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Questions tagged [exact-sequence]

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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Concerning Hartshorne's proof of the Vanishing Theorem of Grothendieck (Hartshorne III 2.7)

At certain point of the proof of the theorem in Hartshorne's book, he mentioned the following in which I don't understand: In Step 3, he said we can reduce our consideration to a sheaf $\mathscr{F}$ ...
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Baer sum, pushback of pushout and pushout of pullback

Let us consider the following constructions in the category of $R-$modules, for some ring $R$. Given a short exact sequence $$ \mathcal{S}: \quad 0 \to A \overset{\alpha}{\to} B \overset{\beta}{\to} ...
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Projective modules in long exact sequences

Let $A$ be a commutative ring (with unit), and let $(P_i)_i$ be projective $A$-modules sitting in a long exact sequence of $A$-modules: $$0 \longrightarrow P_1 \stackrel{f_1}{\longrightarrow} P_2\...
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Short exact sequence of algebras implies bimodule structure

I read a statement in "Algebraic Operads" which I don't understand : Let $$0 \rightarrow M \rightarrow A' \rightarrow A \rightarrow 0 $$ be a short exact sequence of associative algebras over the ...
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methods for computing cohomology from data of an exact sequence

Let X be a topological space and $A$ be a ring. Suppose we have an exact sequence of sheaves of $A$-modules on X, $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, suppose we ...
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Homotopy between unitary element and identity elements, Operator Theory

Let $\mathcal{T}$ be the Toeplitz algebra. I.e. the $C^*$ algebra generated by the shift operator $S\in B(l^2(\Bbb N))$. In page 6, line 8 of a proof we have a unitary element $u \in \mathcal{T} \...
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Betti Numbers' inequality and Mayer-Vietoris sequence

Let $U, V\subset M$ be open. How the Mayer-Vietoris sequence, $$\to H^{i-1}(U \cap V) \to H^i(U \cup V)\to H^i(U)\oplus H^i(V)\to$$ leads immediately to the $$b^i(U \cup V) \leq b^i(U) + b^i(V) + b^{...
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Every short exact sequence with a simple module splits

More concretely, if $R$ is a ring, and $M$ is a simple $R$-module, I want to show that any short exact sequence $0 \rightarrow L \rightarrow M \rightarrow N \rightarrow0$ splits. To this end, I have ...
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Certain exact functor on the Grothendieck group of a module category

Let $A$ be a be a finite dimensional, associative, and unital $\mathbb{C}$-algebra. Let $\mathcal{A}$ be the category of finitely generated $A$-modules. Since $A$ is an Artinian ring, there are only ...
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What do $0$ and $1$ mean in a extension (exact sequence) of group?

I'm self-studying homological algebra but stuck in a little place. What does $0$ and $1$ mean by the author? (I didn't read the book from start to end.) Does it mean the trivial group? Then why we ...
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What is the intuition behind short exact sequences of groups; in particular, what is the intuition behind group extensions?

What is the intuition behind short exact sequences of groups; in particular, what is the intuition behind group extensions? I'm sorry that the definitions below are a bit haphazard but they're how I ...
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Can we deduce the exactness of the pervious modules sequence if the localized exact modules sequence is exact?

My question comes from a proposition: if M is flat, will $S^{-1}M$ be flat? Where $S = R - \mathfrak{p}$, $\mathfrak{p}$ is a prime ideal. Since localization keeps the exactness, I find that what I ...
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1answer
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Does every short exact sequence split?

If we have a short exact sequence $0 \to A \to B \to C \to 0$, then the map $A \to B$ is injective and the map $B \to C$ is surjective. Therefore, there always exists a left inverse for $i$ and a ...
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Exact sequence of groups: proof of injectivity

There must be a duplicate being the question very introductory, but I was not able to find it. We have the following diagram $$\begin{array}{ccccccccc} 1&\to& H &\to & G &\...
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Exact sequence in cohomology while studying orientability of Alexandrov spaces

I'm reading the article Orientability and fundamental classes of Alexandrov spaces with applications by Ayato Mitsuishi. There are the following definitions: for $n\geq 1$, an $n$-dimensional MCS ...
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Looking for the morphisms of $K[X]$-modules

Suppose K is a field and p is a prime number. Find the morphisms that gives the s.e.s $0 \to K \to M \to Q \to 0$ where $K = K[X]/(X)$, $M = K[X]/(X(X-p))$, $Q = K[X]/(X-p)$ and decide if the s.e.s. ...
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How to prove a sum formula about powers of 2 and 3?

