The snake lemma applies to a commutative diagram with exact rows. What are the 'kernel' and 'cokernel' objects in the resulting exact sequence?
Aker(A → A') and coker(C → C')
Bker(f), ker(g), ker(h) and coker(f), coker(g), coker(h), where f, g, h are the vertical maps in the diagram
CThe homology groups of the chain complexes
DThe image and kernel of the horizontal maps
Given a commutative diagram with exact rows 0 → A → B → C → 0 and 0 → A' → B' → C' → 0, and vertical maps f: A → A', g: B → B', h: C → C', the snake lemma produces: ker(f) → ker(g) → ker(h) →^δ coker(f) → coker(g) → coker(h). The connecting homomorphism δ: ker(h) → coker(f) is the 'snake' — it connects the kernel sequence to the cokernel sequence by weaving through the diagram.
Question 2 True / False
The connecting homomorphism δ in the snake lemma is defined by a 'diagram chase.' This means it depends on choices of representatives.
TTrue
FFalse
Answer: False
Although the CONSTRUCTION of δ involves choosing lifts (given c ∈ ker(h), choose b ∈ B mapping to c, then g(b) ∈ B' has image 0 in C', so g(b) comes from some a' ∈ A', and δ(c) = [a'] in coker(f)), the result is independent of the choices made. If we chose a different b' mapping to c, then b - b' is in the image of A → B, and tracking through the diagram shows the two choices give the same element of coker(f). This well-definedness is a key part of the snake lemma's proof.
Question 3 Short Answer
How does the snake lemma produce the long exact sequence of a pair (X, A) in homology?
Think about your answer, then reveal below.
Model answer: The short exact sequence of chain complexes 0 → C_*(A) → C_*(X) → C_*(X)/C_*(A) → 0 gives, for each n, a commutative diagram with the boundary maps as vertical arrows. Applying the snake lemma to this diagram (with appropriate identifications) produces the connecting homomorphism ∂: H_n(X, A) → H_{n-1}(A) and the exact sequence H_n(A) → H_n(X) → H_n(X, A) → H_{n-1}(A) → H_{n-1}(X) → .... Iterating across all dimensions and splicing together gives the full long exact sequence of the pair.
More precisely, the snake lemma is applied to the diagram where the rows are the short exact sequences of cycles and boundaries (derived from the chain complex SES), and the vertical maps come from the inclusion of boundaries into cycles. The connecting homomorphism emerges from chasing elements through the diagram, exactly as in the abstract snake lemma. This is the universal mechanism: every long exact sequence in homological algebra arises from the snake lemma applied to a short exact sequence of chain complexes.
Question 4 True / False
In the movie 'It's My Turn' (1980), the snake lemma is proved on a blackboard. This reflects its status as a foundational result in homological algebra.
TTrue
FFalse
Answer: True
The snake lemma is indeed proved in the opening scene of the 1980 film 'It's My Turn' starring Jill Clayburgh as a mathematics professor. This pop-culture appearance reflects the lemma's status as perhaps the single most important technical result in homological algebra — it is the lemma that makes long exact sequences possible. Every textbook on algebraic topology or homological algebra proves the snake lemma, and it is often the first 'diagram chase' students encounter.