Questions tagged [exact-sequence]

A sequence of morphisms where the image of one is the kernel of the next. It is useful thing to examine in the study of abstract algebra and homological algebra. For sequences of numbers use the tag (sequences-and-series) instead.

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45 views

Is there a way of speaking about exact sequences in a category “nice enough” but without zero object?

I have a concrete complete and cocomplete category $\mathcal{A}$ (in fact, a category of modules over a monoid in a symmetric closed monoidal category) in which all monomorphisms and all epimorphisms ...
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Reverse sequence of a short exact sequence is short exact sequence

Why is the reverse sequence of a given short exact sequence also exact? (I understand that the map on the left will be one-to-one and on the right will be onto but how to see exactness of the module ...
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Split exact sequence of linear maps implies split exact sequence of linear hom maps

If the exact sequence of left $A$-modules $$0\xrightarrow{}F'\xrightarrow{u}F\xrightarrow{v}F''\xrightarrow{}0$$ splits, the sequence $$0\xrightarrow{}\text{Hom}(E,F')\xrightarrow{\bar{u}}\text{...
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Split and non splitting short exact sequence [closed]

Please explain that in this, Why the first sequence is split exact while other is non-split and exact?
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Split exact short sequence

If the given short exact sequence $$ 0\to M'\xrightarrow{f}M\xrightarrow{g}M''\to0 $$ splits then why is $M$ isomorphic to the internal direct sum of $M'$ and $M''$? I know that if this sequence ...
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Additivity of Euler characteristic

It is stated that for $\chi(X)$ the Euler characteristic of a space $X$, and $Z \subset Y \subset X$, it holds that: $$ \chi (X,Z) = \chi(X,Y) + \chi(Y,Z)$$ How does this follow from the exact ...
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Homology with Coefficients of an abelian group

I am trying to see why this is true, this is done in Brown's Cohomology of groups: (In terms of notation $\tau(y)$ is the divided polinomyal algebra). Consider the study of $H_*(G,\mathbb{Z}_p)$, ...
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Short exact sequences of finite abelian groups

I was reading this post, and I was wondering if you had instead the direct sums as such: $$0\rightarrow\mathbb{Z}_{p^{a_1}}\oplus...\oplus\mathbb{Z}_{p^{a_n}}\rightarrow\mathbb{Z}_{p^{b_1}}\oplus...\...
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Properties of an A-module

I must show that the following properties for an $A$-module $P$ are equivalent: 1) The functor $Hom(P,-)$ is exact. 2) There is an $A$-module $Q$ such that $P \oplus Q$ is free. 3) Every short ...
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Commutative Diagram of Vector Spaces with Short Exact rows

I found the following question in an online book of Linear Algebra exercises (page 69) All of the mathematical objects are vector spaces, and thus the maps are linear. It does not specify if the ...
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Suppose that $g$ is isomorphism. Then prove that $f$ is monic and $h$ is epic.

$$ \\ 0 \to A \to B \to C \to 0 \\ 0 \to A' \to B' \to C' \to 0 $$ These are exact and they occur a commutative diagram by homomorphism. $$ g=B\to B'\\ f=A\to A'\\ h=C \to C' $$ Suppose that $g$ is ...
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Prove that there exists the exact sequence (over Z) 0 → Z 2 →Z 4 → Z 4 → Z 2 → 0 .

$$ 0 \to \mathbb Z /2 \to \mathbb Z /4 \to \mathbb Z /4 \to \mathbb Z / 2 \to 0$$ To show that it is exact, I need $f : \mathbb Z / 2 \to \mathbb Z / 4$ monic, and $g : \mathbb Z /4 \to \mathbb Z /2$ ...
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59 views

In the following commutative diagram, with exact rows and columns, is $j_1$ surjective?

I have the following commutative digram of abelian groups and homomorphisms between them. The rows and columns are exact: \begin{array} A & && && & & & & 0 &...
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Clarification in early example of exact sequence in Eisenbud's Commutative Algebra

It feels like a lot of shortcuts are being take here: "The kernel is generated by the class of $a$ modulo $I$." So, is the kernel just $(a+I)$? "Since the kernel is generated by just one element, ...
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Cohomology of a monad.

Definition. A monad over a projective variety $X$ is a complex $$M : 0 \longrightarrow \mathcal{A} \stackrel{f} {\longrightarrow} \mathcal{B} \stackrel{g} {\longrightarrow} \mathcal{C} \longrightarrow ...
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Modica-Mortola functional

I need some help in one part of my exercise about Modica-Mortola functional in calculus of variation. \begin{equation} F(u)=\int_\mathbb{R}u'(x)^2+W(u(x))dx \end{equation} Where $u\in H^1_{loc}$ and ...
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Short exact sequence and the order of groups

Given a short exact sequence $$1 \rightarrow A \xrightarrow{\phi} B \xrightarrow{\psi} C \rightarrow 1, $$ what can I say about the relationships between the order of the groups $A, B$ and $C$? This ...
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Comparing the exactness of $(\prod_{\lambda\in \Lambda}M_{\lambda})$ and $M_{\lambda}$ on a long exact sequence.

