Questions tagged [exact-sequence]
A sequence of morphisms where the image of one is the kernel of the next. It is a 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.
1,465 questions
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Exact sequence of finitely generated free modules, $\sum_{i=0}^n (-1)^i rk(F_i)=0$
This question is actually the exercise 10.33 in Rotman Advanced abstract algebra: Let $0\to F_N\to F_{N-1}\to \cdots \to F_0\to 0$ be an exact sequence of finitely generated free k-modules, where k ...
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Relation between the elementary Rouché-Fröbenius theorem and the more abstract Fröbenius theorem with exact sequences.
In highschool I was taught that the "Rouché-Fröbenius theorem" was that given a homogenous linear system of equations, the solution space has dimension $n-r$ where $n$ is the dimension of ...
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Short sequences exact on left and right, but not in the middle
Hopefully a quick one I'd like to check with you.
Assume $R$ is a commutative unital ring and let $A,B,B^\prime,C$ be $R$-modules of finite rank. Suppose we have two sequences $$0\rightarrow A \...
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The purpose of requiring splitting homomorphism for a section to get a split sequence
I see this from Dummit Foute and want to check my understanding: Now pick any $c \in C$,we can get a corresponding representaive $\bar{\beta} \in B/ker \phi$ and we specify it as one possible $\beta \...
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How can we paste together some finite long exact sequences (LES's) at the top page 14 of Weibel (in the SES of complexes to infinite LES proof)?
The Snake Lemma starting diagram at the bottom of page 13 in this proof is:
Here is a link to the above diagram.
Now at the top of page 14, it states "The kernel of the left vertical is $H_n(A)$ ...
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Tensor product $\mathbb{Z}^n\otimes \mathbb{Z}^m$
For a problem in my PhD I'm using Leray–Hirsch Theorem (as presented in the Hatcher) to compute a cohomology ring. For a particular group $G$, I have the s.e.s.
$$
\def\Inn{\operatorname{Inn}}
\def\...
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Long exact sequence of homotopy groups: connecting map and action of $\pi_1(B)$
Let $F \rightarrow E \rightarrow B$ be a fiber bundle, and let $\delta_n:\pi_n(B) \rightarrow \pi_{n-1}(F)$ be the connecting homomorphism in the associated long exact sequence of homotopy groups.
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Projective module on finite dimensional algebra
My question: let A be a finite-dimensional algebra, $X_{A}$ be a right A-module, if $X \otimes_{A} -$ is exact, then $X_{A}$ is projective.
I think we need to use the tensor-hom adjunction, and at the ...
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Group isomorphic to semidirect product quotient s.t. the exact sequence does not split
The problem that I have is to find an example for a group $G$ with normal subgroup $N$ s.t. $G\simeq N \rtimes G/N$ but the direct sequence $$1\to N\overset{\iota}{\to} G \overset{\pi}{\to}G/ N\to 1$$ ...
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Tensor product not left exact, but does it preserve inclusions of submodules?
The question is in the title: i know that tensor products generally don't preserve injectivity (of module maps, in the case of my question), but what about inclusions of submodules? This is a special ...
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Understanding exact sequences
Im reading "Introduction to Conmutative algebra" by Atiyah and Macdonald. They define a a sequence of $A$-modules and $A$-homomorfism, where $A$ is a abelian ring with identity and $0$ is ...
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Exact sequence of subspace
Let $V$ be a finite dimensional vector space over $k$ and $V_1, V_2 \subseteq V$ be two subspace of $V$. Suppose that $W \subseteq V$ is a subspace such that $V_1, V_2 \subseteq W$.
We have two ...
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Is there an automorphism of $\mathbb{S}^2\times\mathbb{S}^2$ with $(x,0)\mapsto(x,x)$?
The question:
Does there exist a diffeomorphism $\alpha:\mathbb{S}^2\times\mathbb{S}^2\rightarrow\mathbb{S}^2\times\mathbb{S}^2$ such that for some $x_0\in\mathbb{S}^2$ and every $x\in\mathbb{S}^2$ we ...
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When does a short exact sequence with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on overflow.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and $f:A\...
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Let $ 0 \to M \to M' \to M'' \to 0 $ be a short exact sequence of $R$-modules. If $M$ and $M''$ are torsion free, then $M'$ is torsion free.
Assume $R$ be a domain and $M,M'$ and $M''$ are $R$-module.
Let $ 0 \to M \to^{\varphi} M' \to^{\phi} M'' \to 0 $
be a short exact sequence of $R$-modules. If $M$ and $M''$ are torsion free, then $M'$ ...
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Confusion about Left split implying a direct product of groups [duplicate]
I was doing a homework problem that has a left split exact sequence $$ 1 \to H \xrightarrow{i} G \xrightarrow{\pi} G/H \to 1,$$ ($H$ is a subgroup of $G$, $i$ is the inclusion map). Assume we have a ...
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What are the group extensions of $\mathbb{Z}_2$ by $U(1)$?
