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Five lemma

In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is valid not only for abelian categories but also works in group theory, for example.

The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual[?] to each other.

Statement

Consider the following commutative diagram in any Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field), or in the category of groups.

The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism.

The 2 four lemmas state:

If m and p are epimorphisms and q is a monomorphism, then n is a monomorphism.
If m and p are monomorphisms and l is an epimorphism, then n is an epimorphism.

Proof

The method of proof we shall use is commonly referred to as diagram-chasing[?]. Although it may boggle the mind at first, once one has some practice at it, it is actually fairly routine. We shall prove the five lemma by individually proving each of the 2 four lemmas.

To perform diagram chasing, we assume that we are in a category of modules over some ring, so that we may speak of elements of the objects in the diagram and think of the morphisms of the diagram as functions (in fact, homomorphisms) acting on those elements. Then a morphism is a monomorphism if and only if it is injective, and it is an epimorphism iff it is surjective. Similarly, to deal with exactness, we can think of kernels and images in a function-theoretic sense. The proof will still apply to any Abelian category because of Mitchell's embedding theorem[?], which states that any Abelian category can be represented as a category of modules over some ring. For the category of groups, just turn all additive notation below into multiplicative notation, and note that commutativity is never used.

So, to prove (1), assume that m and p are surjective and q is injective.

Let c' be an element of C'.
Let d be an element of the inverse image under p of t(c'); d exists since p is surjective.
By commutativity of the diagram, u(p(d)) = q(j(d)).
Since im t = ker u by exactness, 0 = u(t(c')) = u(p(d)) = q(j(d)).
Since q is injective, j(d) = 0, so d is in ker j = im h.
Let c in C be such that h(c) = d.
Then t(n(c)) = p(h(c)) = t(c'), so t(c' − n(c)) = 0.
By exactness, c' − n(c) must be in the image of s; let b' be an element of the inverse image of c'.
Since m is surjective, we can find b in B such that b = m(b').
By commutativity, n(g(b)) = s(m(b)) = c' − n(c).
Since n is a homomorphism, n(g(b) + c) = n(g(b)) + n(c) = c' − n(c) + n(c) = c'.
Therefore, n is surjective.

Then, to prove (2), assume that m and p are injective and l is surjective.

Let c in C be such that n(c) = 0.
t(n(c)) is then 0.
By commutativity, p(h(c)) = 0.
Since p is injective, h(c) = 0.
By exactness, there is an element b of B such that g(b) = c.
By commutativity, s(m(b)) = n(g(b)) = n(c) = 0.
By exactness, there is then an element a' of A' such that r(a') = m(b).
Since l is surjective, there is a in A such that l(a) = a'.
By commutativity, m(f(a)) = r(l(a)) = m(b).
Since m is injective, f(a) = b.
So c = g(f(a)).
By exactness, ker g = im f, so c = 0.
Therefore, n is injective.

Combining the 2 four lemmas now proves the entire five lemma.