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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Analysing Exact Sequence

I have the following exact sequence $\mathbb{Z}\xrightarrow{f}\mathbb{Z} \xrightarrow{g} K_0(\mathcal{T})\xrightarrow{h}\mathbb{Z}\xrightarrow{0}0$. From here I want to conclude that ...
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46 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
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26 views

Do we have a “short five lemma” for any two of the isomorphisms?

$\require{AMScd}$ The "short" Five Lemma concerns the famous form of exact commutative diagram: $$\begin{CD}0@>>>A@>>>B@>>>C@>>>0\\&@VV\simeq ...
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17 views

Diagram with exact sequences of modules

I am trying to solve these two problems on diagrams of module morphisms: 1) Let $$\begin{array}{c} M' & \xrightarrow{f_1} & M & \xrightarrow{f_2} & M'' \\ & & ...
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36 views

Where is sheafification in the definition of exact sequence of sheaves?

I am reading Andreas Gathmann's notes on Algebraic geometry,http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf Def 7.1.14(iv)says the following As usual, a sequence of sheaves ...
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20 views

Exact sequence of vector bundles

I'am working on a shorter proof of a theorem but to manage it I need to know if a lemma is true. Conjecture: Given a manifold $M$ and an short exact sequence of vector bundles $$ 0 \rightarrow E' ...
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40 views

Flat modules and their relationship with short exact sequences

I recently came across the following result on a Wikipedia page: Suppose $0\to A\to B\to C\to 0$ is a short exact sequence where $B,\,C$ are flat modules; then $A$ is a flat module. I wanted to ...
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1answer
18 views

Understanding splitting lemma

Read this. If the sequence is short exact, then it already has arrows $0 \to \dots$ and $\dots \to 0$ which imply injectivity and surjectivity respectively, right? So in that case there is already ...
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58 views

$M_1 \to M_2 \to M_3 \to 0$ exact iff $0 \to \text{Hom}_A(M_3, N) \to \text{Hom}_A(M_2, N) \to \text{Hom}_A(M_1, N)$ is exact. [duplicate]

Let $M_1 \to M_2 \to M_3 \to 0$ be a sequence of homomorphisms of $A$-modules. Is this sequence exact if and only if the induced sequence of abelian groups$$0 \to \text{Hom}_A(M_3, N) \to ...
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22 views

$\mathbb{Q}$ is a flat $\mathbb{Z}$-module

I am working on the following problem: Show that $0\rightarrow\mathbb{Q}\otimes A\rightarrow\mathbb{Q}\otimes B$ is exact where the map is $1\otimes i$ with $i$ being the inclusion map $A\subset B$. ...
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52 views

Split exact sequences: a basic question.

I am a bit confused regarding the definition of a split exact sequence, whose definition is for example available here (http://ncatlab.org/nlab/show/split+exact+sequence). Let's work in an abelian ...
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17 views

Probability of loosing a consecutive sequence

Use Case A device needs need to receive keep alive messages which keeps the device connection open for 30 seconds. The line should be able to cope with $5$% continuous packet loss. Configuration ...
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3answers
592 views

Short exact sequence - Why doesn't this argument work?

What is wrong with this "proof"? If the sequence of $\Bbb{Z}$-modules $$0\to M \to N \to \Bbb{Z}/2 \to 0$$ is exact, then $N\cong M \oplus \Bbb{Z}/2$. Call the first map $f$, the second $g$. By ...
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48 views

Getting fiber bundles from short exact sequences

Are there conditions that guarantee that a split short exact sequence of groups $$ 1 \rightarrow K \rightarrow G \rightarrow Q \rightarrow 1 $$ gives rise to a fiber bundle $$ F \rightarrow E ...
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149 views

Proving exactness of the conormal sequence

Problem: Let $\phi \colon A \to B$ be a surjective homomorphism of $R$-algebras with kernel $I$. I want to show that the conormal sequence $$ I/I{}^2 \longrightarrow B \otimes_A \Omega_{A/R} ...
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68 views

For what Abelian Group $A$ is there the following exact sequence?