Given the group extension $1\rightarrow U(1)\overset{f}{\rightarrow}G\overset{g}{\rightarrow}\mathbb{Z}_2\rightarrow 1$, what are the possibilities for $G$?
I understand how to solve this for $1\...
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Equivalent conditions of injective module
My question: Let $R$ be a commutative ring with unity. Then an $R$-module $E$ is injective module if and only if, for any ideal $I$ of $R$, every exact sequence $0 \rightarrow E \rightarrow B \...
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Image of a splitting of short exact sequence of vector bundles is a subbundle
I am currently learning about principal $G$-bundles from Tu's Differential Geometry: Connections, Curvature and Characteristic Classes.
Proposition 27.20 of my edition states the following:
"Let $...
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How to prove the quotient of long exact sequences is a long exact sequence?
Let $f^{\cdot}: A^{\cdot}\to B^{\cdot}$ be an injective morphism of two long exact sequences of an abelian category. Then by taking cokernel, naturely we will have a new complex $K^{^{\cdot}}$. I want ...
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Example 5.2.6 in Weibel's Homological Algebra
Let $\{E^r_{pq}\}$ be a spectral sequence, such that $E^r_{pq}$ is zero unless $p,q\ge 0$. (I.e. it is bounded in the first quadrant).
Let's consider a term of the form $E^n_{0y}$; since the ...
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About the definition of integer-valued additive function on the category of modules
Let $A$ be a commutative ring with unity. In Atiyah-MacDonald (chapter 2, right before proposition 2.11), an integer-valued additive function on a class $\mathcal{C}$ of $A$-modules is defined as a ...
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Question regarding when ideals of $\Bbb Z/n\Bbb Z$ are projective modules
Suppose $r, n$ are integers greater than 1, such that $r|n$. If we consider the Ideal $r(\mathbb{Z}/n\mathbb{Z})$ of the ring $\mathbb{Z}/n\mathbb{Z}$, then I need to determine the following ...
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Sequence of $R$-modules is exact if and only if its induced $\operatorname{Hom}$-sequence is exact
Apologies if this is a copy; I didn't find any relevant questions on here.
I am studying for my exam and came across a Proposition which states the following:
"Let $R$ be a (commutative) ring. ...
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Direct product of pure monomorphisms is pure?
I know that any direct sum of pure exact sequences is pure. More generally, the direct limit of any direct system of pure short exact sequences is pure exact. Now, let we have pure submodules and ...
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Quotient map from $\mathbb{S}^1 \times \mathbb{S}^1$ to $\mathbb{S}^2$ is not nullhomotopic
I'm trying to prove that the map $\mathbb{S}^1 \times \mathbb{S}^1 \rightarrow \mathbb{S}^2$ collapsing $\mathbb{S}^1 \vee \mathbb{S}^1$ induces an isomorphism between $H_2(\mathbb{S}^1 \times \mathbb{...
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$S$ splits iff $S\otimes N$ Exact
I have two questions on exactness of hom and tensor.
(1)Let $S$ be a sequence $0\to A\xrightarrow{f} B\xrightarrow{g} C\to 0$ of $R$-modules, $S$ splits iff $\text{Hom}(N,S)$ exact for every $N$, ...
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Additivity of Euler characteristic on long exact sequence [closed]
In the book of Vakil, the Foundation Of Algebraic Geometry, there is an exercise (19.4.A) that I can't do and I'd like to have some help.
The exercise is the following :
We take a projective $k$-...
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Is a homotopy long exact sequence for a pair always isomorphic to a long exact sequence coming from a fibration
My question was already raised a couple years ago (see here). But it wasn't really answered so I am trying my luck again.
Given a fibration $F\to E\to B$ we know that it's long exact sequence is ...
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Let $G$ be a finitely generated abelian group. Does there exist an exact sequence $0\to \Bbb{Z}/2\Bbb{Z}\to G\to G\to 0?$
Let $G$ be a finitely generated abelian group.
My question is, does there exist an exact sequence $0\to \Bbb{Z}/2\Bbb{Z}\to G\to G\to 0$ ?
If $G$ is finite group, from isomorphism theorem, there is ...
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Short exact sequence in the ideal class group
In the answer to the Motivation behind the definition of ideal class group, I have seen one by user Alex Youcis and he/she claimed that the following is a short exat sequence: $$1\to \mathcal{O}_k\to \...
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Homology + Chinese Remainder Theorem =?
Let $M_i, i=1..n$ be a finite collection of pair-wise coprime moduli. The Chinese remainder theorem says that $\Bbb{Z}/M \approx \prod_i \Bbb{Z}/M_i$. Without going into Bezout / Euclidean algorithm,...
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Exactness of the sequence $0\rightarrow\mathbb{Z}_p\xrightarrow{p^n}\mathbb{Z}_p\xrightarrow{\varepsilon_n} A_n\rightarrow 0$
On page 12 of the book A Course in Arithmetic by Serre, the author proves that the sequence $0\rightarrow\mathbb{Z}_p\xrightarrow{p^n}\mathbb{Z}_p\xrightarrow{\varepsilon_n} A_n\rightarrow 0$ is exact,...
