Tensor and Tor
\envlist
-
\(A\otimes_R {-}\) is:
- Covariant
- Right-exact
- Has left-derived functors \(\operatorname{Ext} ^i_R(A, B) \coloneqq L_i \mathop{\mathrm{Hom}}_R({-}, B)(A)\) computed using projective resolutions.
-
\({-}\otimes_R B\) is:
- Covariant
- Right-exact
- Has left-derived functors \(\operatorname{Ext} ^i_R(A, B) \coloneqq L_i \mathop{\mathrm{Hom}}_R({-}, B)(A)\) computed using projective resolutions.
- Tensor is a colimit, and colimits commute with colimits: \((\colim A_i)\otimes_R M = \colim (A_i \otimes_R M)\).
\envlist
- \(\operatorname{Tor}_n^R(A, B) = 0\) for either \(A\) or \(B\) flat.
The most useful SES for proofs here: \begin{align*} 0 \to {\mathbf{Z}}\xrightarrow{n} {\mathbf{Z}}\xrightarrow{\pi} {\mathbf{Z}}/n \to 0 .\end{align*}
\envlist
- \({\mathbf{Z}}/n \otimes_{\mathbf{Z}}G \cong G/nG\)
- \({\mathbf{Z}}/n \otimes_{\mathbf{Z}}{\mathbf{Z}}/m \cong {\mathbf{Z}}/d\).
- \({\mathbf{Q}}\otimes_{\mathbf{Z}}{\mathbf{Z}}/n \cong 0\).
- \(\operatorname{Tor}^{\mathbf{Z}}_1({\mathbf{Z}}/n, G) \cong \left\{{ h\in H {~\mathrel{\Big\vert}~}nh = e }\right\}\)
- \(\operatorname{Tor}^{\mathbf{Z}}_1({\mathbf{Z}}/n, {\mathbf{Q}}) \cong 0\).
- \(\operatorname{Tor}^{\mathbf{Z}}_1({\mathbf{Z}}/n, {\mathbf{Z}}/m) \cong {\mathbf{Z}}/d\).