For $\ZZ/n$, $\Ext^0$ detects $n$-torsion, $\Ext^1$ is quotient by $n$, and higher $\Ext$ vanishes.Suggest a slogan
Let \(n\geq 1\), let \(A\) be an abelian group, and let \(A[n]\coloneqq\ts{a\in A \st na = 0}\). The groups $\Ext^k_{\ZZ}(\ZZ/n, A)$ are \[ \Ext^0_{\ZZ}(\ZZ/n, A)\cong\Hom(\ZZ/n, A)\cong A[n],\qquad \Ext^1_{\ZZ}(\ZZ/n, A)\cong A/nA,\qquad \Ext^k_{\ZZ}(\ZZ/n, A) = 0\quad (k\geq 2) \] [@Hat02].
What is $\operatorname{Ext}^*_{\mathbf{Z}}(\mathbf{Z}/n, A)$?