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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questions
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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 ...
0
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0answers
47 views
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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0answers
25 views
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
$$0\xrightarrow{}\text{Hom}(E,F')\xrightarrow{\bar{u}}\text{...
0
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1answer
43 views
Split and non splitting short exact sequence [closed]
Please explain that in this,
Why the first sequence is split exact while other is non-split and exact?
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0answers
58 views
Split exact short sequence
If the given short exact sequence
$$ 0\to M'\xrightarrow{f}M\xrightarrow{g}M''\to0 $$
splits then why is $M$ isomorphic to the internal direct sum of $M'$ and $M''$?
I know that if this sequence ...
2
votes
1answer
47 views
Additivity of Euler characteristic
It is stated that for $\chi(X)$ the Euler characteristic of a space $X$, and $Z \subset Y \subset X$, it holds that:
$$ \chi (X,Z) = \chi(X,Y) + \chi(Y,Z)$$
How does this follow from the exact ...
2
votes
0answers
39 views
Homology with Coefficients of an abelian group
I am trying to see why this is true, this is done in Brown's Cohomology of groups:
(In terms of notation $\tau(y)$ is the divided polinomyal algebra).
Consider the study of $H_*(G,\mathbb{Z}_p)$, ...
0
votes
1answer
46 views
Short exact sequences of finite abelian groups
I was reading this post, and I was wondering if you had instead the direct sums as such:
$$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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0answers
24 views
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 ...
0
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1answer
16 views
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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1answer
65 views
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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0answers
58 views
Prove that there exists the exact sequence (over Z) 0 ā Z 2 āZ 4 ā Z 4 ā Z 2 ā 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 $g : \mathbb Z /4 \to \mathbb Z /2$ ...
1
vote
1answer
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 &...
0
votes
1answer
58 views
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, ...
2
votes
1answer
88 views
Cohomology of a monad.
Definition. A monad over a projective variety $X$ is a complex $$M : 0 \longrightarrow \mathcal{A} \stackrel{f} {\longrightarrow} \mathcal{B} \stackrel{g} {\longrightarrow} \mathcal{C} \longrightarrow ...
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0answers
43 views
Modica-Mortola functional
I need some help in one part of my exercise about Modica-Mortola functional in calculus of variation.
\begin{equation}
F(u)=\int_\mathbb{R}u'(x)^2+W(u(x))dx
\end{equation}
Where $u\in H^1_{loc}$ and ...
1
vote
1answer
41 views
Short exact sequence and the order of groups
Given a short exact sequence
$$1 \rightarrow A \xrightarrow{\phi} B \xrightarrow{\psi} C \rightarrow 1, $$
what can I say about the relationships between the order of the groups $A, B$ and $C$?
This ...
1
vote
0answers
28 views
Comparing the exactness of $(\prod_{\lambda\in \Lambda}M_{\lambda})$ and $M_{\lambda}$ on a long exact sequence.
Let $R$ be a ring with unity and $S=\text{End}_R(M), {}_{S}M_R$ be an $(S,R)$-bimodule and $\{M_\lambda\}_{\lambda\in \Lambda}$ be a family of modules. Define the map $\phi:M\to M,\phi(m)= mr,r\in R)$...
0
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0answers
38 views
How to write down presentation of R-Modules
For the following modules I should give presentations by generators & relations (finitely many each if possible):
The $\mathbb{Z}$-module $\mathbb{Q}$
The $\mathbb{C}[X]$-module $\mathbb{C}^4$, ...
0
votes
1answer
33 views
Why is $C_G(A)$ a normal subgroup of $B$ in this context?
Let's consider the group extension $G$ s.t.
$$1 \to A \to G \to B \to 1$$ where $A$ and $B$ are finite, non-abelian and simple groups. Let $C_G(A)$ be the centralizer of $A$. Since $A$ is normal, $C_G(...
