Chain Complexes and Exact Sequences — Mathematics Knowledge System

Overview

Chain complexes and exact sequences are the central combinatorial and algebraic machinery of homological algebra. A chain complex organizes modules and linear maps into a sequence where consecutive compositions vanish; its homology measures the 'holes' or 'obstructions'. Exact sequences are the special case where there are no holes at all. The long exact sequence of a short exact sequence of complexes is one of the most powerful tools in algebra and topology.

Intuition

Think of a chain complex as a pipeline where material flows in one direction and each stage perfectly captures what the previous stage produced — but doesn't perfectly process it. The homology groups measure the backlog: how much material accumulates at each stage. In an exact sequence, there is no backlog — everything that arrives is immediately consumed.

Formal Definition

Definition

A chain complex $(C_\bullet, \partial)$ is a collection of $R$-modules $\{C_n\}_{n \in \mathbb{Z}}$ with boundary maps $\partial_n: C_n \to C_{n-1}$ satisfying $\partial_{n-1} \circ \partial_n = 0$. A cochain complex has maps going up in degree. Exact sequences are chain complexes with zero homology.

\partial_{n-1} \circ \partial_n = 0 \quad \Leftrightarrow \quad \text{im}(\partial_n) \subseteq \ker(\partial_{n-1})

Chain complex condition

\cdots \to A \xrightarrow{f} B \xrightarrow{g} C \to \cdots \text{ exact at } B \Leftrightarrow \ker(g) = \text{im}(f)

Exactness at a term

H_n(C_\bullet) = 0 \text{ for all } n \Leftrightarrow C_\bullet \text{ is an exact complex}

Exact complex = acyclic

Worked Examples

Check exactness at $\mathbb{Z}$ (first copy): the map from 0 has image $\{0\}$, and $\ker(2\cdot) = \{0\}$. Exact.
$\ker(n \mapsto 2n) = \{0\} = \text{im}(0 \to \mathbb{Z})$
Check exactness at $\mathbb{Z}$ (second copy): $\ker(\mathbb{Z} \to \mathbb{Z}/2) = 2\mathbb{Z} = \text{im}(\cdot 2)$. Exact.
$\ker(\pi) = 2\mathbb{Z} = \text{im}(\cdot 2)$
Check exactness at $\mathbb{Z}/2\mathbb{Z}$: the projection $\pi: \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ is surjective, and the map to 0 has kernel all of $\mathbb{Z}/2\mathbb{Z} = \text{im}(\pi)$. Exact.
$\text{im}(\pi) = \mathbb{Z}/2\mathbb{Z} = \ker(\mathbb{Z}/2\mathbb{Z} \to 0)$

Answer: All three exactness conditions hold; the sequence is exact.

Practice Problems

Difficulty 7/10

Define a morphism of chain complexes and explain when two chain maps induce the same map on homology (chain homotopy).

Difficulty 8/10

Use the five lemma to prove: if $f, h$ are isomorphisms and $e, i$ are isomorphisms in a commutative diagram with exact rows, then $g$ is an isomorphism.

Difficulty 9/10

Define the cone of a chain map $f: A_\bullet \to B_\bullet$ and describe its role in the associated long exact sequence.

Common Mistakes

Common Mistake

If $g \circ f = 0$ in a sequence $A \xrightarrow{f} B \xrightarrow{g} C$, the sequence is exact at $B$.

$g \circ f = 0$ only shows $\text{im}(f) \subseteq \ker(g)$. Exactness requires equality: $\ker(g) = \text{im}(f)$. The difference is measured by homology.

Common Mistake

A chain map that is zero on homology must be chain homotopic to zero.

This is false in general. A chain map can induce zero on all homology groups without being null-homotopic. Counterexamples arise in stable homotopy theory.

Quiz

Summary

A chain complex has boundary maps $\partial_n$ with $\partial_{n-1}\partial_n = 0$; its homology $H_n = \ker\partial_n/\text{im}\partial_{n+1}$ measures 'holes'.
An exact sequence is a chain complex with all $H_n = 0$ (every kernel equals the previous image).
A short exact sequence $0 \to A \to B \to C \to 0$ of chain complexes induces a long exact sequence in homology.
Chain homotopic maps induce the same maps on homology.
The mapping cone is a key tool encoding the homological information of a chain map.

References

WebsiteWikipedia — Exact sequence