Ext groups between polynomial-ring residue modules (${\mathrm{Ext}}_{\mathbb{Z}[X]}^{\ast }(\mathbb{Z}/p,\mathbb{Z}/q)$)

When $X$ acts as zero, resolve the first module with the Koszul resolution on $(p,X)$. Writing $N=\mathbb{Z}/q$, the resulting groups are $\ker (p:N\to N)$ in degree zero, $(N/pN)\oplus \ker p$ in degree one, and $N/pN$ in degree two, with zero higher groups. This works for composite as well as prime moduli.

For an ideal generated by a finite regular sequence in a commutative ring, the self-Ext functor algebra is exterior on the degree-one dual generators. Contraction chain maps on the Koszul resolution represent those generators, square to zero, and anticommute. The exterior square-zero relations also apply in characteristic two.