In the snake lemma, the connecting morphism δ maps between which objects?
AFrom coker α to ker γ, providing a bridge from the top row to the bottom row
BFrom ker γ to coker α, crossing from the bottom-right kernel to the top-left cokernel
CFrom ker β to coker β, staying within the middle column of the diagram
DFrom H_n(C) to H_{n-1}(A), directly producing the long exact sequence in homology
The six-term exact sequence produced by the snake lemma is: ker α → ker β → ker γ →^δ coker α → coker β → coker γ. The connecting morphism δ maps from ker γ (bottom-right) to coker α (top-left) — it 'snakes' diagonally across the diagram, which is why the lemma has its name. Option D describes what δ becomes when applied to chain complexes, but in the raw snake lemma, δ connects ker γ to coker α, not homology groups.
Question 2 Multiple Choice
The well-definedness of δ(x) — that different choices of preimage b ∈ B of x ∈ ker γ yield the same class in coker α — depends on which property of the diagram?
AThe commutativity of the right square (involving β and γ) only
BThe exactness of the top row: any two preimages differ by an element of ker(B → C) = im(A → B), which maps into im(A' → B') via commutativity, giving the same class in coker α
CThe injectivity (monomorphism) of the map B → C in the top row
DThe surjectivity (epimorphism) of the map B' → C' in the bottom row
If b and b' are both preimages of x ∈ ker γ under the surjection B → C, then b - b' ∈ ker(B → C) = im(A → B) by exactness of the top row at B. So b - b' = f(a) for some a ∈ A. Applying β: β(b) - β(b') = β(f(a)) = g(α(a)) by commutativity, where g: A' → B' is the bottom-row injection. So α(a) ∈ A' maps to the same element whether we started with b or b', and the cokernel class [α(a)] ∈ coker α is well-defined. Exactness at B is essential; without it, the preimage difference need not land in im(A → B), and the construction fails.
Question 3 True / False
The six-term sequence ker α → ker β → ker γ →^δ coker α → coker β → coker γ produced by the snake lemma is exact at every term.
TTrue
FFalse
Answer: True
This is the full content of the snake lemma: not only does the connecting morphism δ exist, but the resulting six-term sequence is exact throughout — at ker β, at ker γ, at coker α, and at coker β. Exactness at each term requires a separate argument: at ker γ, one shows im(ker β → ker γ) = ker δ; at coker α, one shows im δ = ker(coker α → coker β); and so on. Together, these exactness conditions are what make the lemma powerful: they guarantee that no information is lost or duplicated as you move through the sequence.
Question 4 True / False
The snake lemma can be applied to any commutative diagram of abelian groups with morphisms between them, even if the rows of the diagram are not exact sequences.
TTrue
FFalse
Answer: False
Exactness of both rows is essential, not optional. Every step of the connecting morphism's construction uses exactness: the existence of the preimage b ∈ B (uses surjectivity B → C, hence exactness of the top row at C), the fact that β(b) lies in ker(B' → C') (uses commutativity and ker γ ∋ x), the existence of a' ∈ A' mapping to β(b) (uses exactness of the bottom row at B'), and the well-definedness argument (uses exactness of the top row at B). Without exact rows, none of these existence or uniqueness claims hold, and the connecting morphism cannot be defined.
Question 5 Short Answer
Why is the snake lemma described as the 'engine' that produces long exact sequences in homology from short exact sequences of chain complexes?
Think about your answer, then reveal below.
Model answer: Given a short exact sequence of chain complexes 0 → A_• → B_• → C_• → 0, at each degree n there is a commutative diagram with exact rows: the top row involves the boundary maps of A_n and B_n, the bottom row involves B_n and C_n. Applying the snake lemma to this diagram produces a connecting morphism δ_n: ker(∂_C at C_n) → coker(∂_A at A_{n-1}). Interpreting these in terms of homology (ker ∂ / im ∂ = H_n), the connecting morphisms become δ_n: H_n(C) → H_{n-1}(A). The snake lemma's exactness then stitches together the fragments H_n(A) → H_n(B) → H_n(C) from each degree with the connecting morphisms into the long exact sequence ⋯ → H_n(A) → H_n(B) → H_n(C) →^δ H_{n-1}(A) → ⋯. Without the snake lemma's existence and exactness guarantee, there would be no systematic bridge between short exact sequences and long exact sequences of invariants.
This application — deriving long exact sequences in homology — is why the snake lemma appears at the very beginning of algebraic topology and homological algebra. It is not merely a technical result; it is the mechanism that makes the theory computationally useful.