A sequence of morphisms where the image of one is the kernel of the next. It is useful in abstract algebra (ring/module/group theory) and homology theory in particular. Do not use this tag for sequences of numbers; use (sequences-and-series) instead.

learn more… | top users | synonyms

3
votes
1answer
39 views

The long exact sequence in reduced K-theory. How to glue together the short ones?

I'm trying to filling the details about the construction of the long exact sequence in K-theory. I'm using the notation of Hatcher's book (pages 52-53). here is a related question, even thought it's ...
2
votes
2answers
69 views

Homology groups equal when $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) \rightarrow 0 \rightarrow \cdots$

I'm reading a set of notes but I don't understand the following concept. We have a long exact sequence $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) ...
7
votes
4answers
355 views

What is a short exact sequence telling me?

Let's take a short exact sequence of groups $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$ I understand what it says: the image of each homomorphism is the kernel of the next one, so the ...
1
vote
2answers
41 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
3
votes
0answers
29 views

Short exact sequence involving mapping cone, cone, suspension of $C^*$-algebras

This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel ...
2
votes
0answers
20 views

Exact sequence of tangent spaces of principal $G$-bundles

Let $P$ be a smooth manifold, $G$ a Lie group, $\alpha:P\times G\to P$ a smooth action and $p:P\to P/G$ a smooth principal $G$-bundle. Then, we have the sequence $$ G \xrightarrow{\alpha_a} P ...
1
vote
1answer
55 views

exact sequence problem

Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} ...
1
vote
1answer
40 views

hom and exact sequence

Let $$ 0 \longrightarrow \operatorname{Hom}(M,Β_1) \stackrel{f^*}\longrightarrow \operatorname{Hom}(M,Β) \stackrel{g^*}\longrightarrow \operatorname{Hom}(M,Β_2) $$ be an exact sequence for any ...
3
votes
1answer
81 views

Proof that the Euler characteristic is additive

I'm reading through a set of notes which assumes that the Euler characteristic is additive, but doesn't give a proof, so I would like to understand why this is. Let $A_n$ be a finitely generated ...
1
vote
0answers
37 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
1
vote
0answers
38 views

Exact sequences of $1 \to A \to SU(N) \to B \to 1$, special unitary group

Inspired by the nice post, I am particularly looking into the exact sequences of SU(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A \to SU(N) \to B \to 1$$ where ...
4
votes
0answers
44 views

Exact sequences of SU(N) and SO(N)

We know that the spin group $Spin(N)$ has a short exact sequence of Lie groups. $$1 \to Z_2 \to Spin(N) \to SO(N) \to 1$$ I wonder whether there are some examples for SU(N) and SO(N) group, such ...
2
votes
0answers
62 views

Splitting of the tangent bundle of a vector bundle and connections

Let $\pi:E\to M$ be a smooth vector bundle. Then we have the following exact sequence of vector bundles over $E$: $$ 0\to VE\xrightarrow{} TE\xrightarrow{\mathrm{d}\pi}\pi^*TM\to 0 $$ Here $VE$ is ...
2
votes
1answer
66 views

About Exact Sequences

Suppose I have the following exact sequence: $$\begin{matrix} A_0& \xrightarrow{\quad\quad} & B_0 \xrightarrow{\quad\alpha\quad} & C_0\\ \uparrow&&&\downarrow \\ C_1 ...
1
vote
0answers
35 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
2
votes
1answer
127 views

Splitting short exact sequence of space groups

I want to prove the following: Assume we have two space groups $G,G^\prime \subseteq \text{Euc}(V) \subseteq \text{Aff}(V)$ which are affinely equivalent, $G \sim G^\prime, \; \text{ i.e. }\; ...
3
votes
2answers
50 views

$mA = 0 = nC, \ \gcd(m,n) = 1 \Rightarrow $ every extension of $A$ by $C$ splits

This is Exercise 7.14(ii) from Rotman, Introduction to homological algebra, and I'm stuck on it. If $A$ and $C$ are abelian groups, with $mA = 0 = nC $ and $\gcd(m,n) = 1$ then every extension of ...
0
votes
1answer
34 views