1
vote
1answer
47 views
Exactness of Inflation restriction sequence, Galois Cohomology
I am trying to prove the following.
Let $K/k$ be a finite Galois extension, $G= G(K/k)$, $k \subset F \subset K$ with $K/k$ normal and $H=G(K/F)$. Then:
$ \rho : C^{2} (G,A) \rightarrow C^{2} (H,A) $ ...
3
votes
3answers
115 views
The only group $G$ with one $A$ and one $B$ as composition factors is $G = A\times B$ (where $A$ and $B$ are non-abelian, finite and simple)
Is it true that if $A$ and $B$ are two non-abelian finite simple groups, then the only finite group $G$ which has one copy of $A$ and one copy of $B$ as composition factors is $G = A \times B$? If not,...
1
vote
0answers
30 views
$A_5$ or the icosahedral group $I$ is not isomorphic to any subgroup of the binary icosahedral group $2I$
I'm trying to show that the extension $1 \to \{\pm 1\} \to 2I \to I \to 1$ does not split. For that, I think it's sufficient to show that $I$ is not isomorphic to any subgroup of $2I$ (?). But I'm not ...
1
vote
1answer
53 views
Determining the homomorphism $\varphi: H \to \mathrm{Aut}(K)$ given a section $s: H \to G$
Per Wikipedia, a split extension is an extension $$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 ...
1
vote
2answers
101 views
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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2answers
42 views
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 $\...
4
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1answer
65 views
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$ ...
5
votes
0answers
455 views
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 ...
1
vote
2answers
62 views
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 ...
2
votes
1answer
88 views
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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0answers
42 views
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$$
I'm trying to analyze the cosets of $\Bbb Z/p$ in $\Bbb Z/p^2$. If $g$ is a generator of $\Bbb Z/p^2$, is it ...
4
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1answer
43 views
Double Complex Where $\mathrm{Tot}^\prod$ is Acyclic but $\mathrm{Tot}^\oplus$ is not
I am working through Weibel's Introduction to Homological Algebra, and am having trouble finding examples for Exercise 1.2.6
A second quadrant double complex $C$ with exact columns such that $\mathrm{...
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1answer
35 views
Visualizing split extensions and central extensions
This may be a bit of a silly question, but I think it is useful to get a visual idea of the concept of group extensions. A short exact sequence $1 \to A \overset{i}{\to} B \overset{\pi}{\to} C \to 1$ ...
0
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1answer
66 views
External power of locally free sheaves
I have a question about exercise II. 5.16 (d) from R. Hartshornes Algebraic Geometry:
Now let $(X,O_X)$ be a ringed space and let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be an ...
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0answers
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Double cover and cohomology
I'm working on this problem, from a past Qual.
Let $p:\tilde X\to X$ be a double cover. Show that there is a LES
$$\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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1answer
34 views
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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vote
1answer
56 views
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 ...
5
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1answer
72 views
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 ...
1
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1answer
34 views
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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0answers
30 views
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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0answers
71 views
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 ...
0
votes
0answers
18 views
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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0answers
41 views
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
$$0\rightarrow A\rightarrow B\rightarrow ...
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0answers
45 views
Why must a short exact sequence of Kahler Differentials exist.
On the Wikipedia page for Kahler Differentials, under examples and basic facts - it is said that given two ring homomorphisms:
$R \rightarrow S \rightarrow T$, there is a short exact sequence $\Omega_{...
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0answers
26 views
Commutative ladder with exact rows
I'm attempting to answer the following:
In a commutative ladder of R-modules with exact rows:
where $\gamma_n$ are isomorphisms. Prove that sequence:
$$ \dots \rightarrow A_n \overset{(\alpha_n,-...
2
votes
0answers
30 views
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 ...
2
votes
2answers
46 views
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 ...
0
votes
1answer
27 views
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 ...
1
vote
1answer
49 views
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}\...
1
vote
1answer
40 views
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)...
