Short five lemma
In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.
It states that for the following commutative diagram, if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.
It follows immediately from the five lemma.
The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B′, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A′ of B′ and also an isomorphism from the factor object B/A to B′/A′, then f itself is an isomorphism. Note however that the existence of f has to be assumed from the start; two objects B and B′ that simply have isomorphic sub- and factor objects need not themselves be isomorphic.
