Tor functor on projective group

Question

Why $Tor(A,B)=0$ if either $A$ or $B$ is projective group? And what exactly is $Tor$ functor? I've read some about $Ext$ and it was more clear for me, how the $Ext$ works and how it is connected with $Hom$, I know that $Tor$ is the same for $\otimes$ but I can't see what exactly it is?

Answer 1

$\text{Tor}^n(A, B)$ can be understood as either the $n^{th}$ right derived functor of the functor $A \otimes (-)$, applied to $B$, or as the $n^{th}$ right derived functor of the functor $(-) \otimes B$, applied to $A$. This means, among other things, that for $n \ge 1$ it vanishes if either of these functors is exact, meaning that if $A, B$ are modules then it vanishes if either $A$ or $B$ is flat.

Now it's an exercise to show that projective modules are flat. You can do this by showing 1) that free modules are flat and 2) that a retract of a flat module is flat.