Let $R$ be a ring with unity and $S=\text{End}_R(M), {}_{S}M_R$ be an $(S,R)$-bimodule and $\{M_\lambda\}_{\lambda\in \Lambda}$ be a family of modules. Define the map $\phi:M\to M,\phi(m)= mr,r\in R)$...
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How to write down presentation of R-Modules

For the following modules I should give presentations by generators & relations (finitely many each if possible): The $\mathbb{Z}$-module $\mathbb{Q}$ The $\mathbb{C}[X]$-module $\mathbb{C}^4$, ...
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Why is $C_G(A)$ a normal subgroup of $B$ in this context?

Let's consider the group extension $G$ s.t. $$1 \to A \to G \to B \to 1$$ where $A$ and $B$ are finite, non-abelian and simple groups. Let $C_G(A)$ be the centralizer of $A$. Since $A$ is normal, $C_G(...
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Exactness of Inflation restriction sequence, Galois Cohomology

I am trying to prove the following. Let $K/k$ be a finite Galois extension, $G= G(K/k)$, $k \subset F \subset K$ with $K/k$ normal and $H=G(K/F)$. Then: $ \rho : C^{2} (G,A) \rightarrow C^{2} (H,A) $ ...
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The only group $G$ with one $A$ and one $B$ as composition factors is $G = A\times B$ (where $A$ and $B$ are non-abelian, finite and simple)

Is it true that if $A$ and $B$ are two non-abelian finite simple groups, then the only finite group $G$ which has one copy of $A$ and one copy of $B$ as composition factors is $G = A \times B$? If not,...
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$A_5$ or the icosahedral group $I$ is not isomorphic to any subgroup of the binary icosahedral group $2I$

I'm trying to show that the extension $1 \to \{\pm 1\} \to 2I \to I \to 1$ does not split. For that, I think it's sufficient to show that $I$ is not isomorphic to any subgroup of $2I$ (?). But I'm not ...
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Determining the homomorphism $\varphi: H \to \mathrm{Aut}(K)$ given a section $s: H \to G$

Per Wikipedia, a split extension is an extension $$1 \to K \overset{\beta}{\to} G \overset{\alpha}{\to} H \to 1$$ with a homomorphism $s: H \to G$ such that going from $H$ to $G$ by $s$ and then back ...
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Additivity of Rank of an Abelian group

I am studying algebraic topology from Rotman and I needed to use rank of an (not necessarily free) Abelian group and in the exercises I faced a problem saying, given an exact sequence $0 \rightarrow A ...
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Short exact sequence and pushout

Let $0\to N\stackrel{f}{\to} E\stackrel{g}{\to} M\to 0$ be a short exact sequence of $R$-modules. Let $\varphi:N\to N'$ be a morphism of $R$-modules and let $E'$ be the pushout of $N\to E$ and $\...
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Explicit description of all extensions of $\mathbf{Z}/n\mathbf{Z}$ by $\mathbf{Z}$

In an exercise in my syllabus on homological algebra, I need to explicitly describe what are the $n$ isomorphism classes of extensions of $\mathbf{Z}/n\mathbf{Z}$ by $\mathbf{Z}$ for the cases $n=p$ ...
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Has anyone seen this generalization of the snake lemma? Is it useful?

While learning about spectral sequences a friend of mine found a proof of the snake lemma using spectral sequences. We noticed that the proof works equally well for larger bicomplexes. Particularly if ...
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How can I show that this diagram is commutative using Mayer-Vietoris sequences?

From Rotman's Algebraic Topology: Assume that $X = X_1^{\circ} \cup X_2^{\circ}$ and $Y = Y_1^{\circ} \cup Y_2^{\circ}$; assume further that $f : X \rightarrow Y$ is continuous with $f(X_i) \subset ...
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Exactness in category theory

In MacLane's 'Category Theory for the working mathematician' there is a definition of exactness (page 200): 'A composable pair of arrows $f: a\rightarrow b$ and $g: b\rightarrow c$ is exact at b if ...
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Cosets of $\Bbb Z/p$ in $\Bbb Z/p^2$: do $g, 2g, 3g, \ldots, pg$ lie in separate cosets where $\langle g \rangle = \Bbb Z/p^2$?

