# 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.

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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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### 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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### 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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### 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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### 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 ...