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.

Filter by
Sorted by
Tagged with
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 \...

1
2 3 4 5
21