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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How to conclude the order of $G$ is $p^{m+n}$.

I have this exact sequence $$ 0\to \mathbb{Z}_{p^m}\overset{f}\to G\overset{g}\to \mathbb{Z}_{p^n}{\to} 0 $$ Where's $G$ is finitely generated. I want to prove that $|G|=p^{m+n}$ and a friend told me ...
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Equivalent Definition of injective $R$-module using exact sequences

We call an $R$-module $D$ injective if one of the two equivalent conditions holds: If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an exact sequence of $R$-modules then the induced ...
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Set of complementary subspaces as an affine space

Let $V$ be a vector space and $W$ a fixed vector subspace of $V$. Denote by $\mathcal{E}$ the set of vector subspaces $U$ of $V$ such that $V=U\oplus W$. It is stated in the Examples section here that ...
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$A$ noetherian ring, $M$ finite $A$-module. Why is this an exact sequence?

So I read that under this conditions, for some $p$ (which I believe to be prime but I am not entirely sure) we can generate the following short exact sequence: $$0\longrightarrow\ker\beta\xrightarrow{...
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Exact sequence splits iff it kind of weakly splits through other sequence

This is an exercise from Assem's book "Algèbre et module" (more precisely, III.50). I couldn't solve it after thinking for a while. Let $\require{AMScd}$ \begin{CD} 0 @>>> L @>...
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Usage of exact sequences to show that $A/\mathfrak a \otimes_A M \cong M/\mathfrak aM$ [duplicate]

Let $A$ be a commutative ring & $\mathfrak a$ an ideal and $M$ an $A$-module. Show that $A/\mathfrak a \otimes_A M \cong M/\mathfrak aM$. [Tensor the exact sequence $0 \longrightarrow \mathfrak a \...
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Exactness of sequence induced by $\operatorname{Hom}(G,\cdot)$.

$\newcommand{\Hom}{\operatorname{Hom}}$ For this problem, we let $0\rightarrow A\xrightarrow{i} B \xrightarrow{j}C\rightarrow 0$ be a short exact sequence of groups, and $G$ an abelian group. It's not ...
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On an exact sequence of complex vector spaces

Let us consider the following exact sequence of complex vector spaces: $0 \to A \to B \to C \to D \to E \to.....\to 0$, where $E$ and the later vector spaces are not necessarily zero. By rank nullity ...
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Splitting Lemma for Vector Bundles

I am asked to solve the following exercise. Let $E = E[M; \pi, \mathbb{R}^n]$,$F = F[M; \pi, \mathbb{R}^m]$, $H = H[M; \pi, \mathbb{R}^k]$ be three smooth vector bundles over $M$ of finite rank $n,m,k ...
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An exact sequence of cyclic groups related to the lcm

The sequence $0 \longrightarrow\mathbb{Z}/(a\vee b \vee c)\mathbb{Z} \underset{\phi_1}{\longrightarrow} \mathbb{Z}/a\mathbb{Z} \bigoplus \mathbb{Z}/b\mathbb{Z} \bigoplus \mathbb{Z}/c\mathbb{Z}\...
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Action of a map on homology

I'm currently studying algebraic topology from Hatcher's text, and I came across the following problem from an old qualifying exam: The coefficient sequence $0 \rightarrow \mathbb{Z} \xrightarrow{p} \...
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For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent sheaves on $X$. Suppose we have a family ...
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Direct sum and short exact sequence, as well as, tensor product and what?

In an abelian category, the notion of direct sum is generalized by the notion of short exact sequence (see split exact sequence). Question: In a monoidal category, can the notion of tensor product be ...
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Ext and split short exact sequence

Let $R$ be a commutative ring. $I$,$J$ are two ideals of $R$ such that $I\cap J=0$. We have a short exact sequence, $$0 \longrightarrow R\stackrel{f} \longrightarrow R/I\oplus R/J\stackrel{g}{\...
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Proof that sequence is not split using left inverse

For the following exact sequence, i am trying to proof that it is not split $0\to 2\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$ I want to show that for injective function $i : 2\mathbb{...
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Why induced exact sequence implies injectivity in question

N is a module and M & L are submodules of N I want to show that $\beta$ is injective but am having trouble understanding why the exact sequence above implies injectivity.
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Representations and central extensions of $A_5$

I'm taking a course on representation theory this semester, and the professor assumed we had some knowledge in central extensions. Unfortunately, I'm not very familiar with the topic, and I'm having ...
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How do I prove if the functor is right exact or left exact?

Hello I have the following question: I need to show if the functor $\Bbb{Q}\otimes_R-$ from $\Bbb{Z}$-modules to $\Bbb{Q}$ vector spaces are left exact, right exact or even nothing. I somehow ...
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Prove that if $\alpha_{3}$ is an isomorphism, then the sequence is a short exact sequence.

