Chain Complexes and Exact Sequences — Mathematics Knowledge System

Overview

A chain complex is a sequence of abelian groups connected by homomorphisms (boundary operators) such that the composition of any two consecutive maps is zero. This algebraic structure underlies both singular and simplicial homology. Exact sequences -- where the image of each map equals the kernel of the next -- are a fundamental tool for computing homology groups and relating the homology of different spaces.

Intuition

Think of a chain complex as a sequence of floors in a building, where the boundary operator takes you from floor n to floor n-1. The rule ∂∘∂ = 0 means: if you go down two floors at once, you always end up at the zero element. Cycles (ker ∂) are the elements that 'look like' they came from one floor above but we cannot tell. Boundaries (im ∂) are the elements that definitely came from one floor above. Homology measures the gap: cycles that are not actually boundaries.

Formal Definition

Definition

A chain complex (C_*, ∂_*) is a sequence of abelian groups ... → C_{n+1} → C_n → C_{n-1} → ... with homomorphisms ∂_n: C_n → C_{n-1} satisfying ∂_{n-1} ∘ ∂_n = 0. A sequence A → B → C is exact at B if im(f) = ker(g). A short exact sequence is 0 → A → B → C → 0 with exactness at A, B, C. The homology of (C_*, ∂_*) is H_n = ker(∂_n)/im(∂_{n+1}).

\cdots \to C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \to \cdots,\quad \partial_{n-1}\circ\partial_n = 0

Chain complex

H_n(C_*) = \ker(\partial_n) / \mathrm{im}(\partial_{n+1})

Homology

0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 \text{ exact}

Short exact sequence

0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 \implies C \cong B/f(A)

Quotient characterization

Notation

Notation	Meaning
$(C_, \partial_)$	Chain complex with groups C_n and boundary maps ∂_n
$H_n(C_*)$	n-th homology group of the chain complex
$Z_n$	Group of n-cycles: ker(∂_n)
$B_n$	Group of n-boundaries: im(∂_{n+1})
$0 \to A \to B \to C \to 0$	Short exact sequence

Theorems

Theorem 1: Snake Lemma

Given a commutative diagram with exact rows: 0 → A → B → C → 0 over 0 → A' → B' → C' → 0, there is a long exact sequence: ker(f_A) → ker(f_B) → ker(f_C) → coker(f_A) → coker(f_B) → coker(f_C), where the connecting homomorphism δ: ker(f_C) → coker(f_A) is natural.

Theorem 2: Long Exact Sequence of a Pair

A short exact sequence of chain complexes 0 → A_* → B_* → C_* → 0 induces a long exact sequence in homology: ... → H_n(A) → H_n(B) → H_n(C) → H_{n-1}(A) → ... where the connecting homomorphisms δ_n: H_n(C) → H_{n-1}(A) are natural.

Theorem 3: Splitting of Short Exact Sequences

A short exact sequence 0 → A → B → C → 0 of abelian groups splits (B ≅ A ⊕ C) if and only if C is free abelian, or equivalently if there exists a section s: C → B with g ∘ s = id_C.

Worked Examples

1
Let f: Z → Z be multiplication by 2 and g: Z → Z/2Z be the quotient map.
$0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\text{mod }2} \mathbb{Z}/2\mathbb{Z} \to 0$
2
Exactness at the first Z: ker(f) = {n : 2n=0} = 0, and im(0→Z) = 0. ✓
3
Exactness at the middle Z: im(f) = 2Z = ker(g) since g(n)=0 iff 2|n. ✓
4
Exactness at Z/2Z: g is surjective (g(0)=[0], g(1)=[1]). ✓

✓ Answer

The sequence is exact; it represents Z/2Z as Z/2Z.

Practice Problems

Mediumproof writing

Prove the Five Lemma: given a commutative diagram with exact rows A→B→C→D→E over A'→B'→C'→D'→E' and isomorphisms at A,B,D,E, the middle map C→C' is also an isomorphism.

Mediumfree response

Given a short exact sequence 0 → Z → Z⊕Z/2Z → Z/2Z → 0, does this split? Find a splitting if so.

Common Mistakes

Common Mistake

g ∘ f = 0 implies exactness at B.

g ∘ f = 0 only implies im(f) ⊆ ker(g). Exactness requires equality im(f) = ker(g). The stronger condition can fail: e.g., A = 0, f = 0, B = Z, g = 0, C = Z gives g∘f=0 but the sequence is not exact at B.

Common Mistake

Every short exact sequence of abelian groups splits.

Splitting requires the quotient (right term) to be free abelian. The sequence 0 → Z → Z → Z/nZ → 0 does not split for n ≥ 2 since Z/nZ is not free.

Quiz

Historical Background

The abstract notion of a chain complex and exact sequence crystallized in the 1940s--1950s as algebraic topology matured. Eilenberg and Mac Lane introduced homological algebra as an independent subject, and the Snake Lemma and Five Lemma became standard tools. Cartan and Eilenberg's 1956 book 'Homological Algebra' systematized the subject, which then spread to algebraic geometry (sheaf cohomology) and representation theory (Ext and Tor functors).

1945
Eilenberg and Mac Lane introduce the general theory of natural transformations and functors
Eilenberg, Mac Lane
1956
Cartan and Eilenberg publish Homological Algebra, systematizing chain complexes
Cartan, Eilenberg
1957
Grothendieck introduces abelian categories for sheaf cohomology
Grothendieck

Summary

A chain complex is a sequence of abelian groups with boundary operators satisfying ∂∘∂ = 0.
Homology H_n = ker(∂_n)/im(∂_{n+1}) measures cycles that are not boundaries.
A sequence is exact at B if im(incoming) = ker(outgoing); a short exact sequence 0→A→B→C→0 expresses B as an extension of A by C.
The Snake Lemma and connecting homomorphisms convert short exact sequences of chain complexes into long exact sequences in homology.
Short exact sequences of abelian groups split if and only if the quotient term is free abelian.

References

BookHatcher, A. Algebraic Topology. Cambridge University Press, 2002. Section 2.1.
BookWeibel, C.A. An Introduction to Homological Algebra. Cambridge University Press, 1994.
WebsiteWikipedia: Chain complex