Questions tagged [exact-sequence]
A sequence of morphisms where the image of one is the kernel of the next. It is a useful thing to examine in the study of abstract algebra and homological algebra. Use this tag if your question is about the general theory of exact sequences, not just because an exact sequence appears in it. For sequences of numbers use the tag (sequences-and-series) instead.
1,381
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Prove that the sequence is not convergent
Given $t>0,r>0$. Let $\{x_k\}$ be given by $x_0=1$
$$ x_{k+1}=x_k-2t(x_k+r\cdot{\rm sign}(x_k)).$$
I would like to prove that the sequence does not converge and does not admit $0$ as a cluster ...
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Short exact sequence of Lie Algebra representations
We have the following SES of a complex Lie Algebra representation,
$$0\to \mathbb{C} \to W \to \mathbb{C} \to 0$$ Now the easy claim would be: $(\rho,W)$ is a two dimension representation and $\rho(x)$...
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Universal coefficient theorem for a surface groups
Let $\Sigma$ be a compact oriented surface of genus $g\ge 1$.
Its fundamental group is given by
$\Gamma:= \langle x_1,y_1,\dots,x_g,y_g\ \lvert\ \prod_{i=1}^{i=g}[x_i,y_i]\rangle$.
It is known that $...
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Suppose $0\to A\to B\to C\to D\to 0$ is exact. Let $0\to A[2]\to B[2]\to C[2]\to X\to 0$ be an exact sequence. Then, $X [2]$ is finite.
Let $A,B,C,D$ be abelian groups and $A$ is finite.
Suppose $D[2]$ is finite.
Is the following true ?
Suppose $0\to A\to B\to C\to D\to 0$ is exact. Let $0\to A[2]\to
B[2]\to C[2]\to X\to 0$ be an ...
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A question on diagram of groups [closed]
Question:
Assume that the rows of the following diagram of abelian groups are exact.
Prove or disprove the following statements:
If $p,q,s,t$ are zero homomorphisms, so is $r.$
If $p,q,s,t$ are ...
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1
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1
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Proving injectivity in a commutative diagram
I have been given the following exercise:
Consider the following commutative diagram of modules over a ring R with exact rows:\
Let $\alpha$ and $g$ be surjective. Fix $b′ \in B′$ and $c \in C$ ...
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1
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1
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35
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Meaning of projection onto one factor in $0\to A^{r-1}\to A^r\to A\to 0$
The following is taken from: $\textit{Partial Differential Control Theory Vol 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Definition 1.50.}$ $M$ is call a $\textit{...
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Which one is the right morphism in the following exact sequence: $0\to A^{r-1}\to A^r\to A\to 0$
The following is taken from: $\textit{Partial Differential Control Theory Vol 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Definition 1.50.}$ $M$ is call a $\textit{...
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Unclear about the notation $A^n\to M:(1,0,\ldots,0)\to x_1,\ldots,(0,\ldots,0,1)\to x_n$ in Proposition 1.55
The following is taken from: $\textit{Partial Differential Control Theory Vol 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Definition 1.50.}$ $M$ is call a $\textit{...
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Free chain complex and exactness
If $C_*$ is exact sequence of free $R$-modules, and $F:_R\text{Mod}\to _S\text{Mod}$ is additive functor, $FC_*$ is in general not exact. But if $R$ is a field, is $FC_*$ is exact?
(I think since $C_*$...
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What does the following notation mean in the enclosed commutative diagram for exact sequence?
The following is taken from: $\textit{Partial Differential Control Theory Vol 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Proposition 1.59.}$ Let
$$0\to M'\...
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2
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Why a nonzero finite abelian group is not projective?
Here is the question I am trying to solve (I know it is answered here $A$ be a nonzero finite abelian group then $A$ is not a projective or injective $\Bbb Z$ module. but the answer is not very clear ...
- 3,653
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Question about the maps: $K\to F$ and $L\to F$ in proof of proposition 1.59 concerning finitely presented module.
The following is taken from: $\textit{Partial Differential Control Theory Vo 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Definition 1.49.}$ If $M$ is a module over ...
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Notaional question about the projection map $A^n\to M$ and short exact sequence $A^m\to A^n\to M\to 0.$
The following is taken from: $\textit{Partil Difgferential Control Theory Vo 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Definition 1.49.}$ If $M$ is a module over ...
