The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. 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.
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:
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: ker ( f ) ⟶ ker a ⟶ ker b ⟶ ker c ⟶ d coker a ⟶ coker b ⟶ coker c ⟶ coker g ′ {\displaystyle \ker(f)~{\color {Gray}\longrightarrow }~\ker a~{\color {Gray}\longrightarrow }~\ker b~{\color {Gray}\longrightarrow }~\ker c~{\overset {d}{\longrightarrow }}~\operatorname {coker} a~{\color {Gray}\longrightarrow }~\operatorname {coker} b~{\color {Gray}\longrightarrow }~\operatorname {coker} c~{\color {Gray}\longrightarrow }~\operatorname {coker} g'} where d is a homomorphism, known as the connecting homomorphism. Furthermore, if the morphism f is a monomorphism, then so is the morphism ker a ⟶ ker b {\displaystyle \ker a~{\color {Gray}\longrightarrow }~\ker b} , and if g' is an epimorphism, then so is coker b ⟶ coker c {\displaystyle \operatorname {coker} b~{\color {Gray}\longrightarrow }~\operatorname {coker} c} . The cokernels here are: coker a = A ′ / im a {\displaystyle \operatorname {coker} a=A'/\operatorname {im} a} , coker b = B ′ / im b {\displaystyle \operatorname {coker} b=B'/\operatorname {im} b} , coker c = C ′ / im c {\displaystyle \operatorname {coker} c=C'/\operatorname {im} c} .
Explanation of the name To see where the snake lemma gets its name, expand the diagram above as follows:
and then 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
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