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Establish a short exact sequence $0\to Z_p\to Z_{p^2}\to Z_p\to 0$ does not split, every submodule of the projective module need not be projective. [closed]
Establish a short exact sequence $0\to Z_p\to Z_{p^2}\to Z_p\to 0$ does not split, every submodule of the projective module need not be projective.
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Screwing up basic short exact sequence $0\to\mathbb Z\leftrightarrow\mathbb Z\to0$
Represent an isomorphism by $\leftrightarrow$.
HAVE exact sequence.
$$ 0 \rightarrow \mathbb Z \leftrightarrow \mathbb Z \rightarrow 0 $$
Then
$$ \text{img} \left( 0 \rightarrow \mathbb Z \right) = 0 =...
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Factorization of a map because composition is $0$
I apologize for the vagueness of the object I'm referring to. This question comes from considering sheaf of $\mathcal{O}_X$-modules homomorphisms, but I feel that the answer is something more general ...
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Suppose that $0\to A\xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. Prove that $0\to A\xrightarrow{f} B\to Im(g)\to 0$ is exact
Suppose that $0 \to A \xrightarrow{f} B \xrightarrow{g} C$ is exact in an abelian category. I am trying to prove that $0 \to A\xrightarrow{f} B \to Im(g)\to 0$ is exact. Here, I use the definitiom ...
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Can Nakayama Lemma apply to complex
Let $R$ be a Noetherian commutative local ring with maximal ideal $\mathfrak{m}$. Consider a complex of finitly generated projective $R$-mod (so it is free)
$$0\rightarrow P_1\rightarrow P_2\...
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Homotopy exact sequence of a covering map
I wanted to compute a homotopy group using the homotopy exact sequence of a covering but I am missing something obvious.
Consider the covering $\mathbb{C} - \{0, \pm 1\} \rightarrow \mathbb{C} - \{0, ...
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Understanding the definition of left exact functors
I am studying category theory, and in particular exact sequences. I am stuck on proving that the following three conditions are equivalent in an abelian category $\mathcal{C}$:
(a) The sequence $0 \to ...
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what should be the group $B$?
Here is the exact sequence of abelian groups I am studying:
$$0 \to \mathbb Z/2 \to B \to \mathbb Z/2 \to 0 $$
Can I say that $B \cong \mathbb Z/2$ or $B \cong \mathbb Z/2 \oplus \mathbb Z/2$? Is $B \...
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Example where the last arrow in the sequence is not surjective.
Consider the exact sequence $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$ and its induced sequence $0\rightarrow T(M')\rightarrow T(M)\rightarrow T(M'')$, where $T$ denotes the torsion ...
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$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ exact, then the tensorized sequence is also exact
I want to demonstrate that if $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence, then the induced sequence $0\rightarrow M'\otimes_A N\rightarrow M\otimes_A N\rightarrow ...
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Long exact sequence of intersected prime ideals
I'm considering a commutative local noetherian ring $R$ (in fact, $R$ is even Cohen-Macaulay) along with a number of prime ideals $I_i$. I can then construct the exact sequence
\begin{equation}
0\...
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Sequence involving plane curves is exact
I'm reading through Andreas Gathmann's Algebraic Curves and there is this affirmative on Proposition 2.10 at page 14:
Let $P\in\mathbb{A}^2$, and $F, G, H$ curves (or polynomials). If $F$ and $G$ have ...
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Extensions of $G$-modules parametrized by $H^1$
Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess we can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
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Elegant Proof of Snake Lemma
I'm considering the following diagram
\begin{array}{ccccccccc}
&&&&0&&0&&\\
&&&&\downarrow& &\downarrow& &\\
& && & \...
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proving that $A_n/\operatorname{ker} \beta \cong \operatorname{ker} \alpha_{n - 1}.$
Prove that in any exact sequence $$\dots \xrightarrow{\alpha_{n+3}}A_{n+2} \xrightarrow{\alpha_{n+2}} A_{n+1} \xrightarrow{\alpha_{n+1}} A_{n} \xrightarrow{\alpha_{n}} A_{n -1}\xrightarrow{\alpha_{n-...
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Is this torsion submodule sequence an exact sequence?
Let $0\rightarrow M'\xrightarrow{f} M\xrightarrow{g} M''\rightarrow 0$ be an exact sequence. Then, the sequence $0\rightarrow T(M'')\rightarrow T(M)\rightarrow T(M')$, where T is the torsion submodule ...
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Right split exact sequence for a Kan fibration with fiber a group complex
I'm stuck on the proof of Lemma 23.4 from May's Simplicial Objects in Algebraic Topology (p.99) on a seemingly harmless (and so left to reader's proof) step. Some context:
given a Kan complex $K$ with ...
