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.

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Determine if the following short exact sequence is split.

Do the following short exact sequences split? $$0\longrightarrow A\longrightarrow B\longrightarrow \mathbb{Z}^2 \longrightarrow 0$$ $$0\longrightarrow\mathbb{Z}\longrightarrow A\longrightarrow B\...
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finitely generated projective R-modules

Let $L$, $M$ and $N$ be $R$-modules. Then I know that there is a natural homomorphism from $Hom(L,M) \otimes N \to Hom(L,M \otimes N)$ defined by $f \otimes n \to \tilde{f}$ where $\tilde{f}(\ell)=f(\...
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An exact sequence of compact topological groups.

Let $A, B, C $ be abelian topological groups such that we have the following exact sequence : $$0\to A \to B \to C \to 0. $$ Assume also that A, C are compact and all the maps are open. Then it's it ...
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Properties of Short Exact Sequences

Some of the work I have been doing lately is heavily dependent on chasing commutative diagrams so I have been brushing up on short exact sequences since I was not familiar with them. For the most part ...
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31 views

Pushout as a functor on an exact sequence

I'm having some problems with the last question in exercise 2.6.4 from Weibel's "An introduction to Homological Algebra". The exercise asks to show that pushout is not an exact functor in Ab (Abelian ...
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Does tensor product with $L_p$ operator algebra preserve exact sequences?

By $L_p$ operator algebra I mean a closed subalgebra of the algebra of bounded linear operators on some $L_p$ space where $p\in(1,\infty)$. There is a notion of tensor product of $L_p$ spaces (as ...
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$0\to C'\to C\to C''\to0$ splits if $C\cong C'\oplus C''$ as a chain complex?

Question Given a unitary ring $A$ and an exact sequence $$0\to C'\xrightarrow iC\xrightarrow pC''\to0$$ in the Abelian category of chain complexes over $A$, where $C,C',C''$ are chain complexes of ...
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Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
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What is a short exact sequence?

I'll just quote my book here so you can see the definitions I have: Suppose that you are given a sequence of vector spaces $V_i$ and linear maps $\varphi_i: V_i\to V_{i+1}$ connecting them, as ...
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1answer
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homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
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Why solving linear equations is taking a quotient by some subspace?

Linear equation can be represented by a linear form, and its solution space is the same thing as kernel of this form. The same is true for system of linear equations. But this lecture notes suggest ...
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Additivity of trace

Let $A$ be a finitely generated abelian group and $\alpha:A\to A$ be an endomorphism. Since $A=A_{free}\oplus A_{torsion}$, we can induce $\bar \alpha:A_{free}\to A_{free}$, i.e. $\bar\alpha$ is a map ...
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Proof of the exactness of the tensor product

In Atiyah and MacDonald, Prop 2.18 establishes that for any exact sequence $$M'\xrightarrow{f}M\xrightarrow{g}M''\xrightarrow{}0\tag{1}$$ of $A$-modules and homomorphisms, and for any $A$-module $N$, $...
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Understanding a certain proof about $R$-module homomorphisms and split exact sequences

I'm currently reading "Algebra: Chapter 0" by Paolo Aluffi. Before I state my problem, I would like to give here the definition of split exact sequences from the book: A short exact sequence ...
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25 views

Split exact sequence of $C^*$-algebras

Let $\left\{A_i\right\}_{i\in I}$ be a countable set of $C^*$-algebras $A_i$ and $ \bigoplus_{i\in\mathbb{N}}A_i$ the direct sum as a $C^*$-algebra. Let $A_i^1$ the unitization of $A_i$ and $c_0$ the ...
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Top exterior product of exact sequence

Let $M,N,P$ be free $R$-modules of rank $a,a+b,b$ respectively, and that they fit into an exact sequence $0\to M\to N\to P \to 0$. Is it true that $\Lambda^{a+b}N=\Lambda^aM \otimes \Lambda^bP$? (...
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Munkres algebraic topology section 25 question 6 (Mayer Vietoris)

This question is from Munkres Algebraic Topology section 25 question 6 The question, vertabim, says. "We shall study the homology of $X\times Y$ in chapter 7.For the present, prove the following, ...
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What does Hom(M,N) mean? Atiyah Macdonald proposition 2.9