The question is: For what kind of Abelian group $A$ is there a short exact sequence: $$ 0\to\mathbb{Z}\to A\to\mathbb{Z}_{n}\to0. $$ This is an exercise from Allen Hatcher's Algebraic Topology ...
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42 views

Short exact sequence is split iff contractible

Let $0\rightarrow A\overset{f}{\rightarrow} B \overset{g}{\rightarrow} C\rightarrow 0$ be a short exact sequence in an abelian category. I am trying to prove this SES is contractible iff it is split. ...
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38 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!
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64 views

Exact sequence - proof

Let $R$ be a ring. Prove that a sequence of left $R$-modules and homomorphisms $$0 \to N_1 \xrightarrow{f} N_2 \xrightarrow{g} N_3$$ is exact if and only if for all left $R$-modules $M$ sequence ...
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86 views

What does ''$\le$'' mean here?

What does ''$\le$'' mean here? Do you know the meaning of $\le$ in the second last line in the text below? The sequence $0\to N \to M \to M/N \to 0$ is exact, so by Problem 5, the sequence $0 ...
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84 views

Clarification about the Computation of the Homology of the Connected Sum in degree $n-1$.

There are plenty of questions about the homology of the connected sum of two $n$-manifolds, but I didn't find an explicit explanation of the computation done in degree $n-1$. Let's show some examples ...
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34 views

Definition of split of exact sequence

In D&F, they define split as below. (383pg of third edition) Let $R$ be a ring and let $0 \rightarrow A \xrightarrow{\psi} B \xrightarrow{\phi} C \rightarrow 0$ be a short exact sequence of ...
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69 views

Why are functors exact if they preserve all short exact sequences?

If a functor $F\colon \mathcal C → \mathcal D$ of abelian categories preserves short exact sequences, why is it exact? I know the argument is supposed to be that you can split up long exact sequences ...
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18 views

Is the following short exact sequence always split?

Is it always true that the following short exact sequence of modules split? $0\to N\to M\to M/N\to0$
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30 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
5
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33 views

Right-exactness of Kähler-Differential and zeroth relative homology functor

In Commutative Algebra: with a View Toward Algebraic Geometry Eisenbud describes the Kähler-Differential as a functor that assigns $\Omega_{S/R}$ to an $R$-Algebra $S$ and to a commutative diagramm $$ ...
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1answer
54 views

How to prove tensor product is exact when acted on split short exact sequence?

I know tensor product is right exact, but I can't figure out why it's exact when it is acted on a split short exact sequence. In addition, can you give an example that tensor product acts on a short ...
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1answer
31 views

$p_k \colon M_k \to N_k$ is onto for $k>0$, if $p_0$ induces an iso on homology level, prove that $p_0$ is onto

We are working in $\textbf{Ch}_R$, chain complexes of $R-$modules. As the title suggest, I'm given a map (of chain complexes) $p\colon M \to N$ which is onto for $k>0$. It is known that ...
3
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1answer
72 views

Show an ideal is a finitely generated projective module via a split exact sequence

Let $I$ be an ideal of $R$ such that the mapping $f:I\otimes_R\operatorname{Hom}_R (I,R)→R$ defined (on the generators) by $f(i\otimes α)=α(i)$ for all $i∈I$ and $α∈\operatorname{Hom}_R (I,R)$ is ...
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45 views

An inverse limit exact sequence for complete modules

Let $A$ be a commutative complete ring with unit for the $I$-adic topology, where $I$ is the ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such that $I^{n+1}M_n=0$ and that there exist a ...
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50 views

Exact sequences free abelian groups

I have an exact sequence ending with \begin{equation*} \cdots \rightarrow A \xrightarrow{f_1} \mathbb{Z} \xrightarrow{f_2} \mathbb{Z} \oplus \mathbb{Z} \xrightarrow{f_3} \mathbb{Z} \rightarrow (0) ...
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Reduced homology isomorphic to homology relative to a point

I'm going over reduced homology following tom Dieck's Algebraic Topology. I don't understand where the short exact sequence in the excerpt below comes from. $j:(X,\emptyset)\hookrightarrow (X,P)$ is ...
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Are split exact sequences exact in the opposite direction?

