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. 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.
1,020
questions
2
votes
0answers
5 views
Do the central extension of a Loop algebra split
Take $\mathfrak{g}$ to be a simple finite-dimensional Lie algebra og consider the corresponding Loop algebra $L=\mathbb{C}[t,t^{-1}]\otimes \mathfrak{g}$. When constructing the affine Lie algebra, one ...
4
votes
0answers
28 views
Connecting fiber sequences for classifying spaces
My original question was:
Is there a long exact sequence for classifying spaces of topological
groups?
It is solved by a comment from @JHF , since $BA$ is not usually a group. The sequence cannot go ...
7
votes
0answers
59 views
How unique is the connecting morphism in the snake lemma?
This question was inspired by this question. The question whether the connecting morphism in the snake lemma is unique needs to be formulated precisely. Of course, in any specific exact sequence ...
4
votes
0answers
33 views
Manifold and action associated to a central extension of groups
Let $X$ be a manifold equipped with a proper $G$-action. Suppose we have a central extension
$$0\to H\hookrightarrow G'\to G\to 0,$$
where $G\cong G'/H$.
Question 1: Can one construct a manifold $X'$ ...
5
votes
0answers
61 views
The connecting morphism of the mapping cone of $\varphi^\bullet$ is $H^i(\varphi^\bullet)$
Consider the following short exact sequence of complexes in an abelian category
Let $MC(\varphi^\bullet)^\bullet$ be the mapping cone of $\varphi^\bullet$. Recall that the mapping cone fits into an ...
6
votes
1answer
75 views
Uniqueness of the connecting morphism in the snake lemma
Consider the following commutative diagram in an abelian category:$$\require{AMScd}
\begin{CD}
@. A @>{f}>> B @>{g}>> C @>>> 0\\
@. @VV{a}V @VV{b}V @VV{c}V \\
0@>>>...
0
votes
1answer
38 views
Why can we assume surjectivity in short exact sequences?
This is brand new to me, so forgive me if this is simple.
I understand the structure of a short exact sequence (the image of the current homomorphism is the kernel of the next), and I can understand ...
4
votes
1answer
71 views
About the functoriality of the long exact sequence in cohomology
I'm writing some notes on homological algebra and there I proved the long exact sequence in cohomology using the snake lemma. (I can give more details if anyone wants.) The next natural step is to ...
1
vote
1answer
109 views
If $G_1, G_2, G_3$ are abelian groups and $0 \to G_1 \to G_2 \to G_3 \to 0$ is exact, then $G_2 \simeq G_1 \oplus G_3$
If $G_1, G_2, G_3$ are abelian groups and $0 \to G_1 \xrightarrow{\varphi_1} G_2 \xrightarrow{\varphi_2} G_3 \to 0$ is exact, then $G_2 \simeq \ker(\varphi_2) \oplus \text{im}(\varphi_2) \simeq G_1 \...
0
votes
1answer
26 views
Linear algebra vs homological algebra
In linear algebra my book show the five lemma.
In wikipedia I noticed connection with the 9-lemma (=? 3x3 lemma), the snake lemma a.s.o.
In the wikipedia page this lemma are described as being part ...
4
votes
1answer
56 views
Proof of the exact sequence $0\to H^i(M)\to \operatorname{coker} d^{i-1}\to\ker d^{i+1}\to H^{i+1}(M)\to 0$.
Let $M^\bullet$ be a complex in an abelian category. By the universal property of cokernels, there is a unique induced morphism making the diagram
commute. I want to prove that its kernel is $H^i(M^\...
0
votes
1answer
34 views
Exact short sequence [closed]
Given a subspace $V_1$ of $V$, then
$$
\{0\} \xrightarrow{} V_1 \xrightarrow{i} V \xrightarrow{\pi_{V/ V_1}} V/V_1
\xrightarrow{} \{0\}
$$
is a exact short sequence.
Q: Is this something that I ...
