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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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Short exact sequence in the ideal class group

In the answer to the Motivation behind the definition of ideal class group, I have seen one by user Alex Youcis and he/she claimed that the following is a short exat sequence: $$1\to \mathcal{O}_k\to \...
Bowei Tang's user avatar
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2 votes
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Homology + Chinese Remainder Theorem =?

Let $M_i, i=1..n$ be a finite collection of pair-wise coprime moduli. The Chinese remainder theorem says that $\Bbb{Z}/M \approx \prod_i \Bbb{Z}/M_i$. Without going into Bezout / Euclidean algorithm,...
SeekingAMathGeekGirlfriend's user avatar
0 votes
1 answer
87 views

Exactness of the sequence $0\rightarrow\mathbb{Z}_p\xrightarrow{p^n}\mathbb{Z}_p\xrightarrow{\varepsilon_n} A_n\rightarrow 0$

On page 12 of the book A Course in Arithmetic by Serre, the author proves that the sequence $0\rightarrow\mathbb{Z}_p\xrightarrow{p^n}\mathbb{Z}_p\xrightarrow{\varepsilon_n} A_n\rightarrow 0$ is exact,...
Matheus Frota's user avatar
-4 votes
1 answer
62 views

Establish a short exact sequence $0\to Z_p\to Z_{p^2}\to Z_p\to 0$ does not split, every submodule of the projective module need not be projective. [closed]

Establish a short exact sequence $0\to Z_p\to Z_{p^2}\to Z_p\to 0$ does not split, every submodule of the projective module need not be projective.
Haval Mohammed's user avatar
0 votes
1 answer
80 views

Screwing up basic short exact sequence $0\to\mathbb Z\leftrightarrow\mathbb Z\to0$

Represent an isomorphism by $\leftrightarrow$. HAVE exact sequence. $$ 0 \rightarrow \mathbb Z \leftrightarrow \mathbb Z \rightarrow 0 $$ Then $$ \text{img} \left( 0 \rightarrow \mathbb Z \right) = 0 =...
Nate's user avatar
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0 votes
1 answer
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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 ...
ark's user avatar
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1 answer
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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 ...
Squirrel-Power's user avatar
0 votes
0 answers
30 views

Can Nakayama Lemma apply to complex

Let $R$ be a Noetherian commutative local ring with maximal ideal $\mathfrak{m}$. Consider a complex of finitly generated projective $R$-mod (so it is free) $$0\rightarrow P_1\rightarrow P_2\...
An Zhang's user avatar
2 votes
1 answer
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Homotopy exact sequence of a covering map

I wanted to compute a homotopy group using the homotopy exact sequence of a covering but I am missing something obvious. Consider the covering $\mathbb{C} - \{0, \pm 1\} \rightarrow \mathbb{C} - \{0, ...
janullrich's user avatar
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1 answer
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Understanding the definition of left exact functors

I am studying category theory, and in particular exact sequences. I am stuck on proving that the following three conditions are equivalent in an abelian category $\mathcal{C}$: (a) The sequence $0 \to ...
Squirrel-Power's user avatar
1 vote
1 answer
92 views

what should be the group $B$?

Here is the exact sequence of abelian groups I am studying: $$0 \to \mathbb Z/2 \to B \to \mathbb Z/2 \to 0 $$ Can I say that $B \cong \mathbb Z/2$ or $B \cong \mathbb Z/2 \oplus \mathbb Z/2$? Is $B \...
Emptymind's user avatar
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3 votes
1 answer
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Example where the last arrow in the sequence is not surjective.

Consider the exact sequence $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$ and its induced sequence $0\rightarrow T(M')\rightarrow T(M)\rightarrow T(M'')$, where $T$ denotes the torsion ...
claudia's user avatar
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1 vote
1 answer
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$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ exact, then the tensorized sequence is also exact

I want to demonstrate that if $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence, then the induced sequence $0\rightarrow M'\otimes_A N\rightarrow M\otimes_A N\rightarrow ...
claudia's user avatar
  • 101
1 vote
0 answers
42 views

Long exact sequence of intersected prime ideals

I'm considering a commutative local noetherian ring $R$ (in fact, $R$ is even Cohen-Macaulay) along with a number of prime ideals $I_i$. I can then construct the exact sequence \begin{equation} 0\...
A.H's user avatar
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0 answers
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Sequence involving plane curves is exact

