Questions: Abelian Categories and Homological Algebra

Question 1

You want to prove the snake lemma holds for sheaves of abelian groups on a topological space, which is not a module category. You know the proof for abelian groups uses element-chasing. What theorem justifies applying the same element-chasing argument here?

• A. The five lemma, which holds in any category with zero morphisms
• B. The Freyd-Mitchell embedding theorem, which guarantees a full exact embedding of any small abelian category into a module category
• C. The universal coefficient theorem, which reduces sheaf cohomology to module cohomology
• D. The comparison theorem for resolutions, which shows any two resolutions compute the same homology

Answer: (B)

The Freyd-Mitchell embedding theorem states that every small abelian category embeds fully and exactly (preserving and reflecting exactness) into R-Mod for some ring R. This means any diagram-chasing proof valid in R-Mod — including the snake lemma, five lemma, and horseshoe lemma — is automatically valid in every abelian category. You do not need a separate proof for sheaves; you invoke Freyd-Mitchell and transfer the module proof. This is the key that unlocks homological algebra across all abelian categories simultaneously.

Question 2

Which of the following properties is NOT automatically guaranteed by the abelian category axioms, requiring additional hypotheses for certain homological constructions?

• A. Every morphism has a kernel and a cokernel
• B. Every monomorphism is a kernel of some morphism
• C. The category has enough injective objects for right derived functor computation
• D. Short exact sequences 0 → A → B → C → 0 are well-defined

Answer: (C)

The abelian category axioms guarantee kernels, cokernels, the mono=kernel and epi=cokernel conditions, and thus well-defined exact sequences. However, 'enough injectives' — the condition that every object embeds into an injective — is an additional hypothesis required to construct injective resolutions and compute right derived functors. For example, sheaves on a topological space have enough injectives, but this is a theorem, not an axiom. The category of finitely generated modules over a Noetherian ring may lack enough injectives without further assumptions.

Question 3

Most additive category — one where hom-sets are abelian groups and composition is bilinear — is automatically an abelian category.

• T. True
• F. False

Answer: False

Being additive is necessary but far from sufficient for being abelian. An abelian category additionally requires: every morphism has a kernel AND a cokernel, every monomorphism is a kernel (of its cokernel), and every epimorphism is a cokernel (of its kernel). These conditions are non-trivial. The category of free abelian groups is additive but not abelian (cokernels may not be free). The category of Banach spaces with bounded linear maps is additive but not abelian. The kernel/cokernel exactness conditions are what give abelian categories their homological power.

Question 4

In an abelian category, the homology object Hₙ = ker(dₙ)/im(dₙ₊₁) of a chain complex is well-defined because the axioms guarantee that images and kernels exist as subobjects in the required categorical sense.

• T. True
• F. False

Answer: True

In a general category, 'image' and 'kernel' may not be defined or may not be comparable. In an abelian category, every morphism f: A → B has a kernel (an equalizer of f and 0) and an image (defined as the kernel of the cokernel of f). The condition that every monomorphism is a kernel ensures that im(dₙ₊₁) is a legitimate subobject of ker(dₙ) — their categorical quotient Hₙ = ker(dₙ)/im(dₙ₊₁) is then a well-defined object of the category. This is exactly what allows homology to be defined internally without reference to elements.

Question 5

What does the Freyd-Mitchell embedding theorem mean practically for proving a diagram lemma (e.g., the five lemma or horseshoe lemma) in an arbitrary abelian category?

Model answer: It means you can write the proof using element-chasing in R-Mod, as if objects were sets with elements, and the proof is automatically valid in any abelian category. You do not need to reprove the lemma for each new abelian category (sheaves, representations, coherent sheaves on a scheme). The embedding is full and exact, so it preserves and reflects all the categorical structure (kernels, cokernels, exact sequences) needed for the proof to transfer. Essentially: prove it once in R-Mod with elements; it holds everywhere abelian.

Before Freyd-Mitchell, results in abelian categories required abstract, element-free arguments that were significantly harder to write and check. The embedding theorem permits the concreteness of module-theoretic reasoning without losing generality. The price is that the embedding is not canonical and may require enlarging universes for large categories — the theorem applies to small abelian categories — but for any fixed diagram lemma, the relevant category is small, so the theorem applies.