Diagram chasing lemmas
The four lemma is one of the basic diagram chasing lemmas in homological algebra. It follows directly from the salamander lemma. It directly implies the five lemma.
Let be an abelian category.
Consider a commuting diagram in of the form
the rows are exact sequences,
is an epimorphism,
is a monomorphism.
(the image under of the kernel of is the kernel of )
(the preimage under of the image of is the image of )
(the “strong four lemma”) and hence in particular
if is an epimorphism then so is ;
if is a monomorphism then so is
(the “weak four lemma”).
A direct proof from the salamander lemma is spelled out at salamander lemma – implications – four lemma.
The strong/weak four lemma appears as lemma 3.2, 3.3 in chapter I and then with proof in lemma 3.1 of chapter XII of
- Saunders MacLane, Homology (1967) reprinted as Classics in Mathematics, Springer (1995)
Revised on September 24, 2012 23:46:27
by Urs Schreiber