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Notation questions about $A^{r-1}\to A^r, A^n\to M:(1,0,\ldots,0)\to x_1,\ldots,(0,\ldots,0,1)\to x_n$ in Proposition 1.55 concerning free module [closed]
The following is taken from: $\textit{Partial Differential Control Theory Vol 1: Mathematical tools}$ by J F. Pommaret
$\color{Green}{Background:}$
$\textbf{Definition 1.50.}$ $M$ is call a $\textit{...
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Questions about $\text{ker }(\alpha) \xrightarrow{u_1} \text{ker }(\beta)\xrightarrow{v_1}\text{ker }(\gamma)$ in Snake Lemma.
The following is taken from $\textit{Module Theory An Approach to Linear Algebra}$ By: T.S.Blyth
$\color{Green}{Background:}$
$\textbf{Exercise 4.5}$ $\textbf{[The snake diagram]}$ Suppose that the ...
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Let $0\to\ker f\to A \xrightarrow fB\to C \to 0$ be an exact sequence of abelian groups. Can we say that $C\cong\operatorname{coker}f$?
Let $f:A\to B$ be a group homomorphism between abelian groups.
Let $$0\to\ker f\to A\xrightarrow fB\to C\to0$$ be an exact sequence of abelian groups.
Can we say that $C\cong \operatorname{coker}f$ ?
...
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Are both $g', g$ assumed to be surjective in the commutative diagram for $3 \times 3$ lemma?
The following is taken from Module Theory An Approach to Linear Algebra} by T.S. Blyth
$\style{font-family:inherit;}{\color{Green}{\textbf{Background:}}}$
Theorem 3.4:
Consider the diagram
of $R$-...
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Exact Sequence in Algebraic Number Theory
Let $R$ be a Dedekind Ring and $K = \operatorname{Quot}(R)$. Let $\mathcal{I}_K$ be the ideal group and $C \ell_K$ the ideal class group.
In a lecture in algebraic number theory, our professor ...
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Is an exact sequence of sheaves also a complex of sheaves?
A collection of sheaf morphisms $(\phi^k \colon \mathcal{F}^k \to \mathcal{F}^{k+1})_{k \in \mathbb{Z}}$ is called
exact if $\operatorname{im} \phi^k = \ker \phi^{k+1}$, and
a complex if $\phi^{k+1} \...
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Calculating Ext group of an infinite polynomial ring and a quotient ring
Let $R=\mathbb{Z}[x]$, $I=(3,x) \subset R$. I want to calculate Ext$_R^1(R/I,R)$. I'll start with the short exact sequence,
$$0 \to I \to R \to R/I \to 0 $$
where the maps are the obvious choices. ...
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How Should I Prove that the Localization of Quotients Commute from here?
Let S be a multiplicatively closed subset of R, M an R-module, and N a submodule of M. I want to prove that:
S$^{-1}$(M/N) is isomorphic to (S$^{-1}$M)/(S$^{-1}$N).
Attempt: Since $0$$\rightarrow$ N$\...
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Exact short sequences and semidirect products
In several sources I have read that a group is semidirect product $G = N \rtimes_\phi G$ iff there is a short exact sequence related, of the form $0 \rightarrow N \rightarrow \ G \rightarrow K \...
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Why is a short exact sequence the same as a bicartesian square?
Let $N \rightarrowtail G \twoheadrightarrow H$ be a sequence of group homomorphisms, with maps $\alpha: N \rightarrowtail G$ a monomorphism and $\beta: G \twoheadrightarrow H$ an epimorphism. I am ...
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Does this argument use the fact that $\{e_i\}$ is linearly independent at any point?
I'm looking through a proof and I'm trying to tell if it uses a certain fact. If it doesn't, then I think I've figured out a homework problem. Here is the lemma and the proof of it:
Lemma: Every ...
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What does it mean by extending a commutative diagram in Snake Lemma?
The following is taken from $\textit{Module Theory An Approach to Linear Algebra}$ By: T.S.Blyth
$\color{Green}{Background:}$
$\textbf{Exercise 4.5}$ $\textbf{[The snake diagram]}$ Suppose that the ...