In Atiyah Macdonald, "Introduction to commutative Algebra" it says: Proposition 2.9:i) Let $M' \xrightarrow[]{u}M \xrightarrow[]{v} M'' \rightarrow 0$ be a sequence of A-modules and homomorphisms. ...
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Appropriate Generalization of Statement about Pure Subgroups to Pure Submodules

I have been working in a book on Homology by Hilton & Stammbach, wherein they introduce the idea of a "pure sequence of Abelian groups", which is a short exact sequence of Abelian groups $$0\...
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66 views

If $i$ is an inclusion why is the induced $i_*$ an epimorphism

Given the following exact homology sequence of a pair. This is in Example 2 (page 134) from Munkres. This is where I am always stuck computing homology using exact sequence. I cannot grab the last ...
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119 views

Prove $\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$

We took this idea from Simon Plouffe see here $$\ln(2^5)-\pi=8\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{1}{e^{n\pi}+1}-\frac{1}{e^{2n\pi}+1}\right)$$ Can anyone prove this identiy? We found this ...
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Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
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Module of finite length $\implies$ finite direct sum of indecomposable modules

I'm trying to solve the following question. Let $M$ be an $R$-module of finite length (i.e, both Artinian and Noetherian). Prove that it is isomorphic to a finite direct sum of indecomposable ...
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Reference needed for an exact sequence of an ACM curve with a homogeneous coordinate ring.

Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence $$0\to \text{Tor}_i(R,\Bbb{C})_k\to H^1(C,\wedge^{i+1}...
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2answers
252 views

Show that the sequence is exact

We have that $R$ is a commutative ring. Suppose that $0\rightarrow A\rightarrow B\overset{f}{\rightarrow} C\rightarrow 0$ and $0\rightarrow C\overset{g}\rightarrow D\rightarrow E\rightarrow 0$ are ...
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Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ a homomorphism…

I need some help with this problem, it seems easy, but I don't get it. Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ an homomorphism. Suppose that for each $A$-modules $M$ ...
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Left exact functors and long exact sequences

I wonder whether in any Abelian category $\mathcal{C}$ when we have a long exact sequence $0\to M_1\to M_2\cdots\to M_n\to 0$ and a (covariant) left exact functor $F$ we have $0\to FM_1\to FM_2\to \...
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Sum of Hilbert functions of a finite exact sequence of finitely generated graded modules

Let $A = \bigoplus_{n\geq 0} A_n$ be a graded ring that is generated as an $A_0$-algebra by a finite collection of elements of $A_1$, where $A_0$ is artinian. I wish to show that if $$ 0 \to M(1) \...
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Splitness of quotient sequence

Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A', B', C'$ be sub bundles, respectively. Suppose that we have short exact sequences: $$0 \rightarrow A \rightarrow B \...
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Getting an isomorphism from a short exact sequence of inner product spaces

Let $L,M,N$ be finite dimensional inner product spaces and $0 \to L \xrightarrow{\alpha} M \xrightarrow{\beta} N \to 0$ is a short exact sequence. Now let $\beta^* : N \to M $ be the adjoint map (the ...
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Dimension of a sheaf cohomology group on a genus 1 curve

Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...
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Module induced from projective is projective

Let $A,B$ be rings such that $A$ is $B$-module, $P$ be projective $B$-module. I want to prove that $A\otimes_B P$ is projective. I have that $\mathrm{Hom}_A(A\otimes_B P,M) \simeq \mathrm{Hom}_B(P,\...
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The bigger picture the Five Lemma fits into

The Five Lemma is a statement in category theory about certain conditions under which certain maps in exact sequences are isomorphisms. It has a few relatives like the 4 lemmas and maybe the Nine ...
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What exactly is a split of an exact sequence?

I read through Wikipedia and tons of questions of MSE, but I still can't grasp the concept of a split of a short exact sequence. Apparently by definition a sequence $$0\rightarrow A\overset f \...
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Castelnuovo-Mumford regularity and exact sequence.

