Cofree Modules and Essential Extensions

Executive summary

Cofree supplies injectives; essential extensions make them minimal

Free modules are built by taking direct sums of the ring; cofree modules are built by applying Hom_ℤ(Λ, −) to a divisible abelian group. That functor is right adjoint to restriction of scalars, and because restriction is exact, the adjoint carries injectives to injectives — which is how the category of Λ-modules is shown to have enough injectives. Essential extensions then identify, among all injectives containing a module, the smallest one.

Learning objectives

Construct cofree modules and state the defining adjunction.
Explain why the construction yields enough injectives.
Define essential extensions and give examples.
State the existence and uniqueness of the injective hull.

Section 01Cofree modules

For an abelian group D, the cofree Λ-module on D is Hom_ℤ(Λ, D), with Λ acting by shifting the argument. Its defining property mirrors freeness with the arrows reversed:

Hom_Λ(M, Hom_ℤ(Λ, D)) ≅ Hom_ℤ(M, D)

FreeΛ ⊗_ℤ S

Left adjoint to the forgetful functor. Maps OUT of a free module are arbitrary functions on a basis.

CofreeHom_ℤ(Λ, D)

Right adjoint to the forgetful functor. Maps INTO a cofree module are arbitrary group homomorphisms into D.

Why this produces injectives

The forgetful functor is exact, so its right adjoint preserves injective objects. Take D divisible, hence injective over ℤ; then Hom_ℤ(Λ, D) is an injective Λ-module. Embedding M in a divisible group and applying the adjunction embeds M in it.

Section 02Essential extensions

An extension M ⊆ E is essential when every non-zero submodule of E meets M non-trivially — equivalently, no non-zero submodule of E intersects M in zero.

Example

ℤ ⊆ ℚ

Essential: every non-zero subgroup of ℚ contains a non-zero integer multiple, hence meets ℤ.

Example

ℤ/pℤ ⊆ ℤ(p^∞)

Essential: every non-zero subgroup of the Prüfer group contains the unique subgroup of order p.

Non-example

M ⊆ M ⊕ N

Not essential when N ≠ 0, since the summand N meets M in zero. Direct summands are the opposite of essential.

Essential and injective are complementary

A module is injective exactly when it has no proper essential extensions — it cannot be enlarged without adding something detached. This reformulation is what makes the hull construction work.

Section 03The injective hull

AlgorithmExistence and uniqueness of the injective hullin: a module M → out: its injective hull E(M)

Embed M in some injective I, which is possible by the cofree construction.
Consider the essential extensions of M inside I; by Zorn's lemma there is a maximal one, E.
E is injective: a maximal essential extension inside an injective has no proper essential extension.
E is minimal among injectives containing M, and any two such are isomorphic over M. Uniqueness is up to non-canonical isomorphism fixing M.
Write E = E(M), the injective hull.

Uniqueness is not canonicity

The hull is unique up to isomorphism fixing M, but the isomorphism is not unique. Constructions depending on a chosen hull must be checked for independence of that choice — the same discipline that resolutions require.

ReferenceFrequently asked questions

Is there a projective analogue of the injective hull?

The projective cover, but it does not always exist. Rings over which every module has one are called perfect, and they are comparatively rare — ℤ is not among them. This is a genuine asymmetry, not a gap in the theory.

Why does the adjunction preserve injectivity?

Because Hom(−, right adjoint applied to D) is naturally isomorphic to Hom(forgetful of −, D), and the forgetful functor is exact. A composite of an exact functor with an exact Hom is exact, which is precisely injectivity of the target.

Are minimal injective resolutions unique?

Yes, up to isomorphism, if each stage is taken to be the injective hull of the preceding cokernel. The resulting Bass numbers are genuine invariants of the module, which is why minimal resolutions are preferred in commutative algebra.

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