The Snake Lemma

Core Idea

The snake lemma is the fundamental diagram-chasing result in homological algebra. Given a commutative diagram of abelian groups with exact rows, it produces a connecting homomorphism between the kernels and cokernels of the vertical maps, forming a long exact sequence. The snake lemma is the algebraic engine that produces every long exact sequence in homology and cohomology — the long exact sequence of a pair, Mayer-Vietoris, the long exact sequence of a fibration in homotopy — all are instances of the snake lemma applied to appropriate short exact sequences of chain complexes.

Explainer

The snake lemma starts with a commutative diagram of abelian groups with exact rows:

0 -> A -a-> B -b-> C -> 0

|f |g |h

0 -> A' -a'-> B' -b'-> C' -> 0

The lemma asserts the existence of an exact sequence: ker(f) -> ker(g) -> ker(h) -delta-> coker(f) -> coker(g) -> coker(h). The maps between kernels and between cokernels are induced by the horizontal maps (restricted or projected). The connecting homomorphism delta : ker(h) -> coker(f) is the new and essential ingredient — it "snakes" from the right side of the kernel row to the left side of the cokernel row, connecting the two halves of the exact sequence.

The construction of delta by diagram chasing is the prototypical example of this technique. Given c in ker(h) (so c in C with h(c) = 0): since b : B -> C is surjective, choose b_0 in B with b(b_0) = c. Now g(b_0) in B' has the property that b'(g(b_0)) = h(b(b_0)) = h(c) = 0 (by commutativity and the assumption c in ker(h)). So g(b_0) is in ker(b') = im(a'). Choose a' in A' with a'(a') = g(b_0). Define delta(c) = [a'] in coker(f) = A'/im(f). The proof that this is well-defined (independent of the choices of b_0 and a') and that the resulting sequence is exact is a series of straightforward but careful diagram chases.

The snake lemma produces long exact sequences from short exact sequences of chain complexes. Given a short exact sequence of chain complexes 0 -> A_* -> B_* -> C_* -> 0 (exact at each level), the snake lemma applies to the diagram formed by the boundary maps. The connecting homomorphism delta : H_n(C) -> H_{n-1}(A) is exactly the snake lemma's delta applied to an appropriate diagram. Splicing these together across all dimensions gives the long exact sequence: ... -> H_n(A) -> H_n(B) -> H_n(C) -> H_{n-1}(A) -> H_{n-1}(B) -> ... This is a universal construction: every long exact sequence in algebraic topology (the LES of a pair, Mayer-Vietoris, the LES of a fibration, the Gysin sequence, the Wang sequence) is an instance of this pattern.

The snake lemma is the workhorse of diagram chasing, a proof technique that manipulates elements through commutative diagrams by following arrows and using exactness to deduce properties. While diagram chasing can feel mechanical, it is powerful: it converts topological questions (how do the homology groups of A, X, and X/A relate?) into routine algebraic verifications. The snake lemma, the five lemma, and the nine lemma form the basic toolkit of diagram chasing. In more advanced settings, these lemmas generalize to abelian categories (where elements may not exist), leading to the theory of derived categories and spectral sequences — but the snake lemma remains the conceptual prototype.