Five lemma

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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 the category of groups, 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.



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 two four-lemmas state:
(1) If the rows in the commutative diagram


are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism.

(2) If the rows in the commutative diagram


are exact and m and p are monomorphisms and l is an epimorphism, then n is a monomorphism.


The method of proof we shall use is commonly referred to as diagram chasing.[1] 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 if and only if 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 (small) abelian category because of Mitchell's embedding theorem, which states that any small 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.

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