Short exact sequence

A short exact sequence is an exact sequenceThus $j$ is injective, $q$ is surjective, and $\mathrm{im}j=\ker q$.

A short exact sequence with specified kernel module $A$ and quotient module $C$. Its equivalence class is determined by the degree-one Ext functor; equivalence must preserve both end modules.

With trivial action on both end modules, an extension has underlying abelian group ${\mathbb{Z}}^r\oplus {\mathbb{Z}}^s$ and generator action $\left(\begin{array}{cc}I& T\\ 0& I\end{array}\right)$. The integer matrix $T$ is invariant under equivalences fixing the ends. The one-element projective resolution with differential $z-1$ computes the same classification over the group ring.

Pushing an inclusion $i:A\to B$ along $f:A\to A\prime$ gives a new module extension with kernel $A\prime$ and the same quotient. This pushout realizes the covariance of the degree-one Ext functor in its second argument.

A commuting isomorphism between two module extensions that is the identity on both end modules. Classifying middle modules up to arbitrary isomorphism is a different and generally coarser problem.

A short exact sequence splits when its injection has a retraction, equivalently when its surjection has a section. In that case the middle module is isomorphic to the direct sum of the two outer modules.

The Five lemma has a short form: in a commutative diagram of short exact sequences, if the maps on both outer terms are isomorphisms, then the map on the middle terms is an isomorphism.