[[Exact sequence]]
# Short exact sequence

A **short exact sequence** is an [[exact sequence]] with no more than three non-trivial objects, #m/def/homology 
i.e. it is of the form
$$
\begin{align*}
\cdots \to 0 \to A \stackrel{f}{\hookrightarrow} B \stackrel{g}{\twoheadrightarrow} C \to 0 \to\cdots
\end{align*}
$$
A **morphism of short exact sequences** is a triple $(\alpha,\beta,\gamma)$ of homomorphisms such that the expected commutative diagram commutes.

## Properties

1. $f$ is guaranteed to be injective and $g$ to be surjective (see [[Exact sequence#Properties]])
2. [[Split short exact sequence]]


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