Questions: Injective Objects and Injective Envelopes

Question 1

An object I is injective in an abelian category. Given a monomorphism i: A ↪ B and a morphism f: A → I, what does injectivity guarantee?

• A. That f is an isomorphism from A onto I
• B. That there exists a morphism f̃: B → I such that f̃ ∘ i = f — the map from the subobject A extends to all of B
• C. That every morphism from B factors through A via the monomorphism i
• D. That I has no proper subobjects other than the zero object

Answer: (B)

Injectivity is precisely the extension property: any morphism into I from a subobject A can always be extended to the ambient object B. The injective object I is an 'extensible target' — it never blocks extending morphisms from subobjects. This is the categorical generalization of divisibility: just as every homomorphism from a subgroup into ℚ can be extended to the whole group, injectivity guarantees extensions in any abelian category.

Question 2

Why is ℚ injective as a ℤ-module, while ℤ itself is not?

• A. ℚ contains ℤ as a subgroup, and containing a non-injective subobject forces injectivity on the ambient object
• B. ℚ is a field, and all fields are automatically injective over their prime subfields
• C. ℚ is divisible — for any x ∈ ℚ and nonzero n ∈ ℤ, there exists y ∈ ℚ with ny = x — which enables extending any ℤ-homomorphism; ℤ fails this because, for example, 1 ∈ ℤ cannot be divided by 2 within ℤ
• D. ℤ is a ring, not a ℤ-module, so the comparison is invalid

Answer: (C)

Over a PID like ℤ, injective modules are exactly the divisible ones. Divisibility is what enables the extension: when you try to extend a homomorphism to a new element b with nb ∈ A for some n, divisibility lets you define f̃(b) = f(nb)/n consistently inside ℚ. ℤ fails injectivity because there is no integer y with 2y = 1 — the extension is blocked in cases requiring division by 2.

Question 3

Every object in a suitable abelian category embeds essentially into a unique (up to isomorphism) injective envelope.

• T. True
• F. False

Answer: True

The injective envelope I(M) is characterized by two properties: (1) it is injective, and (2) M embeds essentially — every nonzero subobject of I(M) meets M nontrivially. This combination makes it the minimal injective extension of M. Injective envelopes exist in categories like R-Mod and Ab, and their uniqueness up to isomorphism is what makes injective resolutions well-defined for every object.

Question 4

Injective objects and projective objects are the same objects in most abelian category, since injectivity and projectivity are categorically dual and defined by reversing most arrows.

• T. True
• F. False

Answer: False

While injectives and projectives are categorically dual — defined by reversing all arrows in the extension/lifting diagrams — they are generally different objects in the same category. In Ab, projective objects are free abelian groups (ℤ, ℤⁿ, etc.), while injective objects are divisible groups (ℚ, ℤ[1/p], ℚ/ℤ, etc.). Duality tells you that theorems about injectives translate to theorems about projectives by reversing arrows, but it does not make them the same objects.

Question 5

Why are injective resolutions essential for defining derived functors like Ext, rather than simply applying Hom(N, −) directly?

Model answer: The Hom functor Hom(N, −) is only left-exact — it preserves exact sequences on the left but fails exactness on the right. To define Ext^n(N, M), you resolve M by an injective resolution 0 → M → I₀ → I₁ → I₂ → ⋯, apply Hom(N, −) to the deleted resolution, and take cohomology. The injective objects in the resolution are chosen precisely because Hom is exact on injectives, making the cohomology groups well-defined. The Ext groups measure the failure of Hom to be right-exact — a failure that only becomes computable by resolving through injectives.

This is the fundamental idea of homological algebra: derived functors measure the failure of ordinary functors to preserve exactness. To compute these derived functors, you need resolutions by objects on which the functor is exact — injective resolutions for right-derived functors (Ext), projective resolutions for left-derived functors (Tor). Without injective objects and the guarantee that every object embeds into one, this entire framework collapses.