A sequence A →^f B →^g C satisfies g∘f = 0 (the composition is the zero map). Does this guarantee the sequence is exact at B?
AYes — g∘f = 0 is the definition of exactness at B
BYes — the zero composition is equivalent to im(f) = ker(g) in any abelian category
CNo — g∘f = 0 only implies im(f) ⊆ ker(g), not necessarily im(f) = ker(g)
DNo — exactness at B additionally requires that f is injective and g is surjective
g∘f = 0 means every element in the image of f is sent to 0 by g, which is exactly the statement im(f) ⊆ ker(g). But exactness requires the stronger condition im(f) = ker(g) — every element that g kills must have come from f. A sequence satisfying g∘f = 0 is called a chain complex; an exact sequence is a chain complex with the additional constraint that there is no 'extra' kernel. The difference matters: the homology of a chain complex measures exactly this gap, H = ker(g)/im(f), which is trivial precisely when the sequence is exact.
Question 2 Multiple Choice
In the short exact sequence 0 → ℤ →^×2 ℤ →^mod2 ℤ/2ℤ → 0 (where ×2 is multiplication by 2 and mod2 is reduction mod 2), which statement best describes what exactness at ℤ (the middle term) tells us?
AThe map ×2 is an isomorphism from ℤ to ℤ
BEvery integer that maps to 0 under mod2 (i.e., every even integer) is in the image of ×2
CThe only integer sent to 0 by ×2 is 0 itself
Dℤ is isomorphic to the direct sum ℤ ⊕ ℤ/2ℤ
Exactness at the middle ℤ requires im(×2) = ker(mod2). The image of ×2 is the even integers {…, -4, -2, 0, 2, 4, …}. The kernel of mod2 is also the even integers (those that map to 0 in ℤ/2ℤ). So im(×2) = ker(mod2) = 2ℤ — confirmed exact. Option B correctly identifies this equality. Note that option D is false: this short exact sequence does not split (there is no homomorphism ℤ/2ℤ → ℤ), so ℤ is not isomorphic to ℤ ⊕ ℤ/2ℤ — a key illustration that extension problems are non-trivial.
Question 3 True / False
In a short exact sequence 0 → A →^f B →^g C → 0, exactness at A forces f to be injective, and exactness at C forces g to be surjective.
TTrue
FFalse
Answer: True
Exactness at A means im(0 → A) = ker(f). The image of the zero map into A is {0}, so ker(f) = {0}, which means f is injective. Exactness at C means im(g) = ker(C → 0). The kernel of the zero map out of C is all of C, so im(g) = C, which means g is surjective. These are not extra assumptions — they are forced by exactness at the endpoints. This is why short exact sequences 0 → A → B → C → 0 are often described by saying 'f is a monomorphism and g is an epimorphism.'
Question 4 True / False
If 0 → A →^f B →^g C → 0 is a short exact sequence, then B is expected to be isomorphic to the direct sum A ⊕ C.
TTrue
FFalse
Answer: False
This is a classic misconception. Exactness tells you that A embeds into B and C is the quotient B/A, but it does not tell you how B is built from these pieces. B is an extension of C by A, and different short exact sequences with the same A and C (but different B) correspond to genuinely non-isomorphic middle objects. B ≅ A ⊕ C holds only when the sequence splits — meaning there exists a section s: C → B with g∘s = id_C. Whether a given exact sequence splits is a non-trivial question and is precisely what Ext¹(C, A) measures.
Question 5 Short Answer
What does it mean for a sequence to be 'exact at B,' and why is this condition strictly stronger than merely requiring that the composition of consecutive maps is zero?
Think about your answer, then reveal below.
Model answer: A sequence A →^f B →^g C is exact at B if im(f) = ker(g): everything that f maps into B is precisely the collection of elements that g sends to 0. The condition g∘f = 0 only requires im(f) ⊆ ker(g) — every image element is killed by g, but there may be elements in ker(g) that are not in im(f). Exactness demands there are no such 'extra' kernel elements. The gap ker(g)/im(f) — which vanishes exactly when the sequence is exact — is what homology groups measure.
The distinction between chain complexes (g∘f = 0) and exact sequences (im = ker) is foundational to homological algebra. A chain complex can fail to be exact, and that failure is measured by its homology groups. Exact sequences are chain complexes with trivial homology everywhere. This is why long exact sequences in algebraic topology are so powerful: they allow you to relate the homology of spaces in a controlled way, and any deviation from exactness would signal a non-trivial topological feature.