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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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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29 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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$E_{p,q}^r$ 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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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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35 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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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 $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
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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 $\partial(id_A)=x$, ...
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Question on syzygies

It is hard to formulate a question, but I want to ask about a reference/recipe for computing syzygies in general. For example, on $\mathbb{P}^1_{(x:y)}$ there is an exact sequence $0\longrightarrow ...
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Determine which abelian groups can be the central term of this exact short sequence

I am trying to solve the following problem: Determine which abelian groups $A$ can appear as central terms in a short exact sequence $\mathbb{Z} \to A \to \mathbb{Z} \oplus \mathbb{Z}_5$ What ...
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Short exact sequence on $\mathbb{P}^1$

Let F be a torsion free sheaf of rank $n+4$ over $\mathbb{P}^1$ which fits in the SES $0\longrightarrow\mathcal{O}_\mathbb{P^1}(-3)^{n+2}\longrightarrow ...
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24 views

Apply the functor $Hom(-,B)$ to the following exact sequence

Source: Weibel, Page 94. Given an ideal $I$ in a ring $R$, we have the exact sequence: $$0\rightarrow I \rightarrow R \rightarrow R/I \rightarrow 0,$$ so if we apply the contravariant left exact ...
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Derived functors on 2 term exact sequence

Suppose we have a not-short exact sequence 0 -> A -> B -> 0 in some abelian category. Now let us apply right exact functor F: F(A) -> F(B) -> 0. So could I consider a left derived functor H to obtain ...
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Definition of exact sequence [closed]

Take a sequence $A \to B \to C \to 0$ of modules over a commutative ring. How would one show that it is exact? I understand the necessary surjectivity of $B \to C$, but what about the first map?
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Nonsplit extension of $\mathbb{Z}$ by itself

I was classifying all split extensions on a list of short exact sequences, when I arrived on this one: $$1 \rightarrow \mathbb{Z} \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1$$ That the ...
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Morphisms between long exakt sequences

I have a commutative diagram of modules of the form $$\require{AMScd} \begin{CD} @. @VVV @VVV @VVV @VVV @. \\ ... @>>> A_n @>>> B_n @>>> C_n @>>> A_{n-1} ...
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Does $\cdots \to G_1\overset f\to G_2 \overset g\to G_3\to \cdots$ exact imply $0\to \ker(g) \to G_2 \to \operatorname{coker}(f)\to 0$ exact?

Given a (part of a) long exact sequence of abelian groups (or modules over some commutative ring) $$ \cdots \to G_1\overset f\to G_2 \overset g\to G_3 \to \cdots $$ we have the short exact sequence $$ ...
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functor $F$ satisfies $kerF(f)\cong F(ker(f))$. $0\to M'\to M\to M''$ exact => $0\to F(M')\to F(M)\to F(M'')$ exact?

$R$ ring, $R-MOD$ is the category of $R-$modules anf $F:R-Mod\to R-Mod$ a functor such that the induced map$$Hom(M,N)\to Hom(F(M),F(N))$$ is a homomorphism of abelian groups. $F$ satisfies ...
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Exactness and conjunction of finite character properties for modules

From Commutative Algebra - Constructive Methods by Lombardi and Quitte: Definition 2.9. A property $\mathsf P$ concerning commutative rings and modules is called a finite character property if it is ...
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Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
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31 views

Proving exact functor preserves kernels

I'm trying to show that if $F$ is a functor on abelian categories that preserves exact sequences, then it also preserves kernels. I have started with $0 \to \text{ker}(f) \overset{k}{\to} A ...
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51 views

Canonical sheaf projective bundle

Given a smooth variety $Y$, and a vector bundle $E$ of rank $r+1$ defined on it, call $X$ the variety associated to the projective bundle given by $E$. Call $\pi: X \rightarrow Y$ the natural map. ...
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Extending coboundary to cocycle

I was reading these notes and got confused at the beginning of section 9.2 (p. 21-22). $i^n$ is the map from $Z^n$ (i.e. ($\ker\partial^n$))to $B^n$ (i.e. $\text{im}(\partial^{n-1})$) of a cochain ...
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Isomorphisms Between some terms of a Long Exact Sequence

