# 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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### Factorization of a map because composition is $0$

I apologize for the vagueness of the object I'm referring to. This question comes from considering sheaf of $\mathcal{O}_X$-modules homomorphisms, but I feel that the answer is something more general ...
• 61
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### Suppose that $0\to A\xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. Prove that $0\to A\xrightarrow{f} B\to Im(g)\to 0$ is exact

Suppose that $0 \to A \xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. I am trying to prove that $0 \to A\xrightarrow{f} B \to Im(g)\to 0$ is exact. Here, I use the definitiom ...
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• 2,087
1 vote
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### Is this torsion submodule sequence an exact sequence?

Let $0\rightarrow M'\xrightarrow{f} M\xrightarrow{g} M''\rightarrow 0$ be an exact sequence. Then, the sequence $0\rightarrow T(M'')\rightarrow T(M)\rightarrow T(M')$, where T is the torsion submodule ...
• 101
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### Right split exact sequence for a Kan fibration with fiber a group complex

I'm stuck on the proof of Lemma 23.4 from May's Simplicial Objects in Algebraic Topology (p.99) on a seemingly harmless (and so left to reader's proof) step. Some context: given a Kan complex $K$ with ...
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### Does this type of short exact sequence always split?

Consider the family of short exact sequences $$0 \to \mathbb{Z}^m \to G \to \mathbb{Z}/n \mathbb{Z} \to 0$$ where $G$ is a finitely generated abelian group. This is known to not always split, as per ...
• 2,077
1 vote
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### How to show the following sequence of group homomorphism to be exact?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz ....a short exact sequence of groups is just another way of talking about a ...
• 3,641
1 vote
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### How to show the following sequence of group homomorphism is exact?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz ....a short exact sequence of groups is just another way of talking about a ...
• 3,641
33 views

### How to show that 1-dimensional affine group over F is a split extension of subgroup of translations by an abelian subgroup.

I am working on this problem where first it was asked to show that the set of translations $t_{1\beta}, \beta \in F$ forms a transitive normal abelian subgroup $T$ of $AGL_1(F)$. This was trivial, I ...
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• 130
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### A divisibility property in a sequence with exponential terms [closed]

Given the sequence $(a_n)_{n\geq 1}$ with $a_n = \frac{{2 \cdot 2^{2^{n}} + 1}}{3}$, prove that $3^n \mid a_{3^n}$ for every $n \geq 1$. I thought about using LTE 2 times for the numerator but I got ...
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### Exercise 12, Section 4.1 of Hungerford’s Algebra

(The Five Lemma). Let $\require{AMScd}$ \begin{CD} A_1 @>{f_1}>> A_2 @>{f_2}>> A_3 @>{f_3}>> A_4 @>{f_4}>> A_5 \\ @V{\alpha_1}VV @V{\alpha_2}VV @V{\alpha_3}VV @V{\...
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• 394
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### Nice example for a splitting short exact sequence of vector bundles [closed]

I am preparing an introduction to banach manifolds and currently I am working on exact sequences of vector bundles of banach manifolds. Are there any nice examples (i.e. easy to understand for people ...
• 695
1 vote
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### Does an exact sequence of holonomic D-modules imply an exact sequence in the solution space?

Here is a probably quite basic question on holonomic D-modules, but I am only a physicist so please bear with me. If I have the weyl algebra $D$ in $n$ variables, and an exact sequence of holonomic $D$...
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1 vote
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### A faulty "proof" regarding exactness of derived functors

I've been trying to wrap my head around derived functors and have come upon the following chain of arguments, which seems to yield the unreasonabel conclusion that all left derived functors are exact. ...
1 vote
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### Understanding why an element mapped to a generator has finite order

Here is the setup of a problem in group theory I can trying to solve. $G$ is a group, and there is a short exact sequence  0 \rightarrow \mathbb{Z}/2 \rightarrow G \rightarrow^{\varphi} \mathbb{Z}/k ...
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• 4,227
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### Interpretation of $B\otimes M$ when $B\subset A$

Let us fix the setting to avoid ambiguity. Let $R$ be a commutative, unital ring. All tensor products will be considered for modules over $R$. Let $A$ be an $R$-module, $B$ a submodule. Now, is \$A\...
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