I discovered this yesterday and I am wondering: Is this a well known formula? How do I go about proving something like this? (Beyond induction.) $$2^{2n} = 3^n+\sum_{k=0}^{n-1}3^{n-k-1}2^{2k}$$ ...
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Isomorphism between injective resolution

When reading homological theory, I confront with a statement as follow. In an exact sequence of $A$-modules: $0\to M\to U_0\to U_1\to \dots\to U_{n-1}\to C\to0$ with all $U_i$ injective, and let $...
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Modifying long exact sequences

Let $$\dots A_i\stackrel {f_i}\to B_i \stackrel {g_i}\to C_i \stackrel {h_i}\to A_{i+1}\to \dots$$ be a long exact sequence of Abelian groups. Is it true that if there are maps $k_i:D_i\to E_i$ such ...
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Is this an exact sequence split?

I am reading about split exact sequence and come out with a question. Consider the following sequence. $0 \to A\to A\oplus B\oplus C\to B\to0$ It seems to me that it's exact. So, is it an split ...
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Neukirch's interpretation of unit group and class group

In page 22 from Neukirch's Algebraic Number Theory, he defines the class group $Cl_K$ of a number field $K$ to be the quotient of group of fractional ideals $J_K$ by the subgroup of principal ideals $...
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Hom(-,N) is left exact if $N \in \mathcal{M}$ and $\mathcal{M}$ is semi simple and indecomposable

I have the following situation: $\mathcal{M}$ is a semi simple, indecomposable module category over a semisimple, rigid monoidal category $\mathcal{C}$ with finitely many irreducible objects and ...
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Split Sequences. What is the Group?

See page 49 of this book. Proposition 3.22 An extension (17) splits if $N$ is complete. In fact, $G$ is then direct product of $N$ with the centralizer of $N$ in $G$. I believe (17) refers to $$...
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An exact sequence of abelian groups

Consider an exact sequence of abelian groups $$ 0 \to A \to \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \to B \to 0, $$ where we make no assumption on the map $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{...
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Proof-verification: Show that there exists an exact sequence

Let $R$ be a ring and let $$ M_1 \stackrel{f_1}{\hookrightarrow} M_2 \\ \alpha_1\downarrow \hspace{1cm} \downarrow \alpha_2 \\ N_1 \stackrel{f_2}{\hookrightarrow} N_2 $$ be a commutative diagram of $...
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Counterexample: surjective module homomorphism in an exact sequence

I need to give an example to show that the exactness of $0 \to M_1 \to M_2 \to M_3 \to 0$ need not imply that the map $\operatorname{Hom}_R(N,M_2) \to \operatorname{Hom}_R(N,M_3)$ is surjective, where ...
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Exact Sequence (Ostrik)

It was very helpful when I asked the first part of the proof, so maybe someone can help me with the second one too. I am reading in https://arxiv.org/abs/math/0111139 Theorem 1 on page 10. The ...
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Show a sequence of R-modules and R-homomorphisms is exact. [duplicate]

Suppose given a exact sequence, $$0 \to A\xrightarrow{\enspace f\enspace} B\xrightarrow{\enspace g\enspace} C\to 0$$ I want to show the sequence,$\DeclareMathOperator{\Hom}{Hom}$ $$0→\Hom_R(N,A) \...
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Asymmetry in the notion of exactness

$\newcommand{\im}{\operatorname{im}}$We all know that, in an abelian category, a sequence $A\xrightarrow{f} B\xrightarrow{g} C$ is called exact if $\im f=\ker g$. The one inclusion $\im f\subseteq\ker ...
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Homology of Klein bottle with Mayer-Vietoris

I'm practicing with using the Mayer-Vietoris sequence, and found this computation. I thought it would be a good exercise to try cutting the Klein bottle into two cylinders, instead of into two mobius ...
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Is there a Mayer-Vietoris sequence for an (uncountably) infinite collection of sets?

In Allen Hatcher's Algebraic Topology, Van Kampen's theorem is stated for a (possibly uncountably infinite) collection of path-connected open sets $A_\alpha$ whose union is some topological space $X$. ...
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$M$ and $N$ flat, then $M\otimes N$ flat

I want to show that if $M$ and $N$ are flat $R$-modules, then $M\otimes_R N$ is flat. By flat we mean that if $0\to A\to B$ is exact, then $0\to A\otimes_RM\to B\otimes_RM$ is exact. I am assuming ...
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Show that $Ax=0$.