In a question on MathOverflow it is said that: It is known that given a short exact sequence of finitely generated graded modules over a polynomial ring over a field:$$0 \to M'' \to M \to M' \to 0$...
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53 views

Lifting homomorphism when module is direct summand of free module [closed]

Let $N,M,M',N'$ be modules such that $F = N \oplus N'$ is a free module. Let further $f: M \rightarrow M''$ be a surjective homomorphism. I would like to show that for any homomorphism $\phi: N \...
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35 views

Applying hom-functor on short exact sequence

Let $N, M, M', M''$ be $R$-modules. Given homomorphisms $f: M' \rightarrow M$ $g: M \rightarrow M''$ $\psi: N \rightarrow M$ with $\operatorname{im} f = \ker g$, $g \circ \psi = 0$ and $g$ ...
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commutative diagram of groups in Lang's algebra

In Lang's Algebra (section I.3), he says that we can describe the third isomorphism theorem $\frac{G}{K}/\frac{H}{K}\cong G/H$ by the following commutative diagram. \begin{array}{c} 0 & \to &...
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25 views

Split Lie algebra extensions?

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras. A Lie algebra extension is a short exact sequence $$0\longrightarrow \mathfrak{h}\stackrel{\jmath}{\longrightarrow} \mathfrak{e}\stackrel{\...
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Equivalence between absolutely pure modules and FP-injective modules

Let $R$ b a ring and $Q_{R}$ a right $R$-module. Show that the following conditions are equivalent for $Q_{R}$ : $(i)$ $Q_{R}$ is FP-injective, that is, given any short exact sequence $0 \rightarrow ...
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Equivalent conditions for a short pure exact sequence

Let $L$, $N$, and $M$ be right $R$-modules and let $\widehat{L}=\mathrm{Hom}_{\mathbb{Z}}\left(L,\mathbb{Q}/\mathbb{Z} \right)$ ($\widehat{N}$ and $\widehat{M}$ are defined analogously). Show that the ...
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Exact sequence of free abelian groups, $\sum_{i=0}^n(-1)^i\text{rank}(F_i)=0$.

This question is from Rotman's Introduction to the Theory of Groups: (i) Suppose we have an exact sequence of free abelian groups $A\to B\to C\to D$ with maps $f,g,h$ in between. Show $B\cong \text{...
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Why functors that preserve cokernels are right exact?

"Let $F$ be a functor from $A-$modules to $A-$modules that preserves cokernels. Then $F$ take exact short sequences into right exact short sequences." I found a mistake in my proof. I also can't find ...
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Contractibility of an exact chain complex

How can one prove that an exact (acyclic) chain complex of projective modules that is trivial in negative degrees is contractible? I would appreciate some nudges in the right direction more than ...
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If $G$ is cyclic of order $2$ show $|H^1(G,\mathbb{Z})|=2$.

Let $G$ be the cyclic group of order $2$ acting by inversion on $\mathbb{Z}$. Show $|H^1(G,\mathbb{Z})|=2$. A hint is provided: if $E=\mathbb{Z} \rtimes G$ then every element in $E - \mathbb{Z}$ has ...
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How to determine probabilities according to a position on a list

I'm working on a SQL system in which i have to retrieve a "winning" user from a users list. The winner is determined randomly, but the users should have more chances if their coeficient number is ...
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Why isn't $1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$ a split extension? [duplicate]

Let $\mathbb{Z}_n$ be the cyclic group of $n$ elements and $1$ the trivial group. I'm looking at the following extension: $$1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$$ Because every ...
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Exact Sequences in algebraic geometry [closed]

A very basic question. I am going to take my first course in Algebraic Geometry next semester and I am now repeating some commutative algebra to be prepared. I just came up to the part of Homological ...
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31 views

Bockstein homomorphism and the universal coefficient theorem

The following statement is given in the third comment of kernel of the mod $2$ Bockstein on the first cohomology group: Statement: Let $X$ be a path-connected finite $CW$-complex. Suppose $$ H_1(X;...
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Find an exact sequence from a module $M$ to direct sum of localizations of $M$ in a finitely generated ring

Suppose $f_1,f_2,\ldots, f_k\in R $ such that $R=(f_1,f_2,\ldots, f_k)$. Let $M$ be an $R$-module. Define $U(f)=(f,f^2,f^3,\ldots)$ and let $$M_i=U(f_i)^{-1}M$$ $$M_{i,j}=U(f_if_j)^{-1}M$$ where ...