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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Partial exactness of the total complex.

The acyclic assembly lemma (pages 59/60 of An introduction to homological algebra, Weibel) establish that if I have a bounded double complex with exact rows (or columns) then the total complex is also ...
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45 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to M'\to ...
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37 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
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47 views

Extensions of short exact sequences and second cohomology group

Let $G=\mathbb{I}_{p}=<g>$ be the cyclic group of order $p$, where $p$ is a prime and $A=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ a $G-$ module with the action $g^{n}(x,y)=(x+ny,y)$. I want to show ...
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Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
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2answers
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Existence of non-split sequence

Let $G$ be an abelian group such that $G$ contains non-zero elements of finite order. Why there exists some short exact non-split sequence: $0 \rightarrow \mathbb{Z} \rightarrow H \rightarrow G ...
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1answer
66 views

Invariance of the Fredholm index under finite-dimensional perturbations

Let's call a linear map $f : V \to W$ between vector spaces over some field Fredholm if $\ker(f)$ and $\mathrm{coker}(f)$ are finite-dimensional. (Equivalently, it represents an isomorphism in the ...
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useful exact sequences [closed]

There are some exact-sequences or long-exact-sequences that are great help in proving to prove some surprising theorem, or have some interesting applications. What's your favorite exact ...
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54 views

Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram ...
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2answers
52 views

Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
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2answers
21 views

Working out values of sequences using partial sums

I have the sequence $a_n =\sqrt{n+1}-\sqrt{n}$ and $b_n=\dfrac{1}{\sqrt{n}}$ let $s_n$=$a_1 + a_2 + a_3 + ... + a_n =\displaystyle \sum_{k=1}^n a_k$ and $t_n$=$b_1 + b_2 + b_3 + ... + b_n = ...
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Why any short exact sequence of vector spaces may be seen as a direct sum?

This is actually the first time I have worked with short exact sequences and I have no clue why the following assertion is true: Any short exact sequence of vector spaces $$ 0 \longrightarrow U ...
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Short exact sequence for topological join: split needed

I am desperately trying to solve the following problem: Let $X$ and $Y$ be topological spaces and $X * Y$ their join. Prove that there is a short exact sequence $$ 0 \to \tilde{H}_k(X * Y) \to ...
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1answer
34 views

Exact short sequence vs exact long sequence?

could anyone explain me what exactly the difference between an exact long sequence and an exact short sequence is? I think it pertains to homology theory, right?
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31 views

Applications of Splitting Lemma and Exactness

I'm looking for nice applications of exact sequences, the splitting lemma, and exact functors in algebra and topology (i.e not using the five lemma to get long sequences in homology etc..). For ...
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131 views

Question on proof of Lefschetz Fixed Point Theorem (from Hatcher Theorem 2C.3)

In Hatcher's statement of the Lefschetz Fixed Point Theorem (2C.3), he has a hypothesis that the space $X$ in question must be a retract of a finite simplicial complex. The first part of the proof ...
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1answer
25 views

Split exact sequences.

Edit: The setting is some abelian category. The splitting lemma says that the following conditions are equivalent for a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ ...
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41 views

What kind of objects are both subobjects and quotients?

Fix an object $B$ in some category. What does the existence of a diagram $A \rightarrowtail B \twoheadrightarrow A$ imply about $A$ and $B$? What if $A \rightarrowtail B \twoheadrightarrow ...
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84 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
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1answer
64 views

counterexample to “symmetric” nine lemma

Consider a commutative diagram of $R$-modules ...
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1answer
48 views

Commutative diagram with exact sequences as columns and rows

Suppose that we have the following commutative diagram of groups and homomorphisms $$\newcommand\twoheaduparrow{\mathrel{\rotatebox{90}{$\twoheadrightarrow$}}} \begin{array} A & A_3 & ...
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2answers
25 views

Non exact sequence of quotients by torsion subgroups

$$0\rightarrow G_{1}\rightarrow G_{2}\rightarrow G_{3}\rightarrow 0$$ is a short exact sequence of finitely generated abelian groups. We call $\bar{G_{i}}$ the quotient of $G_{i}$ by its torsion ...
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154 views

Splitting of an exact sequence

Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that $x \in \mathfrak m \setminus \mathfrak m^2$. Is it true that $$ \frac{\mathfrak m}{x\mathfrak m} \cong \frac{\mathfrak m}{(x)} ...
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83 views

Exactness of Hom Functor

The picture above is from Dummit and Foote, Third Edition, Chapter 10. In the text the authors claim that the sequence given by $ Hom $'s is exact if and only if there is a bijection $ F ...
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35 views

Equivalence of short exact sequences

The above image is from the book of Dummit and Foote, Edition 3. In the fourth paragraph, the authors claim that "equivalences involving the same extension module $ B $ are automorphisms of $ B ...
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1answer
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Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might ...
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173 views

Finding all abelian groups such that there exists certain short exact sequence.

