Snake lemma

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The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. A lemma is a mathematically proven proposition, but it has no particular utility on its own; instead, it is used like a building block in constructing other, more complex proofs.

The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms.

Contents

Statement

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:

SnakeLemma01.png

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:

SnakeLemma02.png

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.

Explanation of the name

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

SnakeLemma03.png

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.

Construction of the maps

The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) 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.

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