In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence ...
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37 views

Isomorphisms in exact sequence imply an object is zero

In any abelian category, let $$\cdots \longrightarrow A\overset{\cong}{\longrightarrow} B\overset{\pi}{\longrightarrow} C \overset{s}{\longrightarrow} D \overset{\cong}{\longrightarrow} E ...
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1answer
53 views

How many non-isomorphic central extensions of a cyclic group of order $2$ by the Lamplighter group exist?

In a comment to the question Finitely many group extensions? Derek Holt mentioned that for the Lamplighter Group $L = C_2\wr\mathbb{Z}$ and the cyclic group $C_2$ with two elements there should be ...
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66 views

Exact sequence of groups to exact sequence sheaves

For a topological group $G$ and a topological space $X$, denote by $\underline{G^X}$ the sheaf of continuous functions from $X$ into $G$. Suppose we have an exact sequence of groups $$ 1\rightarrow ...
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25 views

Partial exactness of the total complex.

The acyclic assembly lemma (pages 59/60 of An introduction to homological algebra, Weibel) establish that if I have a bounded double complex with exact rows (or columns) then the total complex is also ...
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1answer
65 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to ...
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1answer
61 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
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83 views

Extensions of short exact sequences and second cohomology group

Let $G=\mathbb{I}_{p}=<g>$ be the cyclic group of order $p$, where $p$ is a prime and $A=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ a $G-$ module with the action $g^{n}(x,y)=(x+ny,y)$. I want to show ...
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222 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
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2answers
38 views

Existence of non-split sequence

Let $G$ be an abelian group such that $G$ contains non-zero elements of finite order. Why there exists some short exact non-split sequence: $0 \rightarrow \mathbb{Z} \rightarrow H \rightarrow G ...
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1answer
78 views

Invariance of the Fredholm index under finite-dimensional perturbations

Let's call a linear map $f : V \to W$ between vector spaces over some field Fredholm if $\ker(f)$ and $\mathrm{coker}(f)$ are finite-dimensional. (Equivalently, it represents an isomorphism in the ...
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Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram ...
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116 views

Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
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28 views

Working out values of sequences using partial sums

I have the sequence $a_n =\sqrt{n+1}-\sqrt{n}$ and $b_n=\dfrac{1}{\sqrt{n}}$ let $s_n$=$a_1 + a_2 + a_3 + ... + a_n =\displaystyle \sum_{k=1}^n a_k$ and $t_n$=$b_1 + b_2 + b_3 + ... + b_n = ...
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2answers
90 views

Why any short exact sequence of vector spaces may be seen as a direct sum?

This is actually the first time I have worked with short exact sequences and I have no clue why the following assertion is true: Any short exact sequence of vector spaces $$ 0 \longrightarrow U ...
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29 views

Short exact sequence for topological join: split needed

I am desperately trying to solve the following problem: Let $X$ and $Y$ be topological spaces and $X * Y$ their join. Prove that there is a short exact sequence $$ 0 \to \tilde{H}_k(X * Y) \to ...
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1answer
43 views

Exact short sequence vs exact long sequence?

could anyone explain me what exactly the difference between an exact long sequence and an exact short sequence is? I think it pertains to homology theory, right?
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159 views

Question on proof of Lefschetz Fixed Point Theorem (from Hatcher Theorem 2C.3)

In Hatcher's statement of the Lefschetz Fixed Point Theorem (2C.3), he has a hypothesis that the space $X$ in question must be a retract of a finite simplicial complex. The first part of the proof ...