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1
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Trouble with the hint: $g\circ \alpha \circ f'=0$ for an exercise on the $3\times3$ lemma
The following is taken from Module Theory An Approach to Linear Algebra by T. S. Blyth:
$\style{font-family:inherit;}{\color{Green}{\textbf{Background:}}}$
Exercise 3.10: [The 3x3 lemma] Consider the ...
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Evaluating a derived functor on an object where the original functor is 0
I feel like this question has surely already been asked, but I wasn't able to find a formulation which made my search fruitful, so here goes. Let $C, D$ be two abelian categories, and F a (left/right ...
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88
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Explanation needed for a proof about the Four Lemma in Module theory.
The following is taken from $\textit{Module Theory An Approach to Linear Algebra}$ By: T.S.Blyth
$\color{Green}{Background:}$
$\textbf{Theorem 3.9:}$ $\textbf{[The four lemma]}$ Suppose that the ...
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Proving that $M$ is finitely generated from exact sequence $ 0 \to M' \xrightarrow{f} M \xrightarrow{g} M'' \to 0 $
I'm having some trouble understanding the following proof:
We shall only use the assumption that $$ 0 \to M' \xrightarrow{f} M \xrightarrow{g} M'' \to 0 $$ is exact and $M''$ is finitely generated.
...
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Calculating the ext group of a cyclic group and an $A$ module.
I'm learning about ext groups right now, but the example I have to work with is confusing. I was wondering if someone could take a look at it, and help fill in the gaps.
Let $A=\mathbb{Z}/8\mathbb{Z}$,...
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Meaning of: any exact sequence'can be recovered by "composing" the short exact sequences...
The following is taken from: $\textit{Abstract Algebra}$ by: P. A. Grillet
$\color{Green}{Background:}$
$\textbf{Exercise:}$ Explain how any exact sequence $A\xrightarrow{\varphi}B\xrightarrow{\psi}C$...
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How to calculate the general term formula of $t(k)$
I am trying to find general formula of the sequence $t(k)$ defined by:
$t(1)=2,t(2)=8,g(1)=2,g(2)=4$
and
$t(k+1)=t(k) \cdot2^{g(k)}$,
$g(k)=\dfrac{t(k)}{g(k-1)}$.
alternatively, I want to know if $t(k)...
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Commutative square inducing a linear transformation $r:\text{Coker }b\to \text{Coker }c.$
The following is based on an exercise from the book $linear algebra and geometry" by Leung.
$\color{Green}{Background:}$
Given the linear transformation $c:X\to Y$ and the commutative square ...
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Is every short exact sequence the direct limit of short exact sequences between finitely presented modules?
Let $R$ be a unitary, associative ring. It is well-known that every $R$-module is the direct limit (filtered colimit) of finitely presented $R$-modules. Is it also true that every short exact sequence ...
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Examples of pairs of short exact sequences (s.e.s) of abelian groups with all but one pairs of corresponding sets being isomorphic
I am trying to find examples of pairs of short exact sequences (s.e.s) of abelian groups:
$0\rightarrow M\rightarrow M'\rightarrow M'' \rightarrow 0$ and
$0\rightarrow N\rightarrow N'\rightarrow N'' \...
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Is $0 \rightarrow \Bbb Z \xrightarrow{i_1} \Bbb Z \oplus \Bbb Z/2\Bbb Z\xrightarrow{\text{pr}_2} \Bbb Z/2\Bbb Z \rightarrow 0 $ a short exact seq?
A friend od mine stated that
$0 \rightarrow \Bbb Z \xrightarrow{i_1} \Bbb Z \oplus \Bbb Z/2\Bbb Z\xrightarrow{\text{pr}_2} \Bbb Z/2\Bbb Z \rightarrow 0 \tag{*} $
with $i_1$ the inclusion in the first ...
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1
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Similar short exact sequences where the first abelian group is different
I am trying to get a better understanding of short exact sequences of abelian groups. I know that if
$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$
and
$0 \rightarrow A' \rightarrow B' \...
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Let $0\to N\to M\to P\to 0$ be a split exact sequence. Show that $\varphi:N\oplus P\to M$ given by $\varphi(n,p)=f(n)+h_1(p)$ is an isomorphism.