I'm reading through Andreas Gathmann's Algebraic Curves and there is this affirmative on Proposition 2.10 at page 14: Let $P\in\mathbb{A}^2$, and $F, G, H$ curves (or polynomials). If $F$ and $G$ have ...
Mand's user avatar
  • 303
2 votes
0 answers
50 views

Extensions of $G$-modules parametrized by $H^1$

Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess we can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
Conjecture's user avatar
  • 3,270
1 vote
0 answers
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Elegant Proof of Snake Lemma

I'm considering the following diagram \begin{array}{ccccccccc} &&&&0&&0&&\\ &&&&\downarrow& &\downarrow& &\\ & && & \...
一団和気's user avatar
0 votes
1 answer
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proving that $A_n/\operatorname{ker} \beta \cong \operatorname{ker} \alpha_{n - 1}.$

Prove that in any exact sequence $$\dots \xrightarrow{\alpha_{n+3}}A_{n+2} \xrightarrow{\alpha_{n+2}} A_{n+1} \xrightarrow{\alpha_{n+1}} A_{n} \xrightarrow{\alpha_{n}} A_{n -1}\xrightarrow{\alpha_{n-...
Emptymind's user avatar
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1 vote
0 answers
43 views

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 ...
claudia's user avatar
  • 101
5 votes
0 answers
77 views

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 ...
Alberto Avitabile's user avatar
3 votes
1 answer
195 views

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 ...
Paul Cusson's user avatar
  • 2,077
1 vote
1 answer
57 views

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 ...
Seth's user avatar
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1 vote
0 answers
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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 ...
Seth's user avatar
  • 3,641
2 votes
0 answers
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 ...
Rudra_D's user avatar
  • 125
0 votes
0 answers
58 views

Exact sequence of Lie algebras

I'm trying to solve exercise 7.5.1 of Weibel's An Introduction to homological algebra. I need to show that there is a short exact sequence $$H_2(\mathfrak{g}/\mathfrak{h},k)\oplus[\mathfrak{g},\...
Jolia's user avatar
  • 130
0 votes
1 answer
53 views

For modules, $\text{Hom}$ is an exact functor

To paraphrase Serge Lang, Algebra: Proposition 2.1. Let $X$, $X^\prime$, $X^{\prime\prime}$ and $Y$ be modules over some ring $A$. (The ring does not change, and will therefore not be mentioned ...
paulina's user avatar
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1 vote
1 answer
80 views

Finding a basis of relative homology group $H_1(\mathbb R,\mathbb Q)$ .

I am trying to solve a problem from the Homology theory chapter of Hatcher's book.The problem is the following: Show that the relative homology group $H_1(\mathbb R,\mathbb Q)$ is free abelian and ...
Kishalay Sarkar's user avatar
3 votes
1 answer
65 views

Spectral sequence with two non zero rows

I'm trying to solve exercise 5.2.2 of Weibel's Introduction to homological algebra : if a spectral converging to $H$ has $E_{p,q}^2=0$ expect for $q=0,1$ then there is a long exact sequence $$\cdots \...
Jolia's user avatar
  • 130
2 votes
0 answers
69 views

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 ...
math.enthusiast9's user avatar
0 votes
0 answers
56 views

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{\...
user264745's user avatar
  • 4,227
-1 votes
1 answer
66 views

A question about some exact sequence of Lie algebras [closed]

M. Schottenloher in his book "A mathematical introduction to conformal field theorey" in Remark 4.3 says that: For every central extension of Lie algebras $0\to \mathfrak{a}\to \mathfrak{h}\...
Mahtab's user avatar
  • 755
3 votes
1 answer
60 views

Proposition 2.10 Atiyah MacDonald

I have been self studying from Atiyah and Macdonald's Intoduction to Commutative Algebra and am having difficulty with the following proposition: Proposition 2.10 Let $$ \require{AMScd} \begin{CD} 0 @...
MichaelCatliMath's user avatar
0 votes
0 answers
44 views

Hint for a Sequence of Free Modules

Could someone please give me a hint for this problem? The following sequence is exact: $0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$....
Tom's user avatar
  • 1
0 votes
1 answer
77 views

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result. I have no idea on ...
Squirrel-Power's user avatar
2 votes
0 answers
49 views

