# Questions tagged [exact-sequence]

A sequence of morphisms where the image of one is the kernel of the next. It is useful thing to examine in the study of abstract algebra and homological algebra. For sequences of numbers use the tag (sequences-and-series) instead.

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### Is there a way of speaking about exact sequences in a category “nice enough” but without zero object?

I have a concrete complete and cocomplete category $\mathcal{A}$ (in fact, a category of modules over a monoid in a symmetric closed monoidal category) in which all monomorphisms and all epimorphisms ...
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### Reverse sequence of a short exact sequence is short exact sequence

Why is the reverse sequence of a given short exact sequence also exact? (I understand that the map on the left will be one-to-one and on the right will be onto but how to see exactness of the module ...
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### Properties of an A-module

I must show that the following properties for an $A$-module $P$ are equivalent: 1) The functor $Hom(P,-)$ is exact. 2) There is an $A$-module $Q$ such that $P \oplus Q$ is free. 3) Every short ...
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### Commutative Diagram of Vector Spaces with Short Exact rows

I found the following question in an online book of Linear Algebra exercises (page 69) All of the mathematical objects are vector spaces, and thus the maps are linear. It does not specify if the ...
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### Suppose that $g$ is isomorphism. Then prove that $f$ is monic and $h$ is epic.

$$\\ 0 \to A \to B \to C \to 0 \\ 0 \to A' \to B' \to C' \to 0$$ These are exact and they occur a commutative diagram by homomorphism. $$g=B\to B'\\ f=A\to A'\\ h=C \to C'$$ Suppose that $g$ is ...
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### Prove that there exists the exact sequence (over Z) 0 → Z 2 →Z 4 → Z 4 → Z 2 → 0 .

$$0 \to \mathbb Z /2 \to \mathbb Z /4 \to \mathbb Z /4 \to \mathbb Z / 2 \to 0$$ To show that it is exact, I need $f : \mathbb Z / 2 \to \mathbb Z / 4$ monic, and $g : \mathbb Z /4 \to \mathbb Z /2$ ...
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### In the following commutative diagram, with exact rows and columns, is $j_1$ surjective?

I have the following commutative digram of abelian groups and homomorphisms between them. The rows and columns are exact: \begin{array} A & && && & & & & 0 &...
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### Clarification in early example of exact sequence in Eisenbud's Commutative Algebra

It feels like a lot of shortcuts are being take here: "The kernel is generated by the class of $a$ modulo $I$." So, is the kernel just $(a+I)$? "Since the kernel is generated by just one element, ...
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### Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group?

My professor made the following remark while teaching about group extensions: We want to classify finite groups in a manner similar to the fact that every positive integer is uniquely a product of ...
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### Showing that the extension $1 \to C_3 \to C_6 \to C_2 \to 1$ is split by finding complement of $C_3$

I'm trying to solve the following exercise: Show that the extension $1 \to C_3 \to C_6 \to C_2 \to 1$ is split but the extension $1 \to C_2 \to C_4 \to C_2 \to 1$ is not split. I think one way to ...
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### Proving exactness not at the connecting hom in a Snake Lemma subproblem.

Here's a picture of the Snake Lemma from nLab: I'm having trouble showing exactness at $\ker g$ and haven't even got to the connecting hom yet. So let's focus on $\ker g$. Let $q : A' \to B'$ be ...
### Two possibilities of $X$ to make the sequence $0\longrightarrow\mathbb{Z}\longrightarrow X\longrightarrow\mathbb{Z}_{2}\longrightarrow 0$ exact.
I am working on an exercise in homological algebra asking to find the possibilities of $X$ such that the sequence 0\longrightarrow\mathbb{Z}\longrightarrow X\longrightarrow\mathbb{Z}_{2}\...
From Rotman's Algebraic Topology If $A$ is a retract of $X$, then $H_n(X)\approx H_n(A) \oplus H_n(X,A)$ If $r\colon X \rightarrow A$ is the retract with inclusion $i$, then define \$f \colon H_n(X)...