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Snake lemma

In mathematics, especially homological algebra and other applications of Abelian category theory, the snake lemma is a basic tool used in diagram chasing.

In an Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field), consider a commutative diagram

where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c:

Furthermore, if the morphism f is a monomorphism, then so is the morphism ker a → ker b, and if g' is an epimorphism, then so is coker b → coker c.

The maps between the kernels and the maps between the cokernels are induced in a natural manner given the exactness of the rows; the important statement of the lemma is that a connecting homomorphism d exists which completes the exact sequence.

The snake lemma is the crucial tool to construct the long exact sequences of homological algebra.

To see where the snake lemma gets its name, expand the diagram above as follows:

and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering snake.

See also: Five lemma