Suppose we have two long exact sequences of finite dimensional $k$-vector spaces: $$ 0 \to A_1 \to A_2 \to A_3 \to \cdots $$ $$ 0 \to B_1 \to B_2 \to B_3 \to \cdots $$ And assume that $A_6 ...
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Wiki on exact sequences in regular categories

In a regular category, an exact sequence is a diagram which is both a coequalizer and a kernel pair: $$R\overset r{\underset s\rightrightarrows} X\to Y$$ Wiki says that in the abelian case, the above ...
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69 views

Why is the sequence exact?

Bredon states: For $A \subset X$, the sequence $$ 0 \rightarrow \Delta_{*}(A) \otimes G \rightarrow \Delta_{*}(X) \otimes G \rightarrow \Delta_{*}(X,A) \otimes G \rightarrow 0$$ is exact ...
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$\mathbb {Z}/\mathbb{Z}$ isomorphic to $\mathbb{Z}$?

In my lecture notes, it says (talking about short exact sequences) that if you have $$0\to A \to B\to C\to 0$$ and you know it's an exact sequence, then $B/A\cong C$. The notes then give an ...
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Short exact sequences for amalgameted free products and HNN Extensions

If $A$ and $B$ are groups we have the following short exact sequence: $$ 0 \to [A,B] \to A * B \to A \times B \to 0, $$ where the group $[A,B]$ is free (see e.g. Serre's Trees). I am wondering if ...
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A short exact sequence of abelian groups induces a long exact sequence in (co)homology with coefficients

Let $$0\to V'\to V\to V''\to 0$$ be a short exact sequence of abelian groups. Let $X$ be a topological space. How to construct long exact sequences in singular homology and cohomology $$\cdots \to ...
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For finitely generated $B$ all modules in exact sequence are finitely generated in PID

Let there be an exact sequence of $R$-modules: $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ where $R$ is a principal ideal domain and $B$ is finitely generated. Are $A$ and $C$ ...
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Proving that the diagram commutes

Let's consider a continuous map $f: X \rightarrow Y$, so that $Y= U_{1} \cup U_{2}$ -- covering by open sets and $X = f^{-1}(U_{1}) \cup f^{-1}(U_{2}) = V_{1} \cup V_{2}$. How to prove that the ...
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Definition of a pushout of a short exact sequence

What is the definition of pushout of a short exact sequence? In this paper1, page $126$, under the proof of Proposition $2.8$, I don't understand how the author justify the existence of the operator ...
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42 views

$A$-module and free $A$-modules

Let $A$ be a commutative ring with unity and consider an $A$-module $M$. Why do we always have the following exact sequence? $$A^{(J)}\rightarrow A^{(I)}\rightarrow M\rightarrow 0$$ (Here $I,J$ ...
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exact sequence contains kernels

I'm trying to read Iversen, "Cohomology of Sheaves". There, working in a category with zero object and kernels and cokernels, the image is defined as a kernel for a cokernel. A sequence of morphisms ...
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43 views

Functor in $\mathbf{Ban}$ that puts exact sequences into exact sequences

Let $F$ - is functor in category of Banach spaces $\mathbf{Ban}$ with follow property: $f_n : A_n \to A_{n+1}$ exact sequence iff $F f_n : F A_n \to F A_{n+1}$ is exact sequence. Is it true, that $F$ ...
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Exactness of Lie algebra exact sequence

If $G\rightarrow H\rightarrow K$ is an exact sequence of Lie groups, then I want to show that the induced sequence $\mathfrak{g}\rightarrow\mathfrak{h}\rightarrow\mathfrak{k}$ in Lie algebras is ...
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Short Exact Sequence from Single Arrow in Abelian Category

Work in an abelian category. I'm aware that given an exact sequence, one can break it into short exact sequences like so: I was wondering whether it was possible to do derive one of these short ...
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45 views

Horseshoe Lemma

Trying to understand the horseshoe lemma. Suppose I start with the usual exact sequence $0 \to \mathbb{Z}/2\mathbb{Z} \overset{a}{\to} \mathbb{Z}/4\mathbb{Z} \overset{b}{\to} \mathbb{Z}/2\mathbb{Z} ...
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How to prove $\ker \beta + \text{im} \tau = B$ in the definition of split short exact sequence?

