The snake lemma Homological algebra

in abelian category (such category of abelian groups or category of vector spaces on given field), consider commutative diagram:

where rows exact sequences , 0 0 object. there exact sequence relating kernels , cokernels of a, b, , c:

ker
⁡
a



⟶


ker
⁡
b



⟶


ker
⁡
c



⟶
d


coker
⁡
a



⟶


coker
⁡
b



⟶


coker
⁡
c


{\displaystyle \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}

furthermore, if morphism f monomorphism, morphism ker a → ker b, , if g epimorphism, coker b → coker c.