Applying an additive covariant functor to a sequence

Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
2
votes
1answer
74 views

Generalized Snake lemma

I always read the snake lemma with short exact sequences: \begin{eqnarray*} &&\qquad M_1\to M_2\to M_3\to0\\ &&\qquad\ \downarrow\qquad\downarrow\qquad\ \downarrow\\ &&0\to ...
1
vote
2answers
74 views

Help proving a short exact sequence

Show the following sequence is an exact sequence of $\mathbb Z$-modules when $n$ is a positive integer such that $n=rs$: $$ 0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0. $$ ...
0
votes
0answers
38 views

Cokernels of Modules and Exact Sequences

Suppose we have n-dimensional free modules over a dvr $T_1$, $T_2$, $T_3$ (i.e. $T_i \cong R^n$) such that $$0 \to T_1 \to T_2 \to T_3 \to 0$$ is exact. Suppose further that $E_1,E_2,E_3$ are ...
0
votes
0answers
30 views

A proposition on exact sequence of inverse limit (Lang, Algebra, p. 165)

I am trying to understand this proof. My only question is that what are the vertical maps here?
0
votes
1answer
39 views

Exact sequence induced by decomposition of a variety

Let $X=Y_1\cup Y_2$. Denote $I:=\mathcal{I}(X)$, $I_1:=\mathcal{I}(Y_1)$, $I_2:=\mathcal{I}(Y_2)$. Then we have $I=I_1\cap I_2$. In this question, the answer suggests that there is an exact ...
3
votes
1answer
55 views

Why do we have that Hom is an exact functor in the situation described below?

We are given a finite $p$-group $G$ and a finite $G$-module $M$ such that $pM=0$ (therefore $M$ is in particular a $\mathbb{F}_p$-vector space). In addition we have an arbitrary $G$-module $N$ which ...
0
votes
1answer
25 views

Equivalence between exact sequence of module and its induced one.

Let $X,X',X''$ be $A$-modules and denote by $\mbox{Hom}_A(X',X)$ the set of $A$-homomorphisms of $X'$ into $X$. Proposition 2.1 in the Lang's Algebra text states the following: A sequence $$ X' ...
1
vote
1answer
42 views

Does exact sequence of abelian groups split when middle group has a subgroup direct sum of other groups?

If I have an exact sequence of abelian groups, the sequence coming from knowing that $H\cong G/F$, $$0\rightarrow F \rightarrow G \rightarrow H \rightarrow 0$$ where I know that $ F\oplus H\subset ...
0
votes
1answer
35 views

Exactness of sequence of vector spaces with tensor product

Let $k$ be a field and $0\rightarrow V _n \xrightarrow{f_n} V_ {n − 1} \xrightarrow{f_{n−1}} ··· \xrightarrow{f_3} V_ 2 \xrightarrow{f_2} V_ 1 \xrightarrow{f_2} V_ 0 \rightarrow 0$ an exact sequence ...
2
votes
1answer
59 views

If $V/W\cong C$ is it true that $V=W\oplus C$ ?

We know that for a vector space $V$ and its subspace $W$ if $V=W\oplus C$ then the quotient space $V/W$ is isomorphic to a subspace of $V$ (namely, $C$). Is the inverse true?
0
votes
1answer
53 views

Short Exact Sequences

Let $M \ge N \ge P$ be R-modules. Prove that there exist natural (not depending on choices) R-homomorphisms $N/P \to M/P$ and $M/P \to M/N$ for which the sequence $0 \to N/P \to M/P \to M/N \to 0$ is ...
1
vote
0answers
31 views

Thom-Gysin long exact sequence

I have read about the following exact sequence of cohomology: Let $V$ be an algebraic variety over $\mathbb{C}$. If $U\subset V$ is an open subvariety, then there is a long exact sequence for ...
2
votes
0answers
57 views

Long Exact Sequence Cohomology with Compact Support

I found in this topic or (question) a reason for my question, but i do not understand it. As this question is quite old, I hope someone else can help me. Assume $U$ is an open subset of a topological ...
3
votes
0answers
69 views

exact sequence and modules proposition.

I have problems to prove the following proposition: Let's consider $$0 \rightarrow L \stackrel{\alpha}{\rightarrow} M \stackrel{\beta}{\rightarrow} N \rightarrow 0$$ an exact sequence of modules and ...
2
votes
1answer
21 views

Behaviour of $\operatorname{Ext}$ with left exact sequences.

