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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442 views

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 ...
1
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1answer
16 views

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: ...
1
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1answer
33 views

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 ...
2
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0answers
42 views

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 ...
0
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1answer
37 views

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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2answers
60 views

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 ...
2
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1answer
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 ...
1
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2answers
35 views

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 $\Lambda^{a+b}N=\Lambda^aM \otimes \Lambda^bP$? (...
0
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0answers
22 views

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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3answers
80 views

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. ...
1
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0answers
14 views

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\...
1
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1answer
65 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 ...
2
votes
2answers
118 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 ...
1
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2answers
93 views

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 ...
2
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2answers
27 views

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 ...
0
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0answers
10 views

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}...
2
votes
2answers
251 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 ...
0
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1answer
27 views

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 $M$ ...
0
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0answers
28 views

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 \...
0
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1answer
17 views

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) \...
3
votes
1answer
90 views

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 \...
2
votes
1answer
69 views

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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0answers
80 views

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 ...
0
votes
1answer
18 views

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,\...
5
votes
1answer
90 views

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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1answer
28 views

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 \...
4
votes
1answer
70 views

Castelnuovo-Mumford regularity and exact sequence. [closed]

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$...
1
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1answer
48 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 \...
1
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1answer
29 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 $g$ ...
0
votes
1answer
19 views

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 &...
2
votes
1answer
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{\...
0
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0answers
12 views

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 ...
0
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0answers
24 views

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 ...
1
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1answer
27 views

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{...
3
votes
1answer
64 views

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 ...
2
votes
1answer
39 views

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 ...
0
votes
0answers
26 views

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 ...
0
votes
1answer
25 views

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 ...
2
votes
0answers
28 views

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 ...
1
vote
1answer
75 views

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 ...
4
votes
1answer
30 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;...
2
votes
1answer
34 views

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 ...
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2answers
85 views

Show that a Z-module A is flat if and only if it is torsion free?

I found a question in my textbook which really confuses me! Show that a $\mathbb Z$-module $A$ is flat if and only if it is torsion free? Over here, torsion free means if abelian group A is ...
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votes
1answer
57 views

how to prove the following sequence is exact

Suppose E be a right R-module, and E is flat, then how can I prove that for any exact sequence of left R-modules $A\to B\to C$, the sequence $E\otimes A\to E\otimes B\to E\otimes C$ is exact? I am ...
4
votes
0answers
31 views

Rank Nullity Theorem (Short Exact Sequences)

In the book Manifolds, Tensors and Forms, there is the following theorem: We have a short exact sequence $0\rightarrow \ker T\xrightarrow{\iota} V\xrightarrow{T} W\rightarrow 0$ where $\iota$ is the ...
3
votes
2answers
132 views

What is the most general category in which exist short exact sequences?

Let $A,B,C$ be objects, $0$ the final object, and $f:A\to B$ and $g:B\to C$ morphisms in some category. Consider the sequence: $$ 0 \to A \to B \to C \to 0\;. $$ I would like to say something ...
2
votes
1answer
58 views

How to prove the following sequence is exact?

Let R be a ring and $F',F,F'',G',G,G''$ left R-modules. Assume we are given R-module homomorphism $i:F'\to F,p:G'\to G,p':G\to G''$ and $a:F'\to G',b:F\to G,c:F''\to G''$ such that the following ...
1
vote
1answer
31 views

Splitting Lemma where $C=\mathbb{Z}.$

Given a short exact sequence $$ 0 \xrightarrow{\theta_3} A \xrightarrow{\theta_2} B \xrightarrow{\theta_1} \mathbb{Z} \xrightarrow{\theta_0} 0 $$ show that $B \cong A \oplus \mathbb{Z}.$ So far I ...
4
votes
1answer
77 views

Relation between ranks of free sheaves and cohomology

Suppose that $\mathbb{P}^r=\mathbb{P}^r_K$ is the projective space over a field $K$. Let $\mathcal{O}_{\mathbb{P}^r}(-1)^n\longrightarrow \mathcal{O}_{\mathbb{P}^r}^m$ be a morphism of vector bundles....
1
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1answer
65 views

Show that a sequence is a free resolution

Let $I \subset R = k[x_1,\dots,x_n]$ be an ideal and $f \in R$ such that $I = \left < f \right >$ ($k$ is a field, so R is commutative ring). How do I show that (1) $I$ has a free resolution $$...