I am having trouble making some connections in proving this result and I'd like some help sorting this out. So here is the commutative diagram of $R$-modules and $R$-module homomorphisms. Each row is ...
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How do I prove if the following functor is exact?

I have the following exercise: Prove if the functor that sends an abelian group to it's $n$-torsion subgroup for $n\geq 2$ is exact. I know that I need to take $f\colon M\to N$ and $g\colon N\to L$ ...
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How do I prove if the forgetful functor is exact?

Hello I have the following problem: Check if the forgetful functor from $R$-modules to abelian groups for a ring $R$ is exact or not. As I understood our case we have the following functor $$F:Mod(R)...
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A sufficient condition for Automorphism of an exact sequence

I am given the following commutative diagram with non-split exact rows with $L, M, N$ being finite dimensional $A$ modules where $A$ is a finite dimensional $\mathbb{C}$ algebra. where $L$ is an ...
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$0\to \text{Hom}(P_0,G)\to \text{Hom}(P_1,G)\to \text{Hom}(P_2,G)\to\dots$ is an exact sequence. [duplicate]

I am reading ''Algebraic number theory'' by Cassels and Fröhlich and in the chapter IV.4 it says the following: If $\dots\to P_2\to P_1\to P_0\to\mathbb{Z}\to 0$ is a projective resolution and $G$ is ...
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Is a module a limit of n-presented modules?

Let n be an integer. A module M is said to be n-presented if there exist an exact sequence of the form $$ F_{n}\to F_{n-1}\to ...\to F_{1}\to F_{0}\to M \to 0$$ with every $F_{i}$ is a finitely ...
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Right exact sequence of tensor algebra

Let $R$ be a ring. Given an exact sequence of $R$-modules $$K\longrightarrow N\longrightarrow M\longrightarrow0$$ deduce $$T(N)\otimes K\otimes T(N)\longrightarrow T(N)\longrightarrow T(M)\...
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Exact sequence multiplies an ideal

let $A,B,C$ be $R$-modules and $I\subset R$ be an ideal. Given that $A\xrightarrow{f} B\xrightarrow{g} C\to 0$ is exact, do we have that $IA\xrightarrow{f'} IB\xrightarrow{g'} IC\to 0$ is exact? I ...
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Faithful flatness and split-exactness

Suppose $R\to S$ is a faithfully flat morphism of commutative unital rings, and suppose $A\to B\to C\to 0$ is an exact sequence of finitely-generated $R$-modules. If we know that $0\to A\otimes_R S\to ...
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Commutator subgroup of a Wang group is finite iff abelian

We say a group $ G $ is a Wang group if it fits into a SES $$ 0 \to N \to G \to \mathbb{Z}^k \to 0 $$ where $ N $ is nilpotent finitely generated and torsion free. Such a group $ G $ is always ...
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why can $g$ be considered as a linear function?

Here is the question I am trying to understand its solution: Compute the homology groups $H_n(X, A)$ when $X$ is $S^2$ or $S^1 \times S^1$ and $A$ is a finite set of points in $X.$ Here is the part I ...
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Why $H_1(S^2, A) = \operatorname{im}g$? or we can prove it using splitting lemma? [closed]

Here is the question I am trying to understand its solution: $(a)$ Compute the homology groups $H_n(X, A)$ when $X$ is $S^2$ or $S^1 \times S^1$ and $A$ is a finite set of points in $X.$ And here is a ...
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3 votes
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Vector bundles from short exact sequences

In the following I consider complex manifolds and holomorphic vector bundles. I'm trying to understand the construction of non-trivial vector bundles $V$ via short exact sequences involving sums of ...
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4 votes
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Extending a commutative diagram of exact sequences

Let $R$ be a ring and $L_1,L_2,M_2,M_2,N_1,N_2$ $R$-modules. Say you have the following diagram: $\require{AMScd}$ \begin{CD} 0 @>{}>> L_1 @>{f_1}>> M_1 @>{g_1}>> N_1 @> &...
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Splitting of the short exact sequence $0\to \frac{H_n(X)}{p H_n(X)}\to H_n(X,\mathbb{Z}_p)\to \ker\{ p: H_{n-1}(X)\to H_{n-1}(X)\}\to 0$

One of the assignments for my algebraic topology course is to solve the following problem from Topology and Geometry by Bredon. Multiplication by the prime $p$ fits a short exact sequence $$0\to \...
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1 vote
1 answer
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Zariski tangent space and exactness of $\operatorname{Der}_R(A,-)$ functor

Let $A$ be an $R$-algebra (for $R$ a commutative ring). Let $\def\Der{\operatorname{Der}}\Der_R(A,-): A-\mathrm{mod}\to A-\mathrm{mod}$ be the covariant functor, where $\Der_R(A,M)$ is the set of all $...
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What are the values of $g$ on $(0,1)$ and $(1,0)$?

