Extension of trivial modules for an infinite cyclic group (${\mathrm{Ext}}_{\mathbb{Z}[z,z^{-1}]}^1({\mathbb{Z}}^s,{\mathbb{Z}}^r)\cong M_{r\times s}(\mathbb{Z})$)

With trivial action on both end modules, an extension has underlying abelian group ${\mathbb{Z}}^r\oplus {\mathbb{Z}}^s$ and generator action $\left(\begin{array}{cc}I& T\\ 0& I\end{array}\right)$. 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.