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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Extensions and Pushouts using an exact sequence of sets

This might seem a strange way of doing things, that is, inventing a possible example (according to comments, there is no such thing as an exact sequence of sets), but let us try to make one for ...
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34 views

For What Abelian groups $A$ Does There Exist a Short Exact Seqeuence $0\to \mathbb{Z}/p^m\mathbb{Z}\to A\to \mathbb{Z}/p^n\mathbb{Z}\to 0$.

$\newcommand{\Z}{\mathbb Z}$ Question: Let $m$ and $n$ be positive integers. What are all the abelian groups $A$ such that there is a short exact sequence $0\to \mathbb Z/p^m\mathbb Z\to A\to ...
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24 views

What are explicit maps in the following exact sequence?

Let $G$ be a group and $M,N$ be normal subgroups of $G$ such that $G=MN$. Then there is a natural exact homology sequence $Ker(M \wedge N \xrightarrow{\lambda} [M,N]) \xrightarrow{\rho} H_2(G) ...
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1answer
34 views

If $G_3$ is finitely generated abelian group then there is a short exact sequence with $G_2$ and $G_1$ free groups?

Let $G_i$ be abelian Groups. A exact sequence of the form $ 0 \to G_1 \to G_2 \to G_3 \to 0$ is called a short exact sequence. Is the following statement true? If $G_3$ is finitely generated then ...
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56 views

Showing that localization is an exact functor

I'm again in this awfully familiar situation where I'm struggling to prove simple statements mostly because I have no idea how a template of a proof should look like in this specified context. I'm ...
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1answer
74 views

Step in five-lemma's proof.

Consider the following commutative diagram, where rows are exact: $$\require{AMScd} \begin{CD} M_1 @>f_1>> M_2 @>f_2>> M_3 @>f_3>> M_4 @>f_4>> M_5 \\ @Vh_1VV ...
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30 views

Gysin sequence for the sphere bundle $B[O_a \times O_B]^+ \to BO_a \times BO_b$?

I have the sphere bundle $S^0 \hookrightarrow B[O_a \times O_b]^+ \stackrel{p}{\to} BO_a \times BO_b$ that can be thought of like: $B[O_a \times O_b]^+$ as the set of tuples of vector spaces $E^a$ ...
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3answers
412 views

Stanford math qual: Abelian groups $G$ satisfying $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to 0$

I am studying for my qualifying exams and came across the following question: Find all abelian groups $G$ that fit into an exact sequence $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to ...
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44 views

Long exact sequence in homology: naturality=functoriality?

In every book I've looked, the "naturality" of the long exact sequence in homology simply says that every arrow between short exact sequences translates into an arrow between the long exact sequences ...
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55 views

When are direct products exact in the category of quasi-coherent sheaves?

I would like to know if there is a description (or at least some sufficient condition known) of a (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products. I ...
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52 views

Dominant dimension $\geq 2$ implies a certain double centralizer property

let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: $\bullet$ $Ae$-dom.dim.$(A)\geq 2$ ...
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1answer
27 views

Question concerning a faithful module over an Artinian ring

Let $A$ be an Artinian ring and $M$ in $\operatorname{\mathfrak{mod}} A$. Is it true that $M$ is faithful if and only if there is an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for ...
3
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43 views

For an exact sequence $0\to M_1\overset{f_1}\to\cdots\overset{f_r}\to M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$?

For an exact sequence $0\to M_1\overset{f_1}\rightarrow\cdots\overset{f_r}\rightarrow M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$? $M_i$s are modules and $l(M_i)$ ...
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58 views

short exact sequence of algebras over a field

Let $A,B,C$ be algebras over a field $F$ ($F=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime). The height of $A$ is defined to be $$ \mathrm{height}(A)=\sup_{a\in A}\inf\{n(a)\in \mathbb{N}\mid a^{n(a)+1}=0 ...
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1answer
21 views

Exact sequences of projective modules

Let $0{\rightarrow} K \stackrel{g}\rightarrow P\stackrel{f}\rightarrow Q \rightarrow 0$ and $0\rightarrow K' \stackrel{g'}\rightarrow P'\stackrel{f'}\rightarrow Q \rightarrow 0$ be exact sequences ...
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1answer
40 views

Prove that there's a unique morphism that completes the commutative diagram

I have to prove that there's a unique $\gamma : M'' \rightarrow N''$ that completes this diagram considering the rows are exact. $$\begin{array} MM' \stackrel{f_1}{\longrightarrow} & M & ...
3
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2answers
65 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
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1answer
41 views

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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1answer
49 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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1answer
32 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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1answer
21 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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2answers
47 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 ...
2
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1answer
28 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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2answers
45 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
20 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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59 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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1answer
24 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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1answer
56 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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19 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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598 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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1answer
61 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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1answer
156 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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71 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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1answer
52 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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41 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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65 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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89 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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2answers
88 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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45 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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85 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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1answer
20 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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1answer
35 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
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38 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
68 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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32 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 ...
4
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1answer
75 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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49 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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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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31 views

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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62 views

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 ...