Short exact sequence

An exact sequence 0 → A → B → C → 0 capturing a module extension.

Short exact sequence

A short exact sequence is an exact sequence (see exact sequence of modules )

0 \to A \xrightarrow{i} B \xrightarrow{p} C \to 0

such that $i$ is injective and $p$ is surjective, and $\operatorname{im}(i)=\ker(p)$ . In elementary terms, $i$ is an injective map identifying $A$ with a submodule of $B$ , and $p$ is a surjective map with $C\cong B/i(A)$ .

Short exact sequences classify extensions: they encode how $B$ is built from a submodule isomorphic to $A$ and a quotient isomorphic to $C$ .

Examples:

For any module $M$ and submodule $N\le M$ , the canonical sequence $0\to N\to M\to M/N\to 0$ is short exact.
For $n\ne 0$ , the sequence $0\to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathbb Z/n\mathbb Z \to 0$ is short exact.
(Edge case) If $A=0$ , a short exact sequence is just $0\to B\xrightarrow{\sim} C\to 0$ , so $B\cong C$ .