= Extension of trivial modules for an infinite cyclic group
{title2=$\operatorname{Ext}_{\mathbb Z[z,z^{-1}]}^1(\mathbb Z^s,\mathbb Z^r)\cong M_{r\times s}(\mathbb Z)$}
With trivial action on both end modules, an extension has underlying abelian group $\mathbb Z^r\oplus\mathbb Z^s$ and generator action $\left(\begin{smallmatrix}I&T\\0&I\end{smallmatrix}\right)$. 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>.
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