I have to find all abelian groups $A$ such that there exists a short exact sequence $0\rightarrow\mathbb{Z}\rightarrow A\rightarrow\mathbb{Z}\oplus\mathbb{Z}_{6}\rightarrow 0$. I have found ...
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1answer
71 views

Exactness and Naturality

I'm trying to read this blog post about exact functors, and I see mentions of naturality which I have not stumbled upon elsewhere. In particular, in the proof of the Theorem, the author says By ...
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1answer
27 views

Exact sequence of modules exercise

Show that if $$0 \rightarrow M_1 \xrightarrow{f} M_2 \xrightarrow{g} M_3$$ is an exact sequence of $R$-modules, then for all $R$-module $$0 \rightarrow \operatorname{Hom}_R(M,M_1) \xrightarrow{f_*} ...
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1answer
36 views

Exact sequence and Noetherian modules

Let $R$ be a ring, $X,Y,Z$ and $T$ four $R-$ modules such that there exists a short exact sequence $$0 \rightarrow X \xrightarrow{f_1} Y \xrightarrow{f_2} Z \xrightarrow{f_3} T \rightarrow 0 $$ Prove ...
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1answer
40 views

$Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all convergent numerators and denominators. True?

Where $x = A001203$, $Pi = A002486$, $A002485$ $Pi(n) = Pi(n-2) + x\times [Pi(n-1)]$ for all $Pi > n+1 $ Hypothesis: This relation evaluates true for all $A002486$ and $A002485$. Lemma: All ...
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Noetherian property for exact sequence

Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then $M$ is Noetherian is equivalent to $M'$ and $M''$ are Noetherian. For the ...
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$\mathbb Z_{p^2}$ is not a non trivial semidirect product.

I am trying to prove that the group $\mathbb Z_{p^2}$ (p prime) is not a non trivial semidirect product. Since a group $G \cong K \rtimes H$ if and only if for all short exact sequences $$0 ...
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Determining the middle term of an exact sequence of Lie algebras

This is related to my previous question here. Suppose that $A_i, B_i$ are Lie algebras with $A_i$ is a sub-Lie algebra of $B_i$, $i=1,2,3$. Suppose that we have the following commutative diagram where ...
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1answer
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Proof that the connecting morphism in the snake lemma is well defined

The snake lemma says: suppose we have two exact sequences of $R$-modules $M_1 \xrightarrow{f_M} M_2 \xrightarrow{g_M} M_3 \rightarrow 0$ $0\rightarrow N_1 \xrightarrow{f_N} N_2 \xrightarrow{g_N} ...
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1answer
54 views

Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
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Power series modulo polynomials

I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon. $\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and ...
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55 views

A question on short exact sequences

Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting short exact sequence so it exist a morphism $r: M \to L$ such that $r \circ \alpha = Id_L$ and a morphism $s: N \to M$ such ...
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Splitting of short exact sequence of sheaves

Let $X$ be a smooth projective variety over a field, say $k$. Consider the short exact sequence of $k$-modules, $$0 \to A_1 \to A \to A_2 \to 0$$ where $A$ and $A_2$ are $k$-algebras. Since these can ...
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68 views

Question about “M” in a short exact sequence

If we have a splitting short exact sequence $0\to L\stackrel {\alpha}\to M\stackrel{\beta}\to N\to 0$ with $r$ as a retraction of $\alpha$. Is it true then that $\alpha(L)\cup \ker(r)=M$?
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(Split) Exact Sequence of Lie Algebra Associated to Groups

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
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28 views

Exact sequence of sheaves from the exactness of sections

Let $X$ be a topological space, and $\mathcal{F}_1$, $\mathcal{F}_2$ and $\mathcal{F}_3$ be sheaves on $X$. Suppose for all $U$ open in $X$ we have, $0\longrightarrow ...
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31 views

Split exact sequence of Lie algebras

Suppose that we have the following commutative diagram of Lie algebras $$\begin{array} A 0& {\longrightarrow} & A_0 & {\longrightarrow} & A_1 &{\longrightarrow} & A_2 & ...
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1answer
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Boundary Homomorphism

I was studying the proposition 2.10 of Atiyah and MacDonald's Introduction to Commutative Algebra, and have a question. The proposition says: Let $$ \require{AMScd} \begin{CD} 0 @>>> ...
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Why is this map of sheaves surjective?

Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a ...
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Exact sequence of $A$-modules [duplicate]

I was trying to demonstrate the Proposition 2.9 of Atiyah and MacDonald's Introduction to Commutative Algebra. But I couldn't do the following: Let $M$, $M'$, and $M''$ be $A$-modules, $v$ and $u$ ...
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1answer
28 views

Does exactness in each variable coincide with exactness of the product?

Let us restrict to the category of modules. I'm thinking about the definition of exactness of a functor on two variables. The usual definition is that it is exact in each of the two variable, whereas ...
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Kernel of a retraction

I have a couple of questions about a exercise I have: Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N \to 0$ be a splitting exact sequence and let $r$ be a retraction of $\alpha$ such that ...
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1answer
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Help on formalisation proof of the triviality of a kernel in Mayer-Vietoris

Consider the Mayer-Vietoris sequence for $\mathbb{RP}^2$, where the two open sets are $U:= \{ [x;y;z] \in \mathbb{RP}^2 | z \neq 0 \}$ and $V = \mathbb{RP}^2 \setminus [0;0;1]$. I've proved that $U ...
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Directs sum in exact sequences

Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting s.e.s. where $\alpha$ has a retraction $r$. (a) Show that in this case $M=\alpha(L)\oplus\ker(r)$. I am a little bit ...