Module extension ($0\to A\to B\to C\to 0$)

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.