Over $\ZZ$, $\Tor_1(\ZZ/n, \ZZ/m)\cong\ZZ/\gcd(m,n)$, $\Tor_1(A, B) = 0$ when $A$ or $B$ is torsion-free, and $\Tor_k = 0$ for $k\geq 2$.Suggest a slogan
Let \(m, n\geq 1\) and \(d\coloneqq\gcd(m,n)\). For an abelian group \(A\) let \(A[n]\coloneqq\ts{a\in A \st na = 0}\). Then \(\Tor^{\ZZ}_0(\ZZ/n, A)\cong A/nA\), \(\Tor^{\ZZ}_1(\ZZ/n, A)\cong A[n]\), \(\Tor^{\ZZ}_k(\ZZ, A) = 0\) for \(k\geq 1\), \(\Tor^{\ZZ}_1(A, B)\cong\Tor^{\ZZ}_1(B, A)\), and \(\Tor^{\ZZ}_1(A, B) = 0\) whenever \(A\) or \(B\) is torsion-free [@Hat02]. The groups \(\Tor^{\ZZ}_0(V, W) = V\otimes_{\ZZ}W\) are
| \(V \otimes_{\ZZ} W\) | \(W=\ZZ/m\) | \(W=\ZZ\) | \(W=\QQ\) |
|---|---|---|---|
| \(V=\ZZ/n\) | \(\ZZ/d\) | \(\ZZ/n\) | \(0\) |
| \(V=\ZZ\) | \(\ZZ/m\) | \(\ZZ\) | \(\QQ\) |
| \(V=\QQ\) | \(0\) | \(\QQ\) | \(\QQ\) |
and the groups \(\Tor^{\ZZ}_1(V, W)\) are
| \(\Tor^{\ZZ}_1(V, W)\) | \(W=\ZZ/m\) | \(W=\ZZ\) | \(W=\QQ\) |
|---|---|---|---|
| \(V=\ZZ/n\) | \(\ZZ/d\) | \(0\) | \(0\) |
| \(V=\ZZ\) | \(0\) | \(0\) | \(0\) |
| \(V=\QQ\) | \(0\) | \(0\) | \(0\) |
All higher groups \(\Tor^{\ZZ}_k\), \(k\geq 2\), vanish.
What are the groups $\operatorname{Tor}^{\mathbf{Z}}_*$ among $\ZZ/n$, $\ZZ$ and $\QQ$?