Sums map out, products map in
For finitely many modules the direct sum and direct product coincide; for infinite families they differ, the sum consisting of the finitely supported families. The distinction is not pedantic: it is dictated by their universal properties — a map out of a sum is a family of maps, a map into a product is a family of maps — and it governs how Hom behaves on each. A short exact sequence splits exactly when it is isomorphic to the trivial sum decomposition, and there are three equivalent ways to detect that.
Learning objectives
- State the universal properties of the sum and the product.
- Explain when the two constructions differ.
- State the splitting lemma and its three equivalent conditions.
- Describe how Hom interacts with sums and products.
Section 01The two universal properties
Comes with injections. A homomorphism ⊕Mi → N is exactly a family of homomorphisms Mi → N. Elements have finite support.
Comes with projections. A homomorphism N → ∏Mi is exactly a family of homomorphisms N → Mi. Elements are arbitrary families.
For a finite index set the canonical map from the sum to the product is an isomorphism; for an infinite one it is a proper inclusion. Both constructions are determined up to unique isomorphism by their universal properties, which is why the same definitions transplant unchanged into any category possessing them.
| Expression | Equals | Reason |
|---|---|---|
| Hom(⊕i Mi, N) | ∏i Hom(Mi, N) | Maps out of a coproduct are families |
| Hom(M, ∏i Ni) | ∏i Hom(M, Ni) | Maps into a product are families |
| Hom(M, ⊕i Ni) | Not generally a sum | Only for M finitely generated, among other cases |
Section 02Split short exact sequences
- Given 0 → A →μ B →ε C → 0, the following are equivalent.
- There is a retraction ρ: B → A with ρμ = 1A. The subobject is a direct summand.
- There is a section σ: C → B with εσ = 1C. The quotient lifts.
- There is an isomorphism B ≅ A ⊕ C carrying μ and ε to the canonical injection and projection.
- Any one of these implies the other two.
For modules, a retraction exists if and only if a section does. In non-abelian settings — group extensions, for instance — the two conditions come apart, and only the section version survives. That is why group cohomology in degree 2 classifies extensions with a prescribed action rather than direct sums.
Section 03Where sums and products diverge
Exactness
Direct sums are exact in module categories; direct products are exact too, but in general abelian categories products may fail to be exact.
Free modules
A free module is a direct sum of copies of Λ, never a product. ℤℕ is not free as an abelian group.
Derived functors
Ext converts sums in the first variable into products, mirroring the behaviour of Hom. This is used constantly in universal coefficient computations.
ReferenceFrequently asked questions
Why does the coproduct use finite support?
Because a homomorphism out of it must be determined by its restrictions to the factors, and an infinite formal sum would have no well-defined image. The finite-support condition is exactly what makes the universal property work.
Are sums and products interchangeable in a finite direct sum?
Yes, and the resulting object is a biproduct: it carries injections and projections satisfying the expected identities. Additive categories are defined by having finite biproducts.
Does splitting imply the sequence is trivial?
It implies the extension is trivial, which is what the zero element of Ext1 represents. The sequence still carries the information of which submodule was chosen, so splitting is a statement about isomorphism class, not about the maps being canonical.
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