Homological algebra MOC

Five lemma

If the following diagram commutes in with both rows exact

https://q.uiver.app/#q=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

and are isomorphisms, epimorphism, and monomorphism then is an isomorphism.1 #m/thm/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,

#state/tidy | #lang/en | #SemBr

Footnotes

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