# Proof of the five lemma

How to do this using the snake lemma? this is an exercise in Lang's Algebra book. It should somehow be obvious, but I don't see it

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I think usually they are proven independently. – Alexander Grothendieck Dec 8 '13 at 22:08
@OOO I guess you're right; I must have been thinking about something else. – Magdiragdag Dec 8 '13 at 22:10
You can easily do the short five lemma with the snake lemma – Hagen von Eitzen Dec 8 '13 at 22:13
I didn't want to say that spectral sequences are "the" appropriate tool here, but rather wanted to inform you how to correct your statement (which you have deleted meanwhile, so that my comment doesn't make any sense anymore, thanks). – Martin Brandenburg Dec 8 '13 at 23:50
@HagenvonEitzen The usual five lemma follows from the short five lemma: factor each morphism appearing in the rows into an epimorphism followed by a monomorphism. – Zhen Lin Dec 9 '13 at 0:40

## 1 Answer

The proof can be found in Bourbaki's Algèbre homologique, §1, Cor. 3.

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excellent, thanks! I am surprised about the downvote, but this is probably due to a frustrated loser, so don't be discouraged! – user88576 Dec 9 '13 at 19:36