Five lemma

In mathematics, and specifically Abelian category theory, the five lemma is an argument that arises so often that it has been given a name. It states that, for the following commutative diagram, if the rows are exact, m and p are isomorphisms, l is surjective, and q is injective, then n is an isomorphism:

    f      g      h      j    
A  --> B  --> C  --> D  --> E  
|      |      |      |      | 
|l     |m     |n     |p     |q 
v      v      v      v      v 
A' --> B' --> C' --> D' --> E'. 
    r      s      t      u

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 will denote an inverse image by `.

The subclaims of this argument are known as the four lemma:

(1) If m and p are surjective and q is injective, n is injective.

(2) If m and p are injective and l is surjective, n is surjective.

Proof of (1):

    g      h      j    
B  --> C  --> D  --> E  
|      |      |      | 
|m     |n     |p     |q 
v      v      v      v 
B' --> C' --> D' --> E'. 
    s      t      u

Let c be an element of C Let d be an element of the inverse image p`tn(c), which exists since p is surjective. By commutativity, up(d) = qj(d). Since n(c) = 0 and t and u are homomorphisms, 0 = utn(c) = up(d) = qj(d). Since q is injective, j(d) is in ker j = im h. Let e in C be such that h(e) = d. Then t