Let $0 \xrightarrow{}A \xrightarrow{\alpha} B \xrightarrow{\beta} C \xrightarrow{} 0$, be a short exact sequences of module homomorphisms. I understand that if there exists a hom $\tau : C \to B$ ...
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Compact operator in terms of exact sequences

I know a pretty equivalent definition for Fredholm operators in terms of exact sequences. Here is it: We called operator $S : E \to F$ between Banach spaces $E$ and $F$ as Fredholm iff exist exact ...
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Prove that the Localizations of R give Flat Modules over R

Given a multiplicative subset $S\subset R$, how can we show that $S^{-1}R$ is flat over $R$? I don't have any idea where to start.
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Euler characteristic, what is the precise relationship between $\chi(V)$, $\chi(V')$, and $\chi(V'')$? [duplicate]

For a graded vector space $V = \{V_n\}$ with $V_n = 0$ for all but finitely many $n$ and with all $V_n$ finite dimensional, define the Euler characteristic $\chi(V)$ to be $\sum (-1)^n \dim V_n$. Let ...
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How do I show that the two sequences below are short exact sequences of vector spaces?

$0 \rightarrow \mathbb{Z_2}^k \to \mathbb{Z_2}^n \to \mathbb{Z_2}^{n-k} \to 0$ $0 \to \mathbb{Z_2} \to \mathbb{F}_{2^2} \to \mathbb{Z_2} \to 0$, where $\mathbb{F}_{2^2}$ is the Galois field of size ...
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59 views

Exact sequence splits?

I am stuck in the following problem: Show that every group of order 4 is an extension of $\mathbb{Z}_{2}$ by $\mathbb{Z}_{2}$. Which of the exact sequences splits? Ok, i have to consider two cases ...
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Geometric intuition for left/right exactness

Sheaf cohomology measures the obstruction of the global section functor from being exact. Since it's left exact, it is exact iff it preserves epis. In particular, $H^1$ measure the failure to be ...
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Verify that the following sequence of vector spaces is an exact sequence

The sequence is $0\rightarrow V_1 \rightarrow V_1\oplus V_2\rightarrow V_2\rightarrow 0$ and $l:V_1\rightarrow V_1\oplus V_2$ is an inclusion and $p:V_1\oplus V_2 \rightarrow 0$ is a projection. It ...
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40 views

Using the Snake lemma to prove an Extension

I am trying to prove that $E'$ is an extension of $Q$ by $N'$ \begin{array} 00 &\longrightarrow & N & \overset{i_1} \longrightarrow & E & \overset{\pi_1} \longrightarrow& Q ...
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61 views

Find free resolution of ideal $(x,y)$ in $\mathbb{Q}[x,y]$

If we consider the ideal $I=(x,y)$ as a module over the ring $\mathbb{Q}[x,y]$, how can we find a free resolution of $I$? That is, a sequence $$...P_2 \to P_1 \to P_0 \to I \to 0$$ with $P_i$ ...
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Why is $0\to\ker\varepsilon\to P_0\xrightarrow\varepsilon N\to 0$ a projective resolution?

Let $R$ be a commutative ring with $1$. My professor defined $\operatorname{Tor}_i^R(M,N)$ as follows: Tensor a projective resolution $\dots\to P_1\to P_0\to M\to 0$ with $N$ and set ...
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An explicit example of a Pullback of equivelent short exact sequences.

I am trying to construct the Pullback of the example found in an answer Example on Ext Functor. Say I have equivalent exact sequences: of the form $ E=0\rightarrow N\rightarrow E_1\rightarrow ...
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62 views

Connections on principal bundles: Local and Global Formulations.

The standard definition of a connection on a principal $G$-bundle $\pi : P \to X$ is a smooth family of subspaces $H_{p}$ of $T_p P$ such that for every $p \in X$ we have a splitting of vector spaces ...