Right exact, and flat is the condition for more
The tensor product converts bilinear maps into linear ones: homomorphisms out of M ⊗ N correspond exactly to balanced bilinear maps from the pair. Being a left adjoint, it preserves colimits and is right exact; it fails to preserve injections, and the modules for which it does are the flat ones. Projective implies flat, flat implies torsion-free, and over a PID all three coincide for finitely generated modules.
Learning objectives
- State the universal property of the tensor product.
- Prove right exactness from the adjunction.
- Give an example where tensoring destroys injectivity.
- Define flatness and place it relative to projectivity.
- Use extension of scalars.
Section 01Universal property and construction
For a right module M and a left module N over Λ, the tensor product is the abelian group generated by symbols m ⊗ n subject to bilinearity and the balancing relation mλ ⊗ n = m ⊗ λn. Its universal property:
A general element of M ⊗ N is a finite sum of simple tensors, and the representation is not unique. Defining a map on tensors requires checking it is well defined — in practice, defining it as a bilinear map and invoking the universal property, which is exactly what the property is for.
| Tensor product | Result | Reason |
|---|---|---|
| ℤ/m ⊗ ℤ/n | ℤ/gcd(m, n) | Relations force divisibility |
| ℤ/m ⊗ ℤ | ℤ/m | ℤ is the unit |
| ℚ ⊗ ℤ/m | 0 | Every element is divisible by m in ℚ |
| ℚ ⊗ ℚ | ℚ | Localisation is idempotent |
| Λ ⊗Λ N | N | The unit object |
Section 02Right exactness and its failure
Tensoring 0 → ℤ →×2 ℤ → ℤ/2 → 0 with ℤ/2 gives
The first map is zero rather than injective, so exactness fails on the left. The kernel ℤ/2 is Tor1(ℤ/2, ℤ/2) — the first derived functor of tensor, appearing exactly where the injection was destroyed.
Tensor is left adjoint to Hom, and left adjoints preserve colimits, hence cokernels. No computation is needed — right exactness is a consequence of the adjunction, and the same argument gives left exactness of Hom.
Section 03Flat modules
M is flat when M ⊗ − is exact. The hierarchy:
- Stage 01FreeA direct sum of copies of Λ.
- Stage 02ProjectiveA direct summand of a free module.
- Stage 03FlatTensoring preserves injections.
- Stage 04Torsion-freeOver a domain; strictly weaker than flat in general.
| Ring | Flat ⇒ projective? | Torsion-free ⇒ flat? |
|---|---|---|
| PID | For finitely generated, yes | Yes |
| General domain | No — ℚ is flat over ℤ, not projective | No |
| Noetherian local | For finitely generated, yes | Not in general |
| Any ring, finitely presented | Yes | — |
ℚ is flat over ℤ because localisation is exact, but it is not projective — it is not a summand of a free abelian group. Flatness is genuinely weaker, and this example is worth carrying.
Section 04Extension of scalars
For a ring map Λ → Λ′, the functor Λ′ ⊗Λ − carries Λ-modules to Λ′-modules and is left adjoint to restriction of scalars.
Base change
Reduce a module modulo a prime, or extend a real representation to a complex one.
Induced representations
For a subgroup H ≤ G, the functor ℤ[G] ⊗ℤ[H] − is induction, adjoint to restriction — the source of Shapiro's lemma.
Change of rings
Comparing derived functors before and after base change gives the change-of-rings spectral sequences.
ReferenceFrequently asked questions
Why must one module be a right module and the other a left module?
Because the balancing relation moves a scalar across the tensor sign, which requires it to act on the right of the first factor and the left of the second. Over a commutative ring the distinction dissolves and the result is again a module.
Is the tensor product commutative?
Over a commutative ring, yes, up to natural isomorphism. Over a non-commutative ring the expression N ⊗ M need not even be defined, since the sidedness would be wrong.
When does tensoring preserve infinite products?
Rarely. Tensor preserves coproducts always, being a left adjoint, but products only under finiteness conditions — typically when the module is finitely presented. This asymmetry recurs in universal coefficient arguments.
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