful in abstract algebra (ring/module/group theory) and homology theory in particular. Do not use this tag for sequences of numbers; use (sequences-and-series) instead.

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bounded and convergent sub sequences

We are given with a bounded sequence $x_n$ and let $$ y_k = \sup_{n\ge k} x_n= \sup\{x_k,x_{k+1},….\}. $$ How will we prove that sequence $y_k$ is decreasing and bdd?
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1answer
23 views

Does exactness in each variable coincide with exactness of the product?

Let us restrict to the category of modules. I'm thinking about the definition of exactness of a functor on two variables. The usual definition is that it is exact in each of the two variable, whereas ...
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1answer
37 views

Kernel of a retraction

I have a couple of questions about a exercise I have: Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N \to 0$ be a splitting exact sequence and let $r$ be a retraction of $\alpha$ such that ...
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1answer
30 views

Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
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1answer
50 views

Pullback and Kernel

We consider everything in the category of groups. It is known that monomorphisms are stable under pullback; that is, if $$\begin{array} AA_1 & \stackrel{f_1}{\longrightarrow} & A_2 \\ ...
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1answer
11 views

torsion of a module and exactness

Given a PID $A$ and $A$-modules $M$, $M'$, and $M''$. Assume that $$0\rightarrow M'\xrightarrow{f} M\xrightarrow{g} M''$$ is an exact sequence then prove that $$0\rightarrow ...
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1answer
14 views

First function in a short exact sequence

I have a question. Let $0\to L\to M\to N \to0$ be short exact sequence. how does the function $0\to L$ look like? And what does $0$ mean here? Is it the zero of $A$ (a commutative ring s.t. ...
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0answers
27 views

Normal Form of Elements in a Group

Suppose that there is a family of groups $A_n$ with $n\in\mathbb{N}$ and $A_1$ is the trivial group. If there is a split exact sequence $$0\to B_n\to A_n\to A_{n-1}\to 0,$$ where structure of the ...
3
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1answer
36 views

Short exact sequence of modules

I am trying to show that if we have the following left splitting short exact sequence of $R-$modules: $0 \rightarrow M \stackrel{f} \rightarrow N \stackrel{g} \rightarrow S \rightarrow 0$ then there ...
2
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1answer
13 views

Construction of splittings

I consider a graded algebra $\{A_k\}_{k\in\mathbb{Z}}$ and suppose that the sequence $$0\to A_{k-1}\to A_{k}\to A_{k}/A_{k-1} \to 0\qquad(*)$$ splits. I want to show that for all $j\in\mathbb{N}$, ...
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0answers
13 views

Reference request for Homology Gysin sequence.

I am trying to study the Homology Gysin sequence (not cohomology). I am interested in finding references that either use, or explain the Homology Gysin sequence, especially if it gives descriptions ...
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2answers
61 views

relationship between Exact sequences and Universal mapping property

I am stuck in the following question. Show that for any two short exact sequences. $0\overset{}{\rightarrow}K\overset{i}{\rightarrow}V\overset{T}{\rightarrow}U$ and ...
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0answers
53 views

Tensoring an exact sequence of $R$-modules with $R/x$

Let $R$ be a commutative ring with an $R$-module $M$, and let $x \in R$ be an $M$-regular element. Then tensoring any short exact sequence $0 \to B \to A \to M \to 0$ with $R/x$ yields a short exact ...
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0answers
25 views

Central extension: $Hom(Q,Z) \cong$ automorphisms of $G$ acting trivially on subgroup and quotient?

If we have a short exact sequence $1 \rightarrow C \rightarrow G \rightarrow Q \rightarrow 1$ where $C$ is central in $G$ and $Q \cong G/C$, how can I find an isomorphism between $Hom(Q,C)$ (which is ...
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1answer
19 views

Modules that allow an infinite exact sequence on them

I'm looking for a characterization of modules that fit the following property: (*) There's an infinite exact sequence with $\phi_i \neq 0 \space \forall i\in I $ $\require{AMScd}$ \begin{CD}\cdots ...
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1answer
87 views

exact sequence with infinite element

How to prove: Define exact sequence with infinite element from $R$-modules and $R$-homomorphism ,then exact sequence with infinite element from $\Bbb Z$-modules and $\Bbb Z$-homomorphism : ...
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1answer
65 views

Finite cyclic $\Bbb Z$-module and exact sequence

Suppose $M$ is $\Bbb Z$-module, cyclic, finite. How to prove $\require{AMScd}$ \begin{CD} 0 @>>> \Bbb Z @>>> \Bbb Z @>>> M @>>>0\\ \end{CD} is ...
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1answer
47 views

a problem in exact sequence

Suppose $\require{AMScd}$ \begin{CD} @.\acute M @>f>> M @>g>>\check M @>>> O\\ @. @V \alpha V V\ @VV \beta V @VV \gamma V @. \\ O@>>> \acute N ...
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2answers
60 views

Does the short exact sequence $0\rightarrow \mathbb{Z}_3 \rightarrow G \rightarrow \mathbb{Z}_2\oplus \mathbb{Z} \rightarrow 0$ always split? [closed]

Does the short exact sequence $0\rightarrow \mathbb{Z}_3 \rightarrow G \rightarrow \mathbb{Z}_2\oplus \mathbb{Z} \rightarrow 0$ always split?
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1answer
116 views

Tensor Product, Exterior Power and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $$0\to K\to M\otimes M\to M\wedge M\to 0$$ ...
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1answer
44 views

Example of a commutative square without a map between antidiagonal objects?

