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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29 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
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2answers
103 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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2answers
89 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 ...
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2answers
26 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 ...
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9 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 ...
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2answers
195 views
+100

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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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 $M$ ...
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0answers
24 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 ...
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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 ...
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1answer
73 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 ...
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1answer
45 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
77 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 ...
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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 ...
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1answer
87 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
25 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 ...
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1answer
64 views

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 ...
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1answer
36 views

Lifting homomorphism when module is direct summand of free module

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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1answer
27 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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1answer
18 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 ...
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1answer
23 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} ...
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11 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 ...
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20 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 ...
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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 ...
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1answer
61 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
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1answer
38 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 ...
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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
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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
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0answers
27 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 ...
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1answer
74 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
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1answer
26 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 $$ ...
2
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1answer
32 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 $$M_i=U(f_i)^{-1}M$$ $$M_{i,j}=U(f_if_j)^{-1}M$$ ...
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81 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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1answer
54 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
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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
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2answers
131 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
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1answer
56 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 ...
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1answer
30 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
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1answer
75 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 ...
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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 ...
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1answer
30 views

Why do linear splitting maps of Lie Algebra central extensions induce cocycles?

If we consider a central extension $\mathfrak h$ of a Lie algebra $\mathfrak{g}$ by the abelian $\mathfrak a$: $$0 \longrightarrow \mathfrak a \longrightarrow \mathfrak h ...
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92 views

How does a short exact sequence say something about a group?

I have a follow-up question to my question here: How are simple groups the building blocks? In that question I asked about what it means when we say that the simple (finite) groups are the building ...
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2answers
70 views

Splitting of Short Exact sequence

Suppose $d_1d_2=n$ and let $0 \to d_1\mathbb Z_n \overset {i} \to \mathbb Z_n \stackrel {d_2\cdot} \to d_2\mathbb Z_n\to 0$ be a short exact sequence. Show that sequence splits iff ...
4
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1answer
68 views

$\text{Hom}_{\mathcal{O}_X}(\cdot, \mathcal{G})$ is an exact functor

Suppose that $(X, \mathcal{O}_X)$ is a ring space. If $\mathcal{F}, \mathcal{G}$ are sheaves of $\mathcal{O}_X$-modules then the assignment $$U \mapsto \text{Hom}_{\mathcal{O}_X|U}(\mathcal{F}|U, ...
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1answer
28 views

Exact Sequences of Modules and Rank

Suppose that: $0 \rightarrow M_{1} \rightarrow M \rightarrow M_{2} \rightarrow 0$ Is an exact sequence of $R$-modules (for $R$ commutative integral domain). Show that $\mathrm{rk}(M) = ...
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1answer
54 views

Let $A$ be a local ring, $M$ and $N$ finitely generated $A$-modules. Show that if $M \otimes N = 0$ then $M=0$ or $N=0$ [duplicate]

Let $A$ be a local ring, $M$ and $N$ finitely generated $A$-modules. Show that if $M \otimes_A N = 0$ then $M=0$ or $N=0$ I read one proof from my book and it goes as follow: First, show that ...
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28 views

$E_{p,q}^r$ spectral sequence. Find a l.e.s. $\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} E_{p-1,1}^2\to A_p\to E_{p,0}^2\to\cdots$

Let $E_{p,q}^r$ be a spectral sequence which converges to $A_n$. Let $E_{p,q}^r=0$ for $q\ge 2$. How to construct a long exact sequence $$\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} ...
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1answer
39 views

Intuition behind $Ext^1(A,\,C)$

So I recently asked a question concering $Ext^1(A,\,C)$ regarding the connection between isomorphism and the congruence '$\equiv$' (Where am I making a mistake with $Ext^1(A,C)$?). Suppose, for ...
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1answer
45 views

Short exact sequence of $\mathbb{R}[X]$-modules that does not split [closed]

What is an example of a short exact sequence of $\mathbb{R}[X]$-modules that does not split?
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1answer
36 views

Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
3
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1answer
33 views

For $x\in\operatorname{Ext}_R^1(C,A)$, how to construct an extension $0\to A\to B\to C\to 0$ such that $\partial (id_A)=x$? [duplicate]

Let $R$ be a ring, $C, A$ two $R$-modules. For all $x\in\operatorname{Ext}_R^1(C,A)$ I have to construct a short exact sequence $$0\to A\to B\to C\to 0$$ of $R$-modules such that $\partial(id_A)=x$, ...