A diagram-chasing result which, from a commutative diagram of two exact rows and a column of maps, produces a canonical long exact sequence connecting kernels and cokernels, including a connecting homomorphism often denoted δ.

Principle

Element-chasing in a commutative diagram of short exact sequences identifies how kernels map into kernels and cokernels into cokernels and constructs the connecting morphism that links the kernel sequence to the cokernel sequence, yielding exactness throughout the resulting long sequence.

Demonstration

Given a commutative diagram 0→A'→A→A''→0 over R-modules with vertical maps to 0→B'→B→B''→0, the Snake Lemma produces an exact sequence Ker(A'→B')→Ker(A→B)→Ker(A''→B'')→Coker(A'→B')→Coker(A→B)→Coker(A''→B'') and a connecting map δ:Ker(A''→B'')→Coker(A'→B').

Misapplication

Invoking the Snake Lemma when rows are not exact, diagrams fail to commute, or in categories without kernels/cokernels invalidates the construction; assuming naturality of δ without checking functorial hypotheses can also mislead.

Consequence

Provides the fundamental tool to derive long exact sequences in homology and cohomology, construct connecting homomorphisms, and compare derived functors under exact functors or short exact sequences.

Reversal

One can sometimes reconstruct a commutative short-exact diagram from a given long exact sequence, but this inversion is noncanonical and requires extra choices; the Snake Lemma itself runs from diagram to long sequence, not the reverse.

Boundary

Works in abelian categories or any category with kernels and cokernels where short exact sequences induce long exact connecting sequences; it is not directly available in arbitrary nonabelian contexts.

Semantic Tension

Close to the construction of connecting morphisms in derived functor long exact sequences and to various mapping-cone constructions; tension lies between the elementary element-chasing formulation and more abstract derived-category interpretations.

Synthesis

The Snake Lemma turns a commutative diagram of short exact sequences into a canonical long exact sequence of kernels and cokernels with a connecting morphism, enabling passage from local exactness data to global exact relationships used throughout homological algebra.