Let $R=\mathbb Z[X,Y]$. Find an exact sequence $0\to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0$ with $f(g(x,y)) := g(0,0)$ for all $g \in R$

Let $R = \mathbb Z[X,Y]$. Construct an exact sequence of $R$-modules $$0 \to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0,$$ where $f(g(X,Y)) = g(0,0)$ for all $g \in R$. Here $\mathbb Z$ is ...
Squirrel-Power's user avatar
1 vote
1 answer
52 views

Homotopy sequence of a weak fibration with a section

Suppose that $p:E \to B$ is a weak fibration with a section $s:B \to E$ ($p \circ s=id_B$). Also here $F=p^{-1}(b_0)$. I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,...
givememeth's user avatar
0 votes
0 answers
43 views

If $M$ is a flat $R-$module and $I$ is an ideal of $R$, why is $I\otimes_R M$ isomorphic to $IM$?

This is often given as an equivalent property to $M$ being a flat $R-$module (for instance in one of the answers here Are $I\otimes_{R}J$ and $IJ$ isomorphic as $R$-modules?). Certainly we can always ...
Aaron Andersen's user avatar
2 votes
1 answer
72 views

Surjectivity in inverse limit

Suppose we have two inverse systems $(A_i)$ and $(B_i)$ of abelian groups and we have a homomorphism from one system to other, say $\lambda_i:A_i\rightarrow B_i$ for all $i$ (so that respective ...
Maths Rahul's user avatar
  • 3,047
0 votes
0 answers
33 views

Sequence of direct summands of modules over a ring

Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules: $$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$ I claim that ...
Fabio Lucchini's user avatar
-2 votes
2 answers
68 views

Having trouble finding an exact sequence. [closed]

Find an exact sequence: $\{0\} \rightarrow \mathbb{V} \xrightarrow{{\alpha}} \mathbb{U} \xrightarrow{{\phi}} W \rightarrow \{0\} $, such that $$\dim(\mathbb{V}) = \dim(\mathbb{U}) = \infty$$ $$\dim(\...
Avgustine's user avatar
  • 149
0 votes
0 answers
66 views

Short exact sequence of finitely generated R-modules

Assume that M',M'' are finitely generated R-modules (here R is a commutative unitary ring) and M is another R-module. The initial problem is that if there exists a s.e.s. $0\rightarrow M'\xrightarrow{...
Bigalos's user avatar
  • 394
1 vote
1 answer
74 views

2.3.A Zariski's construction (PAG1 - R. Lazarsfeld), big and nef divisor which is not finitely generated

I'm reading Lazarfeld's book «Positivity in Algebraic Geometry I» and I'm stuck on the construction of a big and nef divisor on a variety $X$ such that its canonical ring/algebra $R(X,D) = \bigoplus_{...
NaNoS's user avatar
  • 553
2 votes
1 answer
88 views

Pushforward commutes with pullback in short exact sequences

In 010I we find the definition of the pullback (resp., the pushforward) of a short exact sequence $0\to A\to E\to B\to 0$ in some abelian category along a morphism $B'\to B$ (resp., along a morphism $...
Elías Guisado Villalgordo's user avatar
-1 votes
1 answer
56 views

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 ...
Flynn Fehre's user avatar
1 vote
1 answer
98 views

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 $D$...
A.H's user avatar
  • 41
1 vote
1 answer
56 views

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. ...
Jeppe Obel's user avatar
1 vote
1 answer
48 views

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 ...
Valor Vaporeon's user avatar
2 votes
0 answers
25 views

Homotopy long exact sequence and image of connecting map in the center of some group [duplicate]

For context I am working on Weibel's K-book, chapter IV, my question comes from the proof of proposition 1.7. In this he claims that for the long exact sequence of a fibration with acyclic fiber $F\...
DevVorb's user avatar
  • 1,495
0 votes
1 answer
82 views

Examples of exact sequence in Hungerford’s Abstract Algebra

Example. Note first that for any module $A$, there are unique module homomorphisms $0\to A$ and $A \to 0$. If $A$ and $B$ are any modules then the sequences $0\to A \xrightarrow{\iota} A \oplus B \...
user264745's user avatar
  • 4,227
3 votes
1 answer
89 views

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\...
Academic's user avatar
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