Torsion

Proposition

1. $\text{Tor}(A,B)\cong \text{Tor}(B,A)$.
2. $\text{Tor}({⨁}_i{A}_i,B)\cong {⨁}_i\text{Tor}(A_i,B)$.
3. $\text{Tor}(A,B)=0$ if $A$ or $B$ is free, or more generally torsion-free.
4. $\text{Tor}(A,B)\cong \text{Tor}(T(A),B)$ where $T(A)$ is the torsion subgroup of $A$.
5. $\text{Tor}({\mathbb{Z}}_n,A)\cong \text{Ker}(A\stackrel{n}{\to }A)$.
6. For each short exact sequence $0\to B\to C\to D\to 0$ there is a naturally associated exact sequence $0\to \text{Tor}(A,B)\to \text{Tor}(A,C)\to \text{Tor}(A,D)\to A\otimes B\to A\otimes C\to A\otimes D\to 0$