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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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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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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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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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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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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