Linear algebra without the guarantee of a basis
A module over a ring is what a vector space becomes when the scalars are no longer a field: addition and scalar multiplication behave as expected, but bases need not exist, submodules need not be direct summands, and a surjection need not split. Every difficulty in homological algebra traces back to one of those three failures. Exact sequences are the notation for keeping track of them, and the snake lemma is the first genuinely powerful consequence.
Learning objectives
- Define modules, submodules and quotient modules over a general ring.
- State exactness and read a short exact sequence correctly.
- Apply the five lemma and the short five lemma.
- Derive the connecting homomorphism of the snake lemma.
- Explain which vector-space facts fail for modules and why.
Section 01Modules and their morphisms
A left module over a ring Λ with unit is an abelian group M with an action Λ × M → M that is associative, distributive on both sides and unital. A homomorphism respects both operations, and Hom(M, N) is itself an abelian group — a Λ-module when Λ is commutative.
| Property over a field | Over a general ring |
|---|---|
| Every module is free | Fails — free modules are special |
| Every submodule is a direct summand | Fails — this is exactly semisimplicity |
| Every surjection splits | Fails — splitting defines projectivity |
| Every injection splits | Fails — splitting defines injectivity |
| Rank is well defined | Usually, but not over every ring |
| Sub and quotient constructions work | Survives unchanged |
Projective modules are those for which surjections onto them split; injective modules are those for which injections out of them split; Ext measures the failure of a specific extension to split. The entire apparatus is a bookkeeping system for these three phenomena.
Section 02Exactness
A sequence A →f B →g C is exact at B when im f = ker g. A short exact sequence
says exactly that μ is injective, ε is surjective, and im μ = ker ε — so C is B/A. It is the basic unit of information in the subject: a statement that B is built from A and C, without saying how.
A short exact sequence records that B is an extension of C by A. It does not say B ≅ A ⊕ C. Deciding when it does is the extension problem, and Ext1 is its answer. Reading exactness as though it implied a direct sum is the most common beginner's error in this material.
Section 03Diagram lemmas
Two lemmas do most of the routine work. Both are proved by diagram chasing, and both hold in any abelian category.
- Given a commutative diagram with exact rows, two rows connected by vertical maps α, β, γ.
- Take c ∈ ker γ and lift it to b ∈ B along the surjection. Possible because the top row is exact.
- Then β(b) maps to 0 in C′, so β(b) comes from a′ ∈ A′.
- The class of a′ in coker α is independent of the lift chosen. This is the content of the lemma.
- Define ∂(c) = [a′]. The resulting sequence ker α → ker β → ker γ → coker α → coker β → coker γ is exact.
Five lemma
In a commutative diagram with exact rows and five columns, if the outer two pairs are isomorphisms then the middle one is too.
Short five lemma
For a map of short exact sequences, if the two outer maps are isomorphisms so is the middle one. Used constantly to recognise equivalent extensions.
Nine lemma
A 3×3 diagram with exact columns and two exact rows has its third row exact. Follows from the snake lemma.
ReferenceFrequently asked questions
Why insist on left modules rather than right?
Convention only. A right Λ-module is a left module over the opposite ring, so nothing is lost. The distinction matters when both act at once — bimodules — and when forming tensor products, where a right module meets a left module.
Does the snake lemma need the rows to be short exact?
The rows need to be exact where the chase happens: surjective on the right of the top row and injective on the left of the bottom row. The commonly stated version with short exact rows is the convenient special case.
Is diagram chasing valid in a general abelian category?
Yes, though elements do not literally exist there. The Freyd–Mitchell embedding theorem lets any small abelian category be embedded in a module category, so element-based proofs transfer. Members or generalised elements give an intrinsic alternative.
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