Maybe is a trivial question but I am not so good in derived functors. Assume we are in the category of abelian groups and we have an exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow ...
1
vote
1answer
39 views

Isomorphism preserves exactness

Let $R$ be a commutative ring with unity. Let $A_i$ be an R-module for every $i$. Consider a sequence of modules $$\xrightarrow{\delta_{i-1}}A_{i-1}\xrightarrow{\delta_{i}} ...
2
votes
1answer
52 views

A Sort of Exact Sequence

I have not given a lot of thought to this question: It may be very easy or very hard or somewhere in between. Suppose we have a sequence of modules and morphisms which looks like $ \ldots \to A_1 ...
4
votes
1answer
49 views

uniqueness of groups in an exact sequence

I was wondering how unique are the groups making up to an exact sequence. Suppose we have three groups $A, B, C$ such that the sequence $$ A \rightarrow B \rightarrow C $$ is exact. I wanted to know ...
1
vote
1answer
45 views

Exact Sequences of R-Modules

In "A Course in Ring Theory by Passman" it is mentioned, "But the kernel of the combined epimorphism $P\rightarrow B\rightarrow C$ is clearly equal to $E$". I don't understand this part. How can the ...
1
vote
0answers
43 views

Existence of a long exact sequence for sheaf cohomology

Let $X$ be a normal variety over $\mathbb{C}$ , and let $U$ be a open subset of $X$, then there is an long exact sequence for singular or De Rham cohomology with compact support that relates the ...
1
vote
1answer
86 views

Problems about exact sequence in Vick's homology theory.

I am self studying algebraic topology through vick's homology theorey, but I don't know how to prove 2.3 proposition on page 38~39. (which the author stated only and said it is easy) Here's the ...
0
votes
1answer
33 views

Why these two propositions have different requirements

Proposition 2.18 is similar to 2.19. Why we need $N$ flat in 2.19? What's the difference between 2.18 and 2.19?
2
votes
1answer
57 views

Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
3
votes
1answer
88 views

Showing $M\cong M'\oplus M''$ given an exact sequence

I am struggling with the following question: $R$ is a ring. $$M'\overset{f}{\longrightarrow} M\overset{g}{\longrightarrow} M''$$ are homomorphisms of $R$-modules such that for any $R$-module $N$, the ...
1
vote
1answer
41 views

If an $A$-module $M$ is locally finitely presented (resp. related) then $M$ is finitely presented (resp. related)

In this question I want to ask for a better proof than the one I am about to give for the statement with finitely presented, and inquiry if the statement is also true for the notion of finitely ...
3
votes
1answer
71 views

Determinant of long exact sequence

Let the following be a long exact sequence of free $A$-modules of finite rank: $$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$ I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where ...
0
votes
2answers
57 views

short exact sequence

Let $0 \rightarrow L \stackrel{\alpha}\rightarrow M\stackrel{\beta}\rightarrow N \rightarrow 0$ be an exact sequence, and $M_1$, $M_2$ be two submodules of $M$; then whether the follwing implications ...
0
votes
1answer
61 views

Homology isomorphism of $H_n(S^d\times X)$ and $H_{n-1}(S^{d-1}\times X)$

$X$ is an arbitrary space, $d\geq 1$. The existence of such isomorphism in the title supposedly follows from the Mayer-Vietoris sequence of $(S^d\times X,S^d_{+}\times X,S^d_{-}\times X)$: ...
1
vote
1answer
26 views

Given ring $A$, ideal $I$, and $A$-module $M$, show that $A/I \otimes_A M$ is isomorphic to $M/IM$.

The question is stated as in the title; the hint I am given is to "tensor the exact sequence" $0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0$, which I take to mean using that sequence ...
3
votes
1answer
38 views

Exact sequence induces exact sequences for free parts and torsion parts?

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...
7
votes
1answer
72 views

Direct proof of non-flatness

Consider $k$ a field and the rings $A=k[X^2,X^3]\subset B=k[X]$. How to prove that $B$ is not flat over $A$ by using only the definition of flatness that it maintains exact sequences after making ...
1
vote
1answer
350 views

Formula for binary sequences of length m with no n consecutive 1s?

Formula for binary sequences of length $m$ with no $n$ consecutive $1$s? I know The number of binary strings of length $m$ without consecutive $1$s is the Fibonacci number $F_{m+2}$. But how about ...