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[parent] proof of 5-lemma (Proof)

First, assume that we are in modules over a ring (this is the most commonly used setting anyways).

The method of proof is what is usually called diagram-chasing.

Let $ a$ be in the kernel of $ \gamma_3$. Then $ d(\gamma_3(a))=\gamma_4(da)=0$, and $ \gamma_4$ is injective, so $ da=0$. By exactness, $ a=da'$ for some $ a'\in A_4$. Now, $ d(\gamma_2a')=\gamma_3a=0$, so $ \gamma_2a'=db$, and by the surjectivity of $ \gamma_1$, $ b=\gamma_1a''$. $ da'''=\gamma_2^{-1}d\gamma_1(a'')=a'$. Thus, $ a=d^2a''=0$. So, $ \gamma_3$ is an injection.

Now, assume $ b$ is not in the image of $ \gamma_3$. $ db\neq 0$, so $ a'=\gamma_4^{-1}db\neq 0$. $ \gamma_5 da'=d^2b=0$, and $ \gamma_5$ is injective, so $ da'=0$, and there exists an $ a''$ such that $ da''=a'$. Thus, $ d(b-\gamma_3a'')=0$. So there is an $ \alpha$ such that $ d\gamma_2 \alpha=b-\gamma_3a''$. Thus, $ \gamma_3(a''+d\alpha)=b$. Thus, $ \gamma_3$ is surjective.

This actually implies the result for all abelian categories, since by the Freyd imbedding theorem, any abelian category is equivalent to a subcategory of modules over a ring. This trick is necessary since the trick above required us to have a notion of elements in the objects of our category, one which doesn't always make sense. The 5-lemma can be proved directly, but the proof is just less enlightening than the one above.



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Cross-references: 5-lemma, category, objects, necessary, subcategory, equivalent, imbedding, abelian categories, implies, surjective, image, injection, injective, kernel, proof, ring, modules
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This is version 2 of proof of 5-lemma, born on 2004-12-12, modified 2007-03-02.
Object id is 6568, canonical name is ProofOf5Lemma.
Accessed 2035 times total.

Classification:
AMS MSC18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)

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