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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Exactness and Naturality

I'm trying to read this blog post about exact functors, and I see mentions of naturality which I have not stumbled upon elsewhere. In particular, in the proof of the Theorem, the author says By ...
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Exact sequence of modules exercise

Show that if $$0 \rightarrow M_1 \xrightarrow{f} M_2 \xrightarrow{g} M_3$$ is an exact sequence of $R$-modules, then for all $R$-module $$0 \rightarrow \operatorname{Hom}_R(M,M_1) \xrightarrow{f_*} ...
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35 views

Exact sequence and Noetherian modules

Let $R$ be a ring, $X,Y,Z$ and $T$ four $R-$ modules such that there exists a short exact sequence $$0 \rightarrow X \xrightarrow{f_1} Y \xrightarrow{f_2} Z \xrightarrow{f_3} T \rightarrow 0 $$ Prove ...
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$Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all convergent numerators and denominators. True?

Where $x = A001203$, $Pi = A002486$, $A002485$ $Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all $Pi > n+1 $ Hypothesis: This relation evaluates true for all $A002486$ and $A002485$. Lemma: All ...
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Noetherian property for exact sequence

Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then $M$ is Noetherian is equivalent to $M'$ and $M''$ are Noetherian. For the ...
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$\mathbb Z_{p^2}$ is not a non trivial semidirect product.

I am trying to prove that the group $\mathbb Z_{p^2}$ (p prime) is not a non trivial semidirect product. Since a group $G \cong K \rtimes H$ if and only if for all short exact sequences $$0 ...
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Determining the middle term of an exact sequence of Lie algebras

This is related to my previous question here. Suppose that $A_i, B_i$ are Lie algebras with $A_i$ is a sub-Lie algebra of $B_i$, $i=1,2,3$. Suppose that we have the following commutative diagram where ...
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1answer
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Proof that the connecting morphism in the snake lemma is well defined

The snake lemma says: suppose we have two exact sequences of $R$-modules $M_1 \xrightarrow{f_M} M_2 \xrightarrow{g_M} M_3 \rightarrow 0$ $0\rightarrow N_1 \xrightarrow{f_N} N_2 \xrightarrow{g_N} ...
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Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
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Power series modulo polynomials

I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon. $\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and ...
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53 views

A question on short exact sequences

Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting short exact sequence so it exist a morphism $r: M \to L$ such that $r \circ \alpha = Id_L$ and a morphism $s: N \to M$ such ...
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Splitting of short exact sequence of sheaves

Let $X$ be a smooth projective variety over a field, say $k$. Consider the short exact sequence of $k$-modules, $$0 \to A_1 \to A \to A_2 \to 0$$ where $A$ and $A_2$ are $k$-algebras. Since these can ...
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1answer
65 views

Question about “M” in a short exact sequence

If we have a splitting short exact sequence $0\to L\stackrel {\alpha}\to M\stackrel{\beta}\to N\to 0$ with $r$ as a retraction of $\alpha$. Is it true then that $\alpha(L)\cup \ker(r)=M$?
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(Split) Exact Sequence of Lie Algebra Associated to Groups

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
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24 views

Exact sequence of sheaves from the exactness of sections

Let $X$ be a topological space, and $\mathcal{F}_1$, $\mathcal{F}_2$ and $\mathcal{F}_3$ be sheaves on $X$. Suppose for all $U$ open in $X$ we have, $0\longrightarrow ...
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27 views

Split exact sequence of Lie algebras

Suppose that we have the following commutative diagram of Lie algebras $$\begin{array} A 0& {\longrightarrow} & A_0 & {\longrightarrow} & A_1 &{\longrightarrow} & A_2 & ...
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Boundary Homomorphism

I was studying the proposition 2.10 of Atiyah and MacDonald's Introduction to Commutative Algebra, and have a question. The proposition says: Let $$ \require{AMScd} \begin{CD} 0 @>>> ...
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Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
2
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1answer
69 views

Exact sequence of $A$-modules [duplicate]

I was trying to demonstrate the Proposition 2.9 of Atiyah and MacDonald's Introduction to Commutative Algebra. But I couldn't do the following: Let $M$, $M'$, and $M''$ be $A$-modules, $v$ and $u$ ...
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27 views

Does exactness in each variable coincide with exactness of the product?

Let us restrict to the category of modules. I'm thinking about the definition of exactness of a functor on two variables. The usual definition is that it is exact in each of the two variable, whereas ...
2
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60 views

Kernel of a retraction

I have a couple of questions about a exercise I have: Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N \to 0$ be a splitting exact sequence and let $r$ be a retraction of $\alpha$ such that ...
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1answer
32 views

Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
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Directs sum in exact sequences

Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting s.e.s. where $\alpha$ has a retraction $r$. (a) Show that in this case $M=\alpha(L)\oplus\ker(r)$. I am a little bit ...
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59 views

Pullback and Kernel

We consider everything in the category of groups. It is known that monomorphisms are stable under pullback; that is, if $$\begin{array} AA_1 & \stackrel{f_1}{\longrightarrow} & A_2 \\ ...
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torsion of a module and exactness

Given a PID $A$ and $A$-modules $M$, $M'$, and $M''$. Assume that $$0\rightarrow M'\xrightarrow{f} M\xrightarrow{g} M''$$ is an exact sequence then prove that $$0\rightarrow ...
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23 views

First function in a short exact sequence

Let $0\to L\to M\to N \to0$ be short exact sequence. How does the function $0\to L$ look like? And what does $0$ mean here? Is it the zero of $A$ (a commutative ring s.t. $L,M,N$ are $A$-modules)?
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28 views

Normal Form of Elements in a Group

Suppose that there is a family of groups $A_n$ with $n\in\mathbb{N}$ and $A_1$ is the trivial group. If there is a split exact sequence $$0\to B_n\to A_n\to A_{n-1}\to 0,$$ where structure of the ...
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1answer
47 views

Short exact sequence of modules

I am trying to show that if we have the following left splitting short exact sequence of $R-$modules: $0 \rightarrow M \stackrel{f} \rightarrow N \stackrel{g} \rightarrow S \rightarrow 0$ then there ...
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Construction of splittings

I consider a graded algebra $\{A_k\}_{k\in\mathbb{Z}}$ and suppose that the sequence $$0\to A_{k-1}\to A_{k}\to A_{k}/A_{k-1} \to 0\qquad(*)$$ splits. I want to show that for all $j\in\mathbb{N}$, ...
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Reference request for Homology Gysin sequence.