2
votes
0answers
45 views
Direct proof that a short exact sequence induces a quasi-isomorphism from the mapping cone
Consider the following exact sequence of complexes in an abelian category:
I want to prove that there exists a quasi-isomorphism $\rho^\bullet:\operatorname{MC}(\varphi^\bullet)^\bullet\to N^\bullet$,...
2
votes
0answers
52 views
Künneth formula for cohomology
I see that most authors prove only the Künneth formula for Homology, and get the formula for cohomology as a consequence. My question here is how to get from one to the other. I assume you need to use ...
0
votes
0answers
22 views
Splitting of short exact sequences.
I need to show that every short exact sequence of finitely generated modules over direct product of two fields split.
My thoughts on this:
I know that it is true over a field because any basis for a ...
1
vote
1answer
44 views
Long exact sequence for mapping torus from Mayer-Vietoris
Note: I have actually done the thing I am trying to do. My question is whether there is an easier way to do it.
Question:
Let $f:X\to X$ be a map of topological spaces, and define the mapping torus of ...
2
votes
0answers
51 views
How do I understand the ratios between $\pi_1(U(N))=\mathbb{Z}$, $\pi_1(U(1))=\mathbb{Z}$, and $\pi_1(PSU(N))=\mathbb{Z}/N$?
We know that $$\pi_1(SU(N))=0, \tag{1}$$
$$\pi_1(PSU(N))=\pi_1(SU(N)/(\mathbb{Z}/N))=\pi_0(\mathbb{Z}/N)=\mathbb{Z}/N.\tag{2}$$
Also that
$$\pi_1(U(N))=\pi_1(\frac{SU(N)\times U(1)}{\mathbb{Z}/N})=\...
2
votes
0answers
51 views
Is an exact-by-nuclear extension of $C^*$-algebras again exact?
I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ ...
1
vote
1answer
29 views
What are the implications of a surjection between free modules?
My professor said the following in his lecture:
Suppose $M$ and $P$ are free modules and $\alpha: M\rightarrow P$ is a surjection. Then $\alpha$ splits and the kernel of $\alpha$ is a summand of $M$.
...
0
votes
0answers
21 views
Splitting in the category of representations of a reductive group
Let $G$ be a reductive group over a local field $K$ of characteristic zero. Denote by $\mathrm{Rep}_K(G)$ the category of representations of $G$. I do not specify here algebraic, smooth or locally ...
4
votes
2answers
63 views
Does the short exact sequence $0\mapsto \mathbb{Z}^k\mapsto \Gamma\mapsto \mathbb{Z}^\ell\mapsto 0$ split?
I've seen in Nonsplit extension of $\mathbb{Z}$ by itself that every short exact sequence of the form $0\mapsto \mathbb{Z}\mapsto G\mapsto \mathbb{Z}\mapsto 0$ splits.
I wonder if this is still valid ...
2
votes
0answers
53 views
Short Exact Sequence of a Hyperplane in the Projective Space
Let $H$ be a hyperplane in $\mathbb P^n$ where $f:H\rightarrow \mathbb P^n$ is the closed immersion. Let $\mathcal F$ be a coherent subsheaf of $\mathcal E = \oplus\mathcal O_{\mathbb P^n}(l)$.
There ...
-1
votes
1answer
75 views
$\lim_{n\to\infty}\sqrt[3]{3n}\sigma_n=1$ [closed]
We put :
$S_n=\displaystyle\sum_{k=1}^{k=n}\sigma^2_k$
Where $(\sigma_n)_{n\geq 1}$ is a real sequence, its boundaries are positive.
_Assume that
$(\lim_{n\to\infty}\sigma_n S_n=1)$
Prove that :$\...
1
vote
1answer
72 views
Tensor with an Exact Sequence of Free Module Still Exact?
Let $F\rightarrow G\rightarrow H$ be an exact sequence of free $R$-modules and $M$ be any $R$-module.
Is that true that $F\otimes_RM\rightarrow G\otimes_RM\rightarrow H\otimes_RM$ is exact?
If the ...
0
votes
0answers
26 views
Decision problem for the splitting type of rank 2 bundle from the given exact sequence
Let $\mathcal{F}$ be a rank 2 vector bundle over some algebraic variety.
