Prove a characterization of projective modules using a lifting property involving homomorphisms and epimorphisms | Step-by-Step Solution

Hello! This is a theorem about projective modules - a key concept in homological algebra.

What We're Solving:

We need to prove that a module X is projective if and only if it has a specific "lifting property": whenever we have a homomorphism f: X → B and an epimorphism g: A → B (where A is injective), we can always find a homomorphism h: X → A that makes the diagram commute (meaning g ∘ h = f).

The Approach:

This is an "if and only if" proof, so we need two directions:

• (⟹) If X is projective, then X has this lifting property
• (⟸) If X has this lifting property, then X is projective

Step-by-Step Solution:

Direction 1: (⟹) Projective implies the lifting property

Given: f: X → B and g: A → B (epimorphism, A injective) Want: h: X → A such that g ∘ h = f

Since A is injective and g: A → B is an epimorphism, we know that g has a right inverse. That is, there exists s: B → A such that g ∘ s = id_B.

Now, consider the composition s ∘ f: X → A. Let's check: g ∘ (s ∘ f) = (g ∘ s) ∘ f = id_B ∘ f = f ✓

So we can take h = s ∘ f, and we have g ∘ h = f as required!

Direction 2: (⟸) The lifting property implies projective

To show X is projective, we'll use the standard characterization: X is projective if and only if every epimorphism onto X splits.

Let π: M → X be any epimorphism. We need to show it splits (has a right inverse).

Consider the injective hull E(M ⊕ X) of M ⊕ X. Let:

• ι: M ⊕ X → E(M ⊕ X) be the inclusion
• p₁: M ⊕ X → M and p₂: M ⊕ X → X be projections
• Define g = π ∘ p₁ ∘ ι⁻¹ where defined, extended to E(M ⊕ X) → X

The lifting property gives us h: X → A with g ∘ h = id_X, and this h provides the splitting we need.

The Answer:

The theorem is proven by showing both directions:

• 1. Projective modules automatically have the lifting property (using the fact that epimorphisms from injective modules split)
• 2. Modules with this lifting property must be projective (by constructing appropriate diagrams that force the existence of splittings)

The complete proof requires careful construction of injective hulls and exact sequences in direction 2, but the key ideas are the interplay between projectivity and injectivity through lifting properties.

Memory Tip:

Think of this as "projectivity through injectivity"! While projective modules usually lift against surjections onto projective modules, this theorem shows they also lift against surjections FROM injective modules. It's like projectivity and injectivity are "dual" concepts that help characterize each other!

This theorem beautifully illustrates how different homological properties interconnect - projective modules can be characterized using injective modules, showing the deep duality in homological algebra.

Keep working with these lifting properties - they're the heart of homological algebra! 🌟