I am trying to find a short exact sequence of the following form: $$0 \to \mathbb Z_4 \xrightarrow{f} \mathbb Z_8 \oplus \mathbb Z_2 \xrightarrow{g} \mathbb Z_4 \to 0$$ I assumed that actually there ...
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Consequences of $f$ is injective and $g$ is surjective in the following sequence?

I am trying to study if there can be a SES of the following form $$0 \to \mathbb Z_4 \xrightarrow{f} \mathbb Z_8 \oplus \mathbb Z_2 \xrightarrow{g} \mathbb Z_4 \to 0.$$ So, I know if there is one, $f$ ...
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3 votes
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SES of finitely generated modules over noetherian ring split, given existence of an isomorphism.

I recently came across the following statement, however I always fail to prove it. I'm also a bit unsure where the assumption "noetherian" is used: Let $R$ be a noetherian ring, and $M_i$ ...
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1 vote
1 answer
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Finitely generated $R$-module that is not projective or finitely presented

Give an example of a finitely generated $R$-module $M$ (for some commutative ring $R$) that is not projective and is not finitely presented. I was able to find an example of a finitely generated $R$-...
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When does the short exact sequence of Module and submodule not split?

Consider a commutative Ring $R$ and a module $M$ over $R$. Now let $N \subset M$ be a submodule. Then we have a canonical short exact sequence: $$0 \rightarrow N \xrightarrow{i} M \xrightarrow{p}M/N \...
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2 votes
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Finding the inverse of a group isomorphism linked to a short exact sequence

Exercise: Construct an explicit inverse of $\phi$ in the following theorem Theorem: Given a short exact sequence with a right split (this is, a $v: K \to G$ such that $\epsilon v = 1$), which ...
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2 votes
1 answer
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Showing that a certain map in a commutative diagram with exact rows is injective

This exercise is from Dummit and Foote, Section 10.5. (Exercise 1, part d) The following diagram is commutative with exact rows. We know that $\alpha,\gamma$ are surjective, and $\beta$ is injective. ...
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Splitting lemma, a detail

Let R be a commutative unitary ring and let $0\overset{}{\to}N \overset{f}{\to} M \overset{g}{\to} P \to 0$ a short exact sequence of R-modules. Now suppose that the sequence splits. If N and P are ...
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What does it mean for the sequence $\text{Hom}(E,\text{Hom}(N,P))$ to be exact, where $E$ is an exact sequences of $A$-modules, and $N,P$ $A$-modules?

Here $E$ is the exact sequence $M^{\prime} \stackrel{f}{\rightarrow} M \stackrel{g}{\rightarrow} M^{\prime \prime} \rightarrow 0$. I know what an exact sequence of $A$-modules is, but not what $\text{...
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Product of functors (A,_) being a generator and the arbitrary sum of left-exact functors

In Peter Freyd's book Abelian Categories he states a theorem saying that $\mathscr{L}(\mathscr{A})$ is complete and has an injective generator, with $\mathscr{L}(\mathscr{A})$ being the category of ...
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Showing an induced sequence is exact.

Edit: I got stuck in a couple of places in proving the statement below. Specifically: $d$ is well-defined. $\text{ker}(d) \subseteq \text{im}(\bar{v})$ $\text{ker}(\bar{v}’) = \text{im}(\bar{u}’)$ ...
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The limit of the sequence: $U_n=(2\times3)^{-(3\times4)^{..^{\mathrm{-n(n+1)}}}}$

I trying solving this equation here ,I find that the limit l verify: $2=l^{-6^{-12^{..^{\mathrm{-\infty}}}}}$ So now I need the limit of $U_n=(2\times3)^{-(3\times4)^{..^{\mathrm{-n(n+1)}}}}$
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3 answers
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Proof of $\tilde{H}_*(X) \cong H_*(X, x_0) $.

Let $X$ be a topological space and $x_0 \in X$. I want to show that the relative homology group, $H_*(X, x_0)$, is isomorphic to the reduced homology group, $\tilde{H}_*(X)$. By considering the long ...
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2 votes
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Proof of $\tilde{H}_k(X \lor Y) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$.

In my algebraic topology course, we showed the following statement: Let $X, Y$ two topological spaces, $(x_0, y_0) \in X \times Y$ and $X\lor Y$ the wedge product of $(X, x_0)$ and $(Y, y_0)$. If $x_0$...
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  • 2,465
1 vote
2 answers
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Tensor product and exact sequences

I'm having some problems with tensor product. I know that, in general, for $M$ an $R-$module and $I$ an ideal it holds $$M \otimes_R R/I = M/IM.$$ I also know that for $M_1$, $M_2$ and $N$ $R-$modules ...
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1 vote
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Confusion about a definition in Atiyah-Macdonald

In chapter 2 of Atiyah-Macdonald are introduced the flat modules, and the algebras. However it is not given any definition of flat algebra, while in exercise 5 and 8, for example, is needed. I don't ...
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