After restrict to the given line $L\cong\mathbb{P}^1$, it fits into an exact sequecne
$$
0 \rightarrow \mathcal{O}_L(-2)\...
5
votes
0answers
55 views
Subgroups of semidirect product of two abelian groups
Let $G= N \rtimes_{\varphi} Q$ be a semidirect product coming from a short exact sequence $$ 1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1 $$and assume that $N$ and $Q$ are finitely ...
0
votes
0answers
39 views
Extension of scalars and exact sequences
Let $\rho:K\rightarrow A$ be a homomorphism of a division ring into a ring. For a left $K$-vector space $E$, let $\rho^*(E):=\rho_*(A)\otimes_K E$. Similarly, for each $K$-linear mapping $u:E\...
0
votes
0answers
17 views
How to calculate when a specific value of a logistic curve is reached?
I have a growth curve defined by a logistic term as:
$$N_t = rN_0(1-\frac{N_0}{\kappa})$$
I would like to find when a specific value of N is reached. Let's say I have these parameters:
$$\kappa = 10^{...
1
vote
1answer
72 views
Exact sequence stays exact after tensoring if right Module is free (or projective)
Let $f: R \to A$ be a ring morphism ($R, A$ a commutative rings) and
$$ 0 \to M_1 \to M_2 \to M_3 \to 0 $$
a ex. sequence of $R$ modules. Assume, that $M_3 = R^n$ is free (or
say projective). I want ...
0
votes
1answer
44 views
Do I have to show that elements of $L$ commutes with elements of $N$ (like in case of direct product) and if so, why?
I want to prove the following $a \Longleftrightarrow d$ in the following questoin:
Let $R$ be a commutative ring. For $R-$modules $L,M,N$ show that the following conditions are equivalent.(all ...
0
votes
1answer
53 views
Proving that the existence of a section implies the direct sum.
Here is the question I want to prove so far:
Let $R$ be a commutative ring. For $R-$modules $L,M,N$ show that the following conditions are equivalent.(all functions are $R-$ module homomorphisms.)
a- $...
1
vote
1answer
25 views
Short exact sequence with center right splits
Suppose that $G= \mathbb{Z}^n \times H$ for some group $H$. Then, there is a short exact sequence $$ 1\rightarrow \mathbb{Z}^n \rightarrow G \rightarrow H \rightarrow 1$$ which right splits (clearly).
...
2
votes
0answers
67 views
Regarding Rotman's book on Homological Algebra
My instructor said to look at "Rotman's book" for the proof that direct limit is an exact functor. According to her, this book by Rotman shows details regarding the equality of the kernel ...
1
vote
1answer
51 views
Showing the existence of a function $M \rightarrow L.$
Let $R$ be a commutative ring. For $R-$modules $L,M,N$ show that the following conditions are equivalent.(all functions are $R-$ module homomorphisms.)
a- $M \cong_{R} L \oplus N.$
b- There exists a ...
3
votes
0answers
39 views
Understanding the isometries of $\widetilde{SL(2,\mathbb{R})}$
Let us begin by describing the metric on $\widetilde{SL(2,\mathbb{R})}$.
Let on the tangent space $T\mathbb{H}^2$ of the hyperbolic plane there is a natural metric, called the Sasaki metric, built on ...
0
votes
1answer
88 views
Computing homology groups of some quotient space (Hatcher 2.2.13)
Let $X$ be the $2$-complex obtained from $S^1$ with its usual cell structure by attaching
two $2$-cells by maps of degrees $2$ and $3$, respectively.
Hatcher ask us to compute the homology group of $...
2
votes
0answers
45 views
Exact sequence of vector spaces and sum of dimension
Let
$0\xrightarrow{}E_0\xrightarrow{u_0}E_1\xrightarrow{u_1}E_2\xrightarrow{}\ldots\xrightarrow{}E_{n-1}\xrightarrow{u_{n-1}}E_n\xrightarrow{u_n}0$
be an exact sequence. Then $$\sum_{2k+1\leq
n}\text{...
