Short and Split Exact Sequences

Short exact sequences provide the most fundamental building blocks for understanding the algebraic structure of homology groups and their relationships.

Short Exact Sequences

Definition

A short exact sequence of abelian groups is a sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ where $f$ is injective, $g$ is surjective, and $\ker g = \operatorname{im} f$ . Equivalently, $A$ is isomorphic to a subgroup of $B$ , and $C \cong B / f(A)$ .

Short exact sequences encode extension problems: given $A$ and $C$ , what groups $B$ can fit in the middle? The simplest case is when $B$ is the direct sum $A \oplus C$ .

Definition

A short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split if any of the following equivalent conditions hold:

There exists a homomorphism $s : C \to B$ with $g \circ s = \operatorname{id}_C$ (right splitting)
There exists a homomorphism $r : B \to A$ with $r \circ f = \operatorname{id}_A$ (left splitting)
$B \cong A \oplus C$ via an isomorphism compatible with $f$ and $g$

The Splitting Lemma

Theorem6.3Splitting Lemma

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

The sequence splits on the right: there exists $s : C \to B$ with $g \circ s = \operatorname{id}_C$ .
The sequence splits on the left: there exists $r : B \to A$ with $r \circ f = \operatorname{id}_A$ .
$B \cong f(A) \oplus s(C) \cong A \oplus C$ .

ExampleNon-split sequence

The short exact sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$ does not split. If it did, we would have $\mathbb{Z} \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ , but $\mathbb{Z}$ is torsion-free while $\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ has torsion.

Exact Sequences in Topology

RemarkTopological significance

Short exact sequences arise naturally in algebraic topology when relating homology groups of pairs. For a good pair $(X, A)$ , the long exact sequence of the pair breaks into short exact sequences when certain connecting homomorphisms vanish, often allowing direct computation of homology groups via the splitting lemma.

The utility of short exact sequences lies in their ability to decompose complex algebraic structures into simpler pieces, making computation of homology and cohomology groups tractable through systematic algebraic arguments.