I am trying to study the Homology Gysin sequence (not cohomology). I am interested in finding references that either use, or explain the Homology Gysin sequence, especially if it gives descriptions ...
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relationship between Exact sequences and Universal mapping property

I am stuck in the following question. Show that for any two short exact sequences. $0\overset{}{\rightarrow}K\overset{i}{\rightarrow}V\overset{T}{\rightarrow}U$ and ...
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54 views

Tensoring an exact sequence of $R$-modules with $R/x$

Let $R$ be a commutative ring with an $R$-module $M$, and let $x \in R$ be an $M$-regular element. Then tensoring any short exact sequence $0 \to B \to A \to M \to 0$ with $R/x$ yields a short exact ...
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Central extension: $Hom(Q,Z) \cong$ automorphisms of $G$ acting trivially on subgroup and quotient?

If we have a short exact sequence $1 \rightarrow C \rightarrow G \rightarrow Q \rightarrow 1$ where $C$ is central in $G$ and $Q \cong G/C$, how can I find an isomorphism between $Hom(Q,C)$ (which is ...
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Modules that allow an infinite exact sequence on them

I'm looking for a characterization of modules that fit the following property: (*) There's an infinite exact sequence with $\phi_i \neq 0 \space \forall i\in I $ $\require{AMScd}$ \begin{CD}\cdots ...
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exact sequence with infinite element

How to prove: Define exact sequence with infinite element from $R$-modules and $R$-homomorphism ,then exact sequence with infinite element from $\Bbb Z$-modules and $\Bbb Z$-homomorphism : ...
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69 views

Finite cyclic $\Bbb Z$-module and exact sequence

Suppose $M$ is $\Bbb Z$-module, cyclic, finite. How to prove $\require{AMScd}$ \begin{CD} 0 @>>> \Bbb Z @>>> \Bbb Z @>>> M @>>>0\\ \end{CD} is ...
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a problem in exact sequence

Suppose $\require{AMScd}$ \begin{CD} @.\acute M @>f>> M @>g>>\check M @>>> O\\ @. @V \alpha V V\ @VV \beta V @VV \gamma V @. \\ O@>>> \acute N ...
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Does the short exact sequence $0\rightarrow \mathbb{Z}_3 \rightarrow G \rightarrow \mathbb{Z}_2\oplus \mathbb{Z} \rightarrow 0$ always split? [closed]

Does the short exact sequence $0\rightarrow \mathbb{Z}_3 \rightarrow G \rightarrow \mathbb{Z}_2\oplus \mathbb{Z} \rightarrow 0$ always split?
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Tensor Product, Exterior Power and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $$0\to K\to M\otimes M\to M\wedge M\to 0$$ ...
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1answer
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Example of a commutative square without a map between antidiagonal objects?

In an abelian category, can there be a commutative diagram of (vertical/horizontal) exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> N @>>> M\\ @. @VVV @VVV \\ 0 @>>> X ...
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If the factor of a finitely generated module is free then submodule is also finitely generated

All rings are commutative, associative and with 1. Consider short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ of $R$-modules. How to show that if $M$ is finitely ...
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Confusion in Serre's Local fields book

I read that for right exact functors we consider left derived functors and the resolutions that we consider are projective resolutions... I read that for left exact functors we consider right derived ...
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Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
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A question on short exact sequences.

The following is an excerpt from Atiyah-Macdonald on short exact sequences. I don't understand the part where the author says "Then $d(x'')$ is defined to be the image of $y'$ in Coker ($f'$)". Is ...
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Is there a theory of “derived extensions”?

Given an exact sequence of groups $$1\rightarrow N\rightarrow G\rightarrow K\rightarrow 1$$ we call $G$ a central extension of $K$ by $N$ if the image of $N$ is contained in the center of $G$. Central ...
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Converse to Exactness of localization of modules

It is a standard fact that if $R$ is a ring, and $$\tag{1} 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0 $$ a short exact sequence of $R$-modules, if $S$ is a multiplicative ...
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1answer
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$0\to L\to M\to N\to 0 $ is split if $0\to {\rm Hom}_R(D,L)\to {\rm Hom}_R(D,M)\to {\rm Hom}_R(D,N)\to 0$ is exact for any $D$.

Prove that $0\rightarrow L\rightarrow M\stackrel{\phi}\rightarrow N\rightarrow 0 $ is split if $$0\rightarrow {\rm Hom}_R(D,L)\rightarrow {\rm Hom}_R(D,M)\rightarrow {\rm Hom}_R(D,N)\rightarrow 0$$ ...
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Summation of series- substitution

If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant, can we find a closed form for f(n)?. NB : Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ ...
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Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
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2answers
138 views

Atiyah and Macdonald, Proposition 2.9

The following simple claim is used without proof in Proposition 2.9 of Atiyah and MacDonald (p.23). Although I believe I can prove it with a fairly involved argument, the claim is treated by the ...