Consider the short exact sequence: $$0 \to \Bbb Z/p \to \Bbb Z/p^2 \to \Bbb Z/p \to 0$$ I'm trying to analyze the cosets of $\Bbb Z/p$ in $\Bbb Z/p^2$. If $g$ is a generator of $\Bbb Z/p^2$, is it ...
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Double Complex Where $\mathrm{Tot}^\prod$ is Acyclic but $\mathrm{Tot}^\oplus$ is not

I am working through Weibel's Introduction to Homological Algebra, and am having trouble finding examples for Exercise 1.2.6 A second quadrant double complex $C$ with exact columns such that $\mathrm{...
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Visualizing split extensions and central extensions

This may be a bit of a silly question, but I think it is useful to get a visual idea of the concept of group extensions. A short exact sequence $1 \to A \overset{i}{\to} B \overset{\pi}{\to} C \to 1$ ...
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External power of locally free sheaves

I have a question about exercise II. 5.16 (d) from R. Hartshornes Algebraic Geometry: Now let $(X,O_X)$ be a ringed space and let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be an ...
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Double cover and cohomology

I'm working on this problem, from a past Qual. Let $p:\tilde X\to X$ be a double cover. Show that there is a LES $$\cdots \to H^k(X;\mathbb{Z}/2\mathbb{Z})\to H^k(\tilde X;\mathbb{Z}/2\mathbb{Z})\to ...
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If $0\stackrel{f}\rightarrow V \stackrel{g}\rightarrow 0$ is an exact sequence, then $V = 0$.

Show that if $0\stackrel{f}\rightarrow V \stackrel{g}\rightarrow 0$ is an exact sequence of vector spaces over a field $K$, then $V = 0$. I know that $\ker(g) = \text{im}(f)$, but how can I get that $...
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Short exact sequences envolving syzygies

The following Lemma came from the paper Generalized Igusa-Todorov function and finitistic dimensions - D. Xu, in which is described by the author as a "basic homological fact" and is presented without ...
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Long exact sequence of sheaf cohomology from the normal bundle of $\mathbb{P}^1$ in $\mathbb{P}^2$

Let $i: \mathbb{P}^1 \to \mathbb{P}^2$ denote the inclusion $[x_0,x_1] \mapsto [x_0,x_1,0]$. In other words, we identify $\mathbb{P}^1$ as the subvariety defined by ${x_2 = 0}$ in $\mathbb{P}^2$. The ...
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Short exact sequence with missing functions

A short exact sequence is defined as a sequence of algebraic objects (e.g. groups) of the form $$ 0 \longrightarrow A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} C \longrightarrow 0$$ ...
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Is the fibre-product functor $(-)\times_N M$ exact?

Consider an abelian category $\mathcal C$; if it helps, modules over a sufficiently friendly ring. Let $N\in\mathcal C$. We can consider the over-category $\mathcal C_{/N}$ of objects from $\mathcal C$...
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Understanding the relation between central extensions and 2-cocycles and coboundaries

Context This is basically the lecture note I received on central extensions (we were not introduced to group cohomology before this): If $$1 \to A \to G \to B \to 1$$ is a non-trivial ...
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Excisive triad and long exact sequence

Let $(A\cup B;A,B)$ be an excisive triad (i.e. by axiom of excision inclusion of pairs $(B,A\cap B)\rightarrow (X,A)$ induces an isomorphism $h_*(B,A\cap B)\rightarrow h_*(X,A)$). Let $\Delta_n:h_n(X)\...
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Splitting of short exact sequence of sheaves of vector spaces

It is well-known that any short exact sequence of vector spaces splits. Is it also true that the short exact sequence of sheaves of $\mathbb C$-vector spaces $$0\rightarrow A\rightarrow B\rightarrow ...
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Why must a short exact sequence of Kahler Differentials exist.

On the Wikipedia page for Kahler Differentials, under examples and basic facts - it is said that given two ring homomorphisms: $R \rightarrow S \rightarrow T$, there is a short exact sequence $\Omega_{...
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Commutative ladder with exact rows

I'm attempting to answer the following: In a commutative ladder of R-modules with exact rows: where $\gamma_n$ are isomorphisms. Prove that sequence: $$ \dots \rightarrow A_n \overset{(\alpha_n,-...
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Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group?

My professor made the following remark while teaching about group extensions: We want to classify finite groups in a manner similar to the fact that every positive integer is uniquely a product of ...
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Showing that the extension $1 \to C_3 \to C_6 \to C_2 \to 1$ is split by finding complement of $C_3$

I'm trying to solve the following exercise: Show that the extension $1 \to C_3 \to C_6 \to C_2 \to 1$ is split but the extension $1 \to C_2 \to C_4 \to C_2 \to 1$ is not split. I think one way to ...
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Proving exactness not at the connecting hom in a Snake Lemma subproblem.

Here's a picture of the Snake Lemma from nLab: I'm having trouble showing exactness at $\ker g$ and haven't even got to the connecting hom yet. So let's focus on $\ker g$. Let $q : A' \to B'$ be ...
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Two possibilities of $X$ to make the sequence $0\longrightarrow\mathbb{Z}\longrightarrow X\longrightarrow\mathbb{Z}_{2}\longrightarrow 0$ exact.

I am working on an exercise in homological algebra asking to find the possibilities of $X$ such that the sequence $$0\longrightarrow\mathbb{Z}\longrightarrow X\longrightarrow\mathbb{Z}_{2}\...
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Showing this function between homology groups is well defined

From Rotman's Algebraic Topology If $A$ is a retract of $X$, then $H_n(X)\approx H_n(A) \oplus H_n(X,A)$ If $r\colon X \rightarrow A$ is the retract with inclusion $i$, then define $f \colon H_n(X)...

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