I'm working on the following problem:
Consider a short exact sequence of $R$-modules:
$$0\to N\xrightarrow{f} M\xrightarrow{g} P\to 0. $$
Suppose that $h_1:P\to M$ is an $R$-map such that $g\circ h_1=...
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Why does an injective $R$-module homomorphism $M\to N$ preserve submodules?
Let $R$ be a commutative ring with unity. Let $0\to M\xrightarrow{f} N\xrightarrow{g} P\to 0$ be a short exact sequence of $R$-modules.
I am looking at a proof of the fact that $N$ is Noetherian $\iff$...
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1
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Interpreting $0\xrightarrow{}\text{Im }f, 0\xrightarrow{}\text{Ker }f$ in examples of exact sequences for showing $\text{Im }f_{n+1}=\text{Ker }f_n.$
The following is taken from Modules an approach to linear algebra by Blyth
$\color{Green}{Background:}$
$\textbf{Example:}$ If $f:A\to B$ is a morphism of abelian groups then we have the exact ...
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1
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68
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Assumptions for $\text{Ker}(p)=\text{Im}(i)$ to hold so that $0\xrightarrow{}N\xrightarrow{i}M\xrightarrow{p}P\xrightarrow{}0$ is exact?
The following is taken from Mostly Commutative Algebra by Chambert-Loir
$\color{Green}{\bf Background\!:}$
$\textbf{Proposition:}$ A diagram of $A$-modules
$$
0 \xrightarrow{}
N \xrightarrow{i}
M \...
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A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits. [duplicate]
A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits.
I have seen these similar questions 1, 2, 3, but none contain a proof of this statement above.
Here ...
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0
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Meaning of the phrase "$X'$ can be identified with a subspace of $X$"?
The following is taken from
$\color{Green}{Background:}$
$\textbf{Exercise}$ Show that if a sequence
$$0\xrightarrow{}X'\xrightarrow{}X\xrightarrow{}X''\xrightarrow{}0$$
is exact, then $X'$ can be ...
- 2,989
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0
answers
27
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How to show $\text{Im } i=\text{ker }\pi$ for sequece $0\xrightarrow{}X'\xrightarrow{i}X\xrightarrow{\pi}X/X'\xrightarrow{}0?$
The following is taken from "Linear Algebra and Geometry" by Leung.
$\color{Green}{Background:}$
$\textbf{Exercuse}$ Show that for each subspace $X'$ of a linear space $X,$ the sequence
$$0\...
- 2,989
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0
answers
129
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What does it mean by "$\chi$ is trivial"? [closed]
The following is taken from "A Graduate course in Algebra 1" by Moskowitz and Farmakis
$\color{Green}{Background:}$
$\textbf{Proposition:}$ For an arbitrary short exact sequence of groups
$$...
- 2,989
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votes
1
answer
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Is this Proof of Well-definedness Correct?
I am trying to prove that the function $\bar g$ is well-defined. This is how I did it:
Define $\bar g: \ker\alpha \to \ker\beta$ by $\bar g(m) = g(m)$, for all $m$ in $\ker\alpha$. This means $m$ is ...
- 361
2
votes
1
answer
71
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Relative cohomology of open Mobius strip
Let $M=[0,1]\times (0,1)/\sim$ be the open Mobius strip, and consider the compact subspace $A=[0,1]\times[\frac{1}{3}, \frac{2}{3}]/\sim$. I am trying to compute the relative singular cohomology $H^{\...
- 1,653
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57
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$0\rightarrow N \hookrightarrow M \twoheadrightarrow M/N \rightarrow 0$ is split exact when $M/N$ is free (again!)
I want to show that the ses $0\rightarrow N \hookrightarrow M \twoheadrightarrow M/N \rightarrow 0$ splits when $M/N$ is free (they are all $R$-modules where $R$ denotes a ring). What I want to do is ...
- 1,640
2
votes
1
answer
70
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Schanuel's Lemma
I want to prove the following:
Assume that we have two exact sequences of $R$-modules $$0 \to K \xrightarrow{\psi} P \xrightarrow{\varphi} M \to 0$$ and $$0 \to K' \xrightarrow{\psi'} P' \xrightarrow{...
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