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proof of snake lemma
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(Proof)
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Suppose we are given a commutative diagram
with exact rows. We wish to prove that the sequence
is exact.1
First we claim that if any square
is commutative, then there are well-defined morphisms
 and
 . For example, if
 , then the square
must commute, and so the image of  in the top row must be in  . The proof of the claim for cokernels is similar. Thus we have two sequences,
 and  Suppose
 is sent to
 . By exactness,  has a preimage  . Because the diagram
is commutative and the bottom morphism is injective,  and so
 . So the sequence
is exact. The proof of the claim for the cokernel sequence is similar.
So now all we need to do is find a connecting morphism
 such that the resulting sequence is exact at both of those points.
Suppose
 . Then  has at least one preimage in  . So let  and
 be preimages of  . Thus
 and so by exactness has a preimage  . By commutativity of the diagram,  has a preimage  , which is unique by injectivity of the morphism
 . But we know that the square
is commutative. We wish to define
 by
 . Observe that
and so the choice of preimage of  does not affect which cokernel element we ultimately select. So now we have our connecting morphism. By applying this definition we see that
is a complex.
Suppose
 is sent to 0 by the connecting morphism. Thus we have a diagram
which is commutative. Let
 be the image of  under the morphism
 . Exactness of the diagram implies that
 is a preimage of  . But
 . So the kernel-cokernel sequence is exact at
 . The proof that it is exact at
 is similar.
Footnotes
- 1
- This proof was reconstructed without any notes, but the style of the proof is influenced by a presentation by Edgar Enochs of the zig-zag lemma.
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(view preamble)
Cross-references: implies, points, injective, preimage, diagram, complex, similar, cokernels, image, morphisms, well-defined, zig-zag lemma, proof, commutative diagram
There is 1 reference to this entry.
This is version 1 of proof of snake lemma, born on 2004-02-14.
Object id is 5578, canonical name is ProofOfSnakeLemma.
Accessed 6184 times total.
Classification:
| AMS MSC: | 18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes) |
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Pending Errata and Addenda
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