Problem : I want to prove that ${\rm Tor}$ is symmetric where $1\in R$
Definition : Note that if $B$ has a projective resolution $$ P_n\rightarrow_{d_n}\cdots P_0\rightarrow B\rightarrow 0 $$
By taking $A\otimes $, we have $$ {\rm Tor}_n\ (A,B) : = \frac{ {\rm ker}\ 1\otimes d_n}{{\rm Im}\ 1\otimes d_{n+1}} $$
Proof : We have a claim that $\{ A\otimes P_n \}$ and $\{ P_n'\otimes B\}$ are $cochain\ homotopic$ where $P_n'$ is projective resolution of $A$. If $P_0'$ is free, that is, $$P_0'=\bigoplus Ra_i,\ a_i\in A$$ then define $$ \epsilon_B\otimes f_0 : P_0 \otimes A\rightarrow B\otimes P_0' $$ where $$ f_0(\epsilon_A(a_i))=a_i $$ But I have no idea to proceed anymore.
How can we prove this ?