Chain Complexes, Chain Maps and the Snake Lemma
Handbook guide to chain complexes, chain maps and the snake lemma with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Chain Complexes
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
The Snake Lemma
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
A chain complex (or simply a complex) C∗is a family of R-modules Cn, n ∈Z, along with R-homomorphisms dn : Cn →Cn−1 called differentials, satisfying dndn+1 = 0 for all n. A chain complex with only finitely many Cn’s is allowed; it can always be extended with the aid of zero modules and zero maps. [In topology, Cn is the abelian group of n-chains, that is, all formal linear combinations with integer coefficients of n-simplices in a topological space X. The map dn is the boundary operator, which assigns to an n-simplex an n −1-chain that represents the oriented boundary of the simplex.] The kernel of dn is written Zn(C∗) or just Zn; elements of Zn are called cycles in dimension n. The image of dn+1 is written Bn(C∗) or just Bn; elements of Bn are called boundaries in dimension n.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definition
A chain map f : C∗→D∗from a chain complex C∗to a chain complex D∗ is a collection of module homomorphisms fn : Cn →Dn, such that for all n, the following diagram is commutative. fn Cn → Dn dn ↓ ↓dn Cn−1 → Dn−1 fn−1 We use the same symbol dn to refer to the differentials in C∗and D∗. [If f : X →Y is a continuous map of topological spaces and σ is a singular n-simplex in X, then f#(σ) = f ◦σ is a singular n-simplex in Y , and f# extends to a homomorphism of n-chains. If we assemble the f#’s for n = 0, 1, . . ., the result is a chain map.]
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
A chain map f takes cycles to cycles and boundaries to boundaries. Consequently, the map zn + Bn(C∗) →fn(zn) + Bn(D∗) is a well-defined homomorphism from Hn(C∗) to Hn(D∗). It is denoted by Hn(f).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Result
The Homology Functors One can create a category whose objects are chain complexes and whose morphisms are chain maps. The composition gf of two chain maps
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Chain Homotopy Let f and g be chain maps from C∗to D∗. We say that f and g
Chain Homotopy Let f and g be chain maps from C∗to D∗. Call f and g are chain homotopic and write f ≃g if there exist homomorphisms hn : Cn →Dn+1 such that fn −gn = dn+1hn + hn−1dn; see the diagram below. Cn Cn-1 Dn f -g n n Dn+1 h n h n-1 d n d n+1
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
If f and g are chain homotopic, then Hn(f) = Hn(g).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Lemma Assume that the diagram below is commutative.
Assume that the diagram below is commutative. f A → B d ↓ ↓e C → D g (i) f induces a homomorphism on kernels, that is, f(ker d) ⊆ker e. (ii) g induces a homomorphism on cokernels, that is, the map y+im d →g(y)+im e, y ∈C, is a well-defined homomorphism from coker d to coker e. (iii) If f is injective, so is the map induced by f, and if g is surjective, so is the map induced by g.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Lemma
The first and fourth rows of the enlarged snake diagram are exact.
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Remark Sometimes an even bigger snake diagram is given, with column 1 assumed
Remark Sometimes an even bigger snake diagram is given, with column 1 assumed to be an exact sequence d 0 → A′ → A → C → C′ → 0
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Quick-reference relationships
Problem-solving workflow
State the objects precisely
Write the sets, operations, maps and hypotheses before manipulating them.
Identify the governing definition
Most abstract arguments become shorter once the relevant universal or structural definition is explicit.
Apply results only under their hypotheses
Check finiteness, commutativity, injectivity, surjectivity and independence assumptions.
Keep notation consistent
Distinguish elements, subsets, equivalence classes, maps and quotient objects.
Use a small example as a check
A concrete model can expose a mistaken direction or missing condition.
Return to the structural meaning
State what the calculation proves about the original algebraic object.
Worked-solution emphasis from the supplied source
The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
Common mistakes and boundary conditions
- Skipping a hypothesis because it seems automatic.
- Using notation before defining the ambient structure.
- Treating an illustrative calculation as a universal result.
- Relying on intuition without verifying closure, well-definedness or exactness.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| S1 | Chain Complexes | 226–226 |
| S2 | The Snake Lemma | 227–228 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.