Diagram chasing lemmas
The five lemma is one of the basic lemmas of homological algebra, useful for example in the construction of the connecting homomorphism in the homology long exact sequence.
Let be an abelian category. Consider a commutative diagram in of the form
where the top and bottom rows are exact sequences. For simplicity we denote all the differentials in both exact sequences by .
Proposition (the lemma on five homomorphisms or the five lemma)
sharp five lemma (essentially the weak four lemma)
If and are epi and is mono, then is epi.
If and are mono and is epi, then is mono.
(weak) five lemma (conjunction of the two statements above)
If and are isos, is epi, and is mono, then is iso.
The four lemma follows immediately from the salamander lemma, as discussed at salamander lemma - impliciations - four lemma. Here is direct proof.
By the Freyd-Mitchell embedding theorem we can always assume that the abelian category is Mod (though this requires the category to be small, one can always take a smaller abelian subcategory containing the morphism in the diagram which is small). Then we can do the diagram chasing using elements in that setup. We prove only 1) as 2) is dual.
Suppose . Since is epi, one can choose an element such that . Now . Since is a monomorphism that means that as well. By the exactness of the upper row, that means there is such that , hence also . We would like that be equal to but this is not so, we just see that and hence by exactness of the lower row there is such that . Since is also epi, there is such that . Now is such that
demonstrating that is in the image of .
Hence is an epimorphism.
Short five lemma
(short five lemma)
Let and be two exact sequences. If a homomorphism makes the diagram
commute, then is an isomorphism.
Apply prop. 1 to the diagram
Short split five lemma
A special case of the five lemma is the short five lemma where the objects above are all zero objects. It may hold in more general setups, sometimes with additional assumptions.
The short split five lemma is a statement usually stated in the setup of semiabelian categories:
(short split five lemma)
Given a commutative diagram
where and are split epimorphisms and and are their kernels, if and are isomorphisms then so is .
Early references of the 5-lemma
- (lemma (5,9) in) D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)
- (prop.1.1, page 5) Henri Cartan, Samuel Eilenberg, Homological algebra, Princeton Univ. Press 1956
- (lemma 3.3 in chapter I) S. MacLane, Homology, Springer 1963, 1975
In nonabelian context
The short 5-lemma also appears in various topological algebra contexts; see for example
- Francis Borceux, Maria Manuel Clementino, Topological semi-abelian categories, Adv. Math. 190 (2005), 425-453 (web)