0
votes
0answers
44 views
Showing that a Group Extension is Split
I have a group extension $1\rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ that I think is a split extension (so $G \approx N \rtimes Q$), but I'm having trouble showing this. Is there a ...
1
vote
1answer
61 views
If $G$ is finitely generated and $G/N$ is finitely presented , then $N$ is normal closure of finite set.
Suppose $G$ is a finitely generated group and $N$ is a normal subgroup of $G$ such that $G/N$ is a finitely presented .Then show that $N$ is a normal closure of some finite set of $G$.
[My attempt]
...
2
votes
1answer
62 views
Reason for this sequence being exact?
In Altman-Kleinman "Introduction to Grothendieck Duality Theory", page 105, there is the following theorem:
Why the underlined sequence is exact? And then, why from that we can conclude ...
1
vote
1answer
60 views
For a commutative exact sequence show that $V_1$ and $V_3$ are finite dimensional iff $V_2$ and $V_4$ also are.
For the following commutative exact sequence:
\begin{array}\\
&V_1 & \stackrel{A_1}{\longrightarrow} & V_2\\
& \uparrow_{A_4} &&\downarrow _{A_2}\\
&V_4 & \stackrel{...
1
vote
1answer
38 views
Question about this proof (exactness of a sequence)
In Altman-Kleinman "Introduction to Grothendieck Duality Theory", page 104 there is the following Lema:
I don't understand the underlined part, why the exactness of that sequences implies ...
1
vote
1answer
124 views
Finding the homology of space comprised of $S^2$ and an intersecting torus.
I am reviewing old algebraic topology qualifying exams, and I need to compute the fundamental group and the homology of the following space.
First, a brief calculation of $\pi_1(X)$. Let $C$ be the ...
2
votes
1answer
67 views
natural induced split exact sequence from $ 0\to \Bbb Z\xrightarrow{p}\Bbb Z\to \Bbb Z_p\to 0$
Multiplication by the prime $p:\Bbb Z\to \Bbb Z$ fits in a short exact sequence
$$ 0\to \Bbb Z\xrightarrow{p}\Bbb Z\to \Bbb Z_p\to 0$$
Use this to derive the natural split exact sequence
$$0\to H_n(X)/...
0
votes
0answers
57 views
Does the exact sequence $0\to\ker(N)\to\ker(MN)\to\ker(M)\cap N(V)\to0$ (and another) have a name?
I have seen the following sequences about the kernel and cokernel a few times but I have never seen a name of them. Do they have a name or are they just nameless?
Let $V,W$ finite Vectorspaces $N:V\...
3
votes
2answers
103 views
Restricting split short exact sequence of quasi-coherent sheaves to their coherent sub-sheaves.
Let $X$ be a projective variety, $Z$ a hypersurface section and $U \overset{def}= X \setminus Z$ its complement, an open affine subscheme of $X$. Let $i:U \hookrightarrow X$ be the corresponding open ...
1
vote
1answer
136 views
If $0 \to A \to B \to C \to 0$ is split, we say that $B \cong A \oplus C$. Is there a reason we take the direct sum rather than the product?
I'm studying some commutative algebra and a solution to an exercise(concerning commutative $R$-modules) relies on the isomorphisms $$\text{Hom}(C, A \oplus C) \cong \text{Hom}(C, A \times C) \cong \...
2
votes
2answers
91 views
Why does the short exact sequence for projective module split?
I want to understand the definition of surjective module in terms of splitting sequence. The definition says for a projective $R$-module $P$, the following short exact sequence $$0 \to A \xrightarrow{...
3
votes
1answer
65 views
Dot notation for group extensions
I came across the notation $A.B$ in many occassions while reading papers in Group Theory. However, I have not yet found any description of what this notation is supposed to mean. From context I ...
2
votes
1answer
72 views
Group Extensions and Couplings
In this post, I will be referencing this paper. Here is proposition 1.2 from said paper:
Proposition 1.2 --- Let $G \neq \{1\}$ be a group that decomposes as an extension:
$$1 \longrightarrow K \...