I need a hint to help me get started with this problem: Given the sequence of homomorphisms, $\mathbb{Z}^{m}\to \mathbb{Z}^{n} \to M\to 0$, where $M=\mathbb{Z}^{n}/K$ and $K=im(\phi_{A})\subseteq \...
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Abelian Group In The Middle Of A Short Exact Sequence

Let $p$ be a prime number. Determine all isomorphism classes of abelian groups $A$ that can appear as the middle term of a short exact sequence: $$0 \rightarrow \mathbb{Z}/(p^a) \rightarrow A \...
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Generating a sequence with resticted growth(trying to undering the math notaion and procedure )

let's say we have a sequence has the restricted growth property if it is a sequence of positive integers $a_{1}, a_{2}, a_{3}....a_{n}$ such that: $a_{1}=1.$ $a_{n+1} \leq$ Max$_{1\leq i \leq n}$ ($...
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$R$ has an identity and $D$ is a projective unitary right $R$-module then sequence of corresponding tensor products is a short exact sequence

$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is a short exact sequence of left $R$-modules and $D$ a right $R$ module. $0 \rightarrow D \otimes_R A \xrightarrow{1_D \otimes\ f} D \...
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Homology of the complex of exact sequences of homotopy groups

By naturality of the connecting homomorphism, relative homotopy $\pi_{\bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences ...
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What is $Ext^1(\overline{\mathbb{Q}}^\times, \overline{\mathbb{Q}})$ in abelian groups?

I want to find a way to describe all the extensions of $\overline{\mathbb{Q}}^\times$ by $\overline{\mathbb{Q}}$, i.e., all the abelian groups $A$ (and the maps $\alpha$ and $\beta$) that fit into the ...
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(Solved) Exact functors commute with homology

I want to show that for any exact functor $F\colon {}_R\mathrm{Mod}\rightarrow {}_S\mathrm{Mod}$ there is a natural isomorphism $$F\circ H_n \cong H_n \circ \mathbf{Ch}(F)$$ where $\mathbf{Ch}(F)$ ...
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Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a ...
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C* algebra exact sequences and ideals

if you have C* algebras $A,B$ and $C$ and $\exists$ a short exact sequence as follows $0\rightarrow A\rightarrow B \rightarrow C \rightarrow 0 $ where the functions are $\phi$ and $\psi$ respectively, ...
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Is additivity necessary for a left exact functor to preserve pullbacks?

I'm having a bit of difficulty with exercise 5.16 from Rotman's An Introduction to Homological Algebra (second edition). The exercise (at least the relevant part) reads Prove that every left exact ...
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1answer
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A module $B$ is flat if Tor $= 0$

From Weibel's "An Introduction to Homological Algebra": Exercise 3.2.1: An $R$-module $B$ is flat if Tor$_i^R(A,B) = 0$ for every $R$-module A. It seems to me that the obvious way to do this would ...
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Problem on exact sequences

Let for all $A$-modules $M$, $0\rightarrow \operatorname{Hom}(M,N^{\prime})\xrightarrow{\overline{f}}\operatorname{Hom}(M,N)\xrightarrow{\overline{g}}\operatorname{Hom}(M,N^{\prime\prime})$ be exact. ...
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1answer
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A question on exact sequences [duplicate]

Let $M^{\prime}\xrightarrow{f}M\xrightarrow{g}M^{\prime\prime}\rightarrow 0 \DeclareMathOperator{\Hom}{Hom}$ be a sequence of $A$-modules and homomorphisms. I want to show that the sequence is exact ...
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For which abelian groups $G$ is there a short exact sequence $0 \rightarrow \mathbb{Z}/p^2 \rightarrow G \rightarrow \mathbb{Z}/p^2 \rightarrow 0$?

I am trying to find for which abelian groups $G$ is there a short exact sequence. $0 \rightarrow \mathbb{Z}/p^2 \rightarrow G \rightarrow \mathbb{Z}/p^2 \rightarrow 0$? I have reasoned as follows: ...
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$\forall\, 0\to N\to M\to F\to 0 $ exact, $\forall R$-module $E$, $F$ is flat $\implies 0 \to N\otimes E\to M\otimes E\to F\otimes E \to0$ is exact.

In $R$-mod category, $\forall\, 0\to N\to M\to F\to 0 $ is short exact sequence, $\forall R$-module $E$. $\forall\, 0\to N\to M\to F\to 0 $ exact, $\forall R$-module $E$, $F$ is flat $\implies 0 \to ...
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Pushforward of projection map exact?

Let $$0\to F \to G \to H\to 0$$ be a short exact sequence of coherent sheaves on the product spaces $X\times Y$, and $p$ be the projection to $X$. Is it true that $$0\to p_*F \to p_*G \to p_*H\to 0$$ ...
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Wang exact sequence with base space homology sphere

Let $F\rightarrow E\rightarrow S^n$, $n\geq 2$, be a fibration. Then we have the Wang exact sequence, $$ \cdots\rightarrow H_q(F)\rightarrow H_q(E)\rightarrow H_{q-n}(F)\rightarrow H_{q-1}(F)\...
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sections of regular functions $\mathcal{O}_X$ of the sphere $X = \{x^2 + y^2 + z^2 - w^2 = 0\}$ in projective space $\mathbb{P}^3$ [closed]

I am trying to understand the sheaf of regular functions $\mathcal{O}_X$ in the case of the sphere. The machinery seems rather difficult to set up. Let't try using the following proposition from the ...