In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> N @>>> M\\ @. @VVV @VVV \\ 0 @>>> X ...
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1answer
40 views

If the factor of a finitely generated module is free then submodule is also finitely generated

All rings are commutative, associative and with 1. Consider short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ of $R$-modules. How to show that if $M$ is finitely ...
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1answer
53 views

Confusion in Serre's Local fields book

I read that for right exact functors we consider left derived functors and the resolutions that we consider are projective resolutions... I read that for left exact functors we consider right derived ...
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2answers
64 views

Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
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1answer
46 views

A question on short exact sequences.

The following is an excerpt from Atiyah-Macdonald on short exact sequences. I don't understand the part where the author says "Then $d(x'')$ is defined to be the image of $y'$ in Coker ($f'$)". Is ...
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0answers
62 views

Is there a theory of “derived extensions”?

Given an exact sequence of groups $$1\rightarrow N\rightarrow G\rightarrow K\rightarrow 1$$ we call $G$ a central extension of $K$ by $N$ if the image of $N$ is contained in the center of $G$. Central ...
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1answer
31 views

Converse to Exactness of localization of modules

It is a standard fact that if $R$ is a ring, and $$\tag{1} 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0 $$ a short exact sequence of $R$-modules, if $S$ is a multiplicative ...
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1answer
36 views

$0\to L\to M\to N\to 0 $ is split if $0\to {\rm Hom}_R(D,L)\to {\rm Hom}_R(D,M)\to {\rm Hom}_R(D,N)\to 0$ is exact for any $D$.

Prove that $0\rightarrow L\rightarrow M\stackrel{\phi}\rightarrow N\rightarrow 0 $ is split if $$0\rightarrow {\rm Hom}_R(D,L)\rightarrow {\rm Hom}_R(D,M)\rightarrow {\rm Hom}_R(D,N)\rightarrow 0$$ ...
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1answer
28 views

Summation of series- substitution

If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant, can we find a closed form for f(n)?. NB : Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ ...
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53 views

Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
3
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1answer
49 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
2
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2answers
78 views

Homology groups equal when $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) \rightarrow 0 \rightarrow \cdots$

I'm reading a set of notes but I don't understand the following concept. We have a long exact sequence $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) ...
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4answers
412 views

What is a short exact sequence telling me?

Let's take a short exact sequence of groups $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$ I understand what it says: the image of each homomorphism is the kernel of the next one, so the ...
2
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2answers
48 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
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0answers
55 views

Short exact sequence involving mapping cone, cone, suspension of $C^*$-algebras

This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel ...
5
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0answers
57 views

Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
0
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1answer
57 views

exact sequence problem

Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} ...
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1answer
46 views

hom and exact sequence

Let $$ 0 \longrightarrow \operatorname{Hom}(M,Β_1) \stackrel{f^*}\longrightarrow \operatorname{Hom}(M,Β) \stackrel{g^*}\longrightarrow \operatorname{Hom}(M,Β_2) $$ be an exact sequence for any ...
3
votes
1answer
103 views

Proof that the Euler characteristic is additive

I'm reading through a set of notes which assumes that the Euler characteristic is additive, but doesn't give a proof, so I would like to understand why this is. Let $A_n$ be a finitely generated ...
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0answers
53 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
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0answers
40 views

Exact sequences of $1 \to A \to SU(N) \to B \to 1$, special unitary group

Inspired by the nice post, I am particularly looking into the exact sequences of SU(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A \to SU(N) \to B \to 1$$ where ...
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0answers
54 views

Exact sequences of SU(N) and SO(N)

We know that the spin group $Spin(N)$ has a short exact sequence of Lie groups. $$1 \to Z_2 \to Spin(N) \to SO(N) \to 1$$ I wonder whether there are some examples for SU(N) and SO(N) group, such ...
2
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0answers
67 views

Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is ...
2
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1answer
70 views

About Exact Sequences

Suppose I have the following exact sequence: $$\begin{matrix} A_0& \xrightarrow{\quad\quad} & B_0 \xrightarrow{\quad\alpha\quad} & C_0\\ \uparrow&&&\downarrow \\ C_1 ...
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0answers
76 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
2
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1answer
148 views

Splitting short exact sequence of space groups

I want to prove the following: Assume we have two space groups $G,G^\prime \subseteq \text{Euc}(V) \subseteq \text{Aff}(V)$ which are affinely equivalent, $G \sim G^\prime, \; \text{ i.e. }\; ...
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2answers
58 views

$mA = 0 = nC, \ \gcd(m,n) = 1 \Rightarrow $ every extension of $A$ by $C$ splits

This is Exercise 7.14(ii) from Rotman, Introduction to homological algebra, and I'm stuck on it. If $A$ and $C$ are abelian groups, with $mA = 0 = nC $ and $\gcd(m,n) = 1$ then every extension of ...
0
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1answer
38 views

Applying an additive covariant functor to a sequence

Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
2
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1answer
83 views

Generalized Snake lemma

I always read the snake lemma with short exact sequences: \begin{eqnarray*} &&\qquad M_1\to M_2\to M_3\to0\\ &&\qquad\ \downarrow\qquad\downarrow\qquad\ \downarrow\\ &&0\to ...
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2answers
86 views

Help proving a short exact sequence

Show the following sequence is an exact sequence of $\mathbb Z$-modules when $n$ is a positive integer such that $n=rs$: $$ 0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0. $$ ...