An exact-sequence technique in homological algebra that, given a commutative diagram of short exact sequences, produces a long exact sequence relating kernels and cokernels (or homology groups) via a connecting homomorphism constructed by a diagram chase.

Principle

From a commutative square of modules (or objects in an abelian category) with exact rows, one performs a diagram chase to define a connecting map from the kernel of the right-hand vertical map to the cokernel of the left-hand vertical map; splicing these maps yields a long exact sequence linking kernels and cokernels.

Demonstration

Applied to a short exact sequence of chain complexes, the Snake Lemma yields the long exact sequence in homology: kernels become cycles, cokernels become boundaries modulo images, and the connecting homomorphism identifies classes that obstruct lifting cycles across the short exact sequence.

Misapplication

Using the lemma when rows are not exact, the diagram is noncommutative, or the ambient category lacks kernels and cokernels (i.e., is not abelian) invalidates the construction; sign errors or sloppy chasing can also produce incorrect connecting maps.

Consequence

Supplies a systematic way to relate local algebraic information (kernels and cokernels) across exact diagrams, underpinning many constructions in homological algebra such as the long exact sequence in homology and proofs of five-lemma type results.

Reversal

The dual statement interchanges kernels and cokernels and applies in the dual abelian category, producing analogous long exact sequences for cohomology; thinking in duals clarifies that the lemma is symmetric under categorical duality.

Boundary

Requires a commutative diagram with exact rows in an abelian category (modules, abelian groups, sheaves, etc.); it does not hold verbatim in non-abelian contexts without modifications and fails if exactness or commutativity hypotheses are dropped.

Semantic Tension

Tension exists with simpler diagram lemmas (e.g., the five lemma or nine lemma): the Snake Lemma constructs connecting homomorphisms and long exact sequences, while the five lemma gives isomorphism criteria once long sequences are established—they are complementary tools with overlapping domains.

Synthesis

The Snake Lemma is a diagram-chase device that turns a commutative square of short exact sequences into a long exact sequence connecting kernels and cokernels via a canonical connecting morphism, enabling transfer of algebraic invariants across short exact diagrams.