# Monday, January 23


:::{.remark}
Recall

- $(\wait)\tensor_R B$ is right-exact for any $B\in \rmod$ and $\Hom_R(\wait, B)$ is left-exact.
- For $A\in \modsright{R}$ and $B\in \modsleft{R}$, define $\Tor_*^R(A, B)$ as $H_*(P_A \tensor_R B)$ where $P_A\covers A$ is a projective resolution.
- $\Tor_0^R(A, B) = A\tensor_R B$.
  Here $B[n] \da \ts{b\in B\st nb=0}$.
- $\Ext_R^*(A, B) = H_*(\Hom_R(P_A, B))$.
:::


:::{.example title="?"}
\[
\Tor_*^R(C_n, B) \cong B/nB\cdot t^0 + B[n]\cdot t^1
\]
for any $B\in \zmod$ using $\ZZ\injectsvia{\cdot n} \ZZ\surjects C_n$ to get $P_B = (0\to B\to B\to 0)$.
Similarly, 
\[
\Ext^*_\ZZ(C_m, B) = B[m]\cdot t^0 + B/mB \cdot t^1
.\]
:::