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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9
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49 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.

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 ...
0
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1answer
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 $A\subset B$. ...
0
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1answer
42 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 ...
0
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0answers
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 ...
4
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3answers
578 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 ...
2
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1answer
47 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 ...
3
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1answer
141 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} ...
5
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2answers
63 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 ...
4
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1answer
35 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. ...
0
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0answers
36 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!
0
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2answers
60 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 ...
1
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2answers
82 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 ...
2
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2answers
78 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 ...
0
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0answers
26 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 ...
3
votes
1answer
68 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 ...
0
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1answer
16 views

Is the following short exact sequence always split?

Is it always true that the following short exact sequence of modules split? $0\to N\to M\to M/N\to0$
3
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1answer
29 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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1answer
32 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 $$ ...
1
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1answer
47 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 ...
0
votes
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
votes
1answer
67 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 ...
1
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1answer
42 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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0answers
46 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) ...
1
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0answers
27 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 ...
1
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0answers
53 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 ...
-1
votes
1answer
36 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 ...
2
votes
1answer
49 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 ...
3
votes
1answer
65 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 ...
0
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0answers
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 ...
1
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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 ...
0
votes
1answer
57 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 $A$ ...
1
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1answer
71 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 ...
12
votes
4answers
217 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 ...
4
votes
2answers
35 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 ...
3
votes
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 ...
2
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0answers
74 views

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 ...
4
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1answer
103 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 ...
1
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2answers
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 = ...
1
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2answers
84 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 ...
2
votes
0answers
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 ...
0
votes
1answer
41 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?
5
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1answer
154 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 ...
0
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1answer
39 views

Split exact sequences.

Edit: The setting is some abelian category. The splitting lemma says that the following conditions are equivalent for a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ ...
2
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0answers
49 views

What kind of objects are both subobjects and quotients?

Fix an object $B$ in some category. What does the existence of a diagram $A \rightarrowtail B \twoheadrightarrow A$ imply about $A$ and $B$? What if $A \rightarrowtail B \twoheadrightarrow ...
3
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1answer
103 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
1
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1answer
85 views

counterexample to “symmetric” nine lemma

Consider a commutative diagram of $R$-modules ...
3
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1answer
81 views

Commutative diagram with exact sequences as columns and rows

Suppose that we have the following commutative diagram of groups and homomorphisms $$\newcommand\twoheaduparrow{\mathrel{\rotatebox{90}{$\twoheadrightarrow$}}} \begin{array} A & A_3 & ...
1
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2answers
27 views

Non exact sequence of quotients by torsion subgroups

$$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion ...
5
votes
1answer
171 views

Splitting of an exact sequence

Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that $x \in \mathfrak m \setminus \mathfrak m^2$. Is it true that $$ \frac{\mathfrak m}{x\mathfrak m} \cong \frac{\mathfrak m}{(x)} ...
3
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1answer
90 views

Exactness of Hom Functor

The picture above is from Dummit and Foote, Third Edition, Chapter 10. In the text the authors claim that the sequence given by $ Hom $'s is exact if and only if there is a bijection $ F ...