High Quality Content by WIKIPEDIA articles! The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a connecting homomorphism d exists which completes the exact sequence. In the case of abelian groups or modules over some ring, the map d can be constructed as follows. Pick an element x in ker c and view it as an element of C; since g is surjective, there exists y in B with g(y) = x. Because of the commutativity of the diagram, we have g'(b(y)) = c(g(y)) = c(x) = 0 (since x is in the kernel of c), and therefore b(y) is in the kernel of g' . Since the bottom row is exact, we find an element z in A' with f '(z) = b(y). We then define d(x) = z + im(a). Now one has to check that d is well-defined.

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