Homological algebra MOC

Five lemma

If the following diagram commutes in with both rows exact

and are isomorphisms, epimorphism, and monomorphism then is an isomorphism.¹ homology

Proof

The proof involves proving the two “four lemmata”, by Diagram chasing. We will use additive notation for group operations, but the groups in question need not be abelian.

First we use the fact that are epic and is monic to show that is epic.

1. Let
2. By epi for some
3. By commutativity
4. By exactness
5. By mono
6. By exactness
7. Thus for some
8. Thus
9. Thus
10. By homo
11. By exactness
12. Thus for some
13. By epi for some
14. By commutativity
15. By homo

Therefore is epic. Now we will use the fact that are monic and is epic to show that is monic.

1. Let , so
2. By homo
3. By commutativity
4. By mono
5. By exactness
6. Thus for some
7. By commutativity
8. By exactness
9. Thus for some
10. By epi for some
11. By commutativity
12. By mono
13. By exactness

Therefore is monic.

Every Module is a group, and every abelian category has a representation as a module category (Freyd-Mitchell theorem), so the lemma holds for module and abelian categories,

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Footnotes

1. 2010, Algebraische Topologie, ¶3.1.10, p.130ff ↩