Chain Complexes and Homology - Examples and Constructions

Exact sequences are ubiquitous in homological algebra, encoding important relationships between algebraic structures.

Definition1.6Exact Sequence

A sequence of abelian groups (or $R$ -modules) and homomorphisms: $\cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_n} A_{n-1} \to \cdots$

is exact at $A_n$ if $\text{Im}(f_{n+1}) = \ker(f_n)$ . The sequence is exact if it is exact at every position.

ExampleShort Exact Sequence

A short exact sequence (SES) is an exact sequence of the form: $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$

Exactness means:

$f$ is injective (exactness at $A$ )
$\text{Im}(f) = \ker(g)$ (exactness at $B$ )
$g$ is surjective (exactness at $C$ )

This encodes that $A$ can be viewed as a subobject of $B$ , and $C$ is the quotient $B/A$ .

Definition1.7Split Exact Sequence

A short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ splits if one of the following equivalent conditions holds:

There exists $r: B \to A$ with $r \circ f = \text{id}_A$ (left splitting)
There exists $s: C \to B$ with $g \circ s = \text{id}_C$ (right splitting)
$B \cong A \oplus C$ and the sequence becomes $0 \to A \to A \oplus C \to C \to 0$

TheoremSplitting Lemma

For a short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ , the following are equivalent:

The sequence splits on the left
The sequence splits on the right
$B \cong A \oplus C$ as abelian groups (or $R$ -modules)

ExampleNon-Splitting Sequence

Consider the short exact sequence: $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/2\mathbb{Z} \to 0$

This sequence does not split. If it did, we would have $\mathbb{Z} \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ , which is impossible since the left side is torsion-free while the right side has 2-torsion.

Remark

The failure of a short exact sequence to split often measures important algebraic obstructions. In module theory, split sequences correspond to free summands, while non-split sequences reflect deeper structural properties.

Definition1.8Quasi-isomorphism

A chain map $f: C_\bullet \to D_\bullet$ is a quasi-isomorphism if it induces an isomorphism on homology: $H_n(f): H_n(C_\bullet) \xrightarrow{\cong} H_n(D_\bullet)$ for all $n$ . Quasi-isomorphisms are